1

Contact Classification inCOVID-19 Tracing

Christoph Guumlnther and Daniel Guumlnther

AbstractmdashThe present paper addresses the task of reliablyidentifying critical contacts by using COVID-19 tracing appsA reliable classification is crucial to ensure a high level ofprotection and at the same time to prevent many people frombeing sent to quarantine by the app Tracing apps are basedon the capabilities of current smartphones to enable a broadestpossible availability Existing capabilities of smartphones includethe exchange of Bluetooth Low Energy (BLE) signals and of audiosignals as well as the use of gyroscopes and magnetic sensorsThe Bluetooth power measurements which are often used todaymay be complemented by audio ranging and attitude estimationin the future Smartphones are worn in different ways oftenin pockets and bags which makes the propagation of signalsand thus the classification rather unpredictable Relying on thecooperation of users to wear their phones hanging from theirneck would change the situation considerably In this case theperformance achievable with BLE and audio measurementsbecomes predictable Our analysis identifies parameters thatresult in accurate warnings at least within the scope of validityof the models A significant reduction of the spreading of thedisease can then be achieved by the apps without causing manypeople to unduly go to quarantine The present paper is the firstof three papers which analyze the situation in some detail

Index TermsmdashCOVID-19 corona tracing contacts

I INTRODUCTION

THE COVID-19 pandemic has spread to enormous dimen-sions with 16 Million people affected and more than

644rsquo000 fatalities up to July 26th 2020 Unfortunately therate of increase has only flattened in China and selectedEuropean countries The most important effective method toslow down the pandemic has so far been the enforcement ofquarantine to large portions of the population which led toa massive economic disruption In countries such as ChinaSouth Korea Singapore and a number of European countriesthe reduced infection rates made it possible to alleviate someof the restrictions This involves the obligation to use masksand at least a recommendation to use some form of contacttracing Different proposals for such a tracing have been made[1] and several different approaches are being followed invarious countries The most interesting proposals are thosethat fully focus on the tracing of contacts without tracking themovement of individuals such as the scheme implemented inGermany [2] The associated concepts were developed nearlysynchronously by a number of authors and were published

Christoph Guumlnther is with the German Aerospace Center 82234 Weszliglingand with Technische Universitaumlt Muumlnchen 80330 Munich Germany e-mailKN-COVIDdlrde

Daniel Guumlnther is a student at Technische Universitaumlt Muumlnchen 80330Munich Germany e-mail dguenthertumde

in [3] [4] and [5] A review of associated requirements isfound in [6] and a review of major apps in [7] In view ofthe highly contagious nature of COVID-19 of the lack of avaccine and of the high casualty rates an effective tracingand significant testing capabilities are essential In Germany16 million people have downloaded the associated app on theiriPhone and Android Phones so far

Tracing apps rely on Bluetooth to detect the proximityof other peoplersquos devices These apps generate random IDswhich are broadcast and stored to identify contacts in thecase that the owner of a device is tested positively If theowner is tested positively the list of IDs stored on his deviceis published Conversely each device keeps the IDs of pastcontacts and compares them to the published list of IDs on aregular basis in order to establish whether a critical contact hastaken place Apple published an update of its operating systemto support the development of such apps (iOS 1315) andGoogle updated its Application Programming Interface (API)In the case of a critical contact the person should quarantinehimself and register for testing The outcome might be that heis found to be a carrier of the disease In this case the ownershould trigger the release of his devicersquos list of random IDsThe consequences of positive and negative testing dependson local regulations In Germany a contact is characterizedas critical and is called a Category 1 contact if two peoplewere in a face-to-face meeting at a distance of less than 2meters for more than 15 minutes The present paper relies onthis definition but its parameters can easily be adapted to anyother definition

Several countries have released tracing apps The classifica-tion methods used are typically not discussed publicly Ideallythe classification ensures a minimum missed detection rate atan acceptable level of false alarms In the case of too manyfalse alarms people will be unduly sent to quarantine andthe app-based will be rejected by the public If on the otherside the app fails to identify potential carriers they continueinfecting others and its effectiveness is jeopardized As shall beseen both issues are most critical in the case of a high densitiesof COVID-19 carriers As a consequence the present analysiswill be most pertinent to regions with a high infection rateThis paper is the first of a series of three papers The other twopapers address the particularities of the evaluation of BluetoothRadio Signal Strength Indicators (BT-RSSI) [8] and of audioranging [9] in more detail

Electromagnetic signals such as Bluetooth signals canbe used for time of flight measurements which providesaccurate ranging results Unfortunately this and some otherideas cannot be considered presently since a contact tracing

arX

iv2

008

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ssS

P] 1

0 A

ug 2

020

2

app must rely on existing smartphones and devices Thusonly existing functions provided by the chipsets and evenmore importantly by the APIs of the devices can be usedThe options for Bluetooth on existing equipment are limitedto power measurements The outcome of such measurementsvery much depend on the location of the device which mightbe in a pocket or in a bag often together with keys coinsmetallic business card holders and the like Furthermore thehuman body with a strong water content strongly absorbsBluetooth signals Together these uncertainties greatly influ-ence the power levels measured at a distant receiver The dif-ficulties of tracing contacts by Bluetooth power measurementsare also discussed in [10] The remaining uncertainties abouta potential contact to an infected person could potentiallybe resolved by interrogating the people involved This wouldrequire the disclosure of the location at the time of contactwhich might have been on a commuter train or at lunch in arestaurant for example The people must then identify wherethey sat or stood which they might remember or not In anycase this would be a source of privacy issues discomfort andresidual uncertainty The German app would not support such amanual tracing anyway since it does not collect the necessaryinformation In any case such a human intervention wouldreduce the level of acceptance

As a consequence we propose to carry the smartphone inan exposed manner namely hanging around onersquos neck Insummer time younger people often do that already On thebasis of the present findings this is recommended to everyonealso in a business context see Figure 1 Corresponding casesare available from several vendors This mode of wearingthe smartphone ensures a line of sight situation betweentwo fellows facing each other It leads to measurements thatare a lot easier to interpret using Bluetooth Radio SignalStrength Indication (RSSI) audio ranging as well as gyroand magnetic sensors The paper starts with a description ofthe statistical relationship between individual measurementsand their classification in Section II This section lays thefoundation for evaluating the performance of classification insimulations or experiments The probability of missed detec-tion turns out to be critical for the success of the classificationBluetooth RSSI evaluation is rather sensitive to the manner inwhich measurements are evaluated Section III describes someaspects relating to the modeling of Bluetooth propagation andpower measurements as well as the essential result from themore in-depth study of the situation developed by Dammannet al [8] The following Section IV addresses audio rangingwhich turns out to be an important complementary techniqueSome audio properties of smartphones are summarized in thissection A more detailed study is published by Kurz et al [9]Section V shortly addresses the possibility of using attitudesensing which is not explored in depth Section VI finallydiscusses some basics of classifying contacts using the set ofsensors mentioned

II STATISTICS OF CLASSIFICATION

The success of classifying contacts into Category 1 andother contacts depends critically on our capability of estimat-ing distances As a consequence it is important to understand

Fig 1 Example of a hull for carrying a smartphone hanging from the neck

the influence of under- and overestimating distances from apandemic point of view This requires a study of the associatedstatistics For a Category 1 contact two fellows have to befacing each other at a distance of less than 2 meters for atleast 15 minutes This is called a C1 contact throughout thepaper

Assume that we are the person A and that we monitor thepresence of B We aim at determining whether the contact toB is a C1 contact or not denoted by C1 or notC1 respectivelyFurthermore denote the outcome of the estimation process byC1 and notC1 then there are four different possibilities as listedin Table I (classical hypothesis testing)

Obviously in any good design pmd and pfa are small Thefour cases have to be considered jointly with the possibilitythat B is tested positively which happens with probability piand shall be denoted by B Current values based on datapublished by John Hopkins University on July 26th forpi are15prime100 for Germany and 1113 for the USA If B is eithernot tested or tested negatively this shall be denoted by notBLet finally pC1

be the probability of C1 then this leads to thesituations summarized in Table II

The first and fourth rows of Table II provide the desiredoutcome The probability pC1 of a contact being C1 is drivenby social behavior Social distancing aims at reducing pC1

This is important since many people would otherwise bepotentially infected and sent to quarantine by the first rowin the table whenever pi is significant The product pC1piis the probability that the contact is C1 and that fellow Bis infected at the same time Aiming at a small value ofpmd ensures that few potential carriers continue spreading thedisease (second row) The actual value of pmd is a directmeasure of the containment benefit provided by a tracingapp Since 1 minus pC1 is large it is very important that theprobability pfa of wrongly classifying a contact as being C1

be small Otherwise numerous people would be unduly sentto quarantine by the third row The value of pfa characterizesthe extra load in terms of quarantining and testing generatedby a tracing app This has to be taken into account in the trade-off of pfa versus pmd Also note that the undesired outcomesie the rows 2 and 3 have a probability proportional to piwhich means that they are unlikely to occur in the case ofa small density of infectious people As a consequence a

3

TABLE IC1 CONTACT EVENTS

Event Description ProbabilityC1|C1 C1 is detected and C1 is correct (C1 contact) pdnotC1|C1 C1 is rejected while C1 would have been correct (missed detection) pmd = 1minus pdC1|notC1 C1 is detected but C1 is not correct (false alarm) pfanotC1|notC1 C1 is rejected which is the correct conclusion(no contact) 1minus pfa

TABLE IIC1 CONTACT EVENT PROBABILITIES AND CONSEQUENCES

Event Probability ConsequenceC1|C1 andB pC1pipd sim pC1pi A goes to quarantinenotC1|C1 andB pC1

pipmd A pot spreads the virusC1|notC1 andB (1minus pC1 )pipfa Unnecessary quarantine of A

notB or (notC1|notC1 andB) 1minus pi + (1minus pC1)pi(1minus pfa) none (all other cases)

potential under-performance of an app only becomes apparentin environments with a high number of infectious people

The decision for C1 or notC1 is taken after a substantialnumber of individual measurements They are assumed to beperformed at regular intervals The number of such intervals ina time laps of 15 minutes is denoted by x0 Depending on theassumed behavior of people different methods of analyzingthe measurement data shall be considered

bull Model A People are rather mobile and the environmentis changing quickly - the contact duration is accumulatedover many short intervals Examples of such situationsoccur when people work closely together which is notparticularly critical in terms of classification They occurin underground trains during breaks at conferences atany form of party and the like In these cases a decisionis taken every 15 seconds if x0 = 60 such measurementsindicate that fellow B is in the contact zone of fellowA the contact is classified as being C1 It will turn outthat this model cannot be addressed with the currentcapabilities

bull Model B People come together stay in a given relativepose and then separate again This happens when peopleare seated in a train especially in long-distance inter-city trains in restaurants meeting rooms lecture hallstheaters and the like In this case a single test (x0 = 1)is performed to decide on whether A is in the contactzone of B Specifically in the case of Bluetooth RSSImeasurements a timer is started when the RSSI value ex-ceeds a critical value for the first time From then on thetimes for which the RSSI values are compatible with a C1

contact are accumulated If the time exceeds 15 minutesat the end of the contact a C1 contact is declared Thereare many different options for the implementation of thismodel They will not be further discussed however sincethey assumes a static constellation of people which is notthe most common case

bull Model C Is an intermediate model which allows for slowchanges in the distribution of people In this model theRSSI is accumulated over time like in Model B but onlyover intervals of 3 or 5 minutes Contrary to Model Bthere is no further condition on the accumulation The

accumulated RSSI-values are evaluated against a thresh-old at the end of the interval Exceeding the thresholdx0 = 5 times (3 minutes of accumulation) or 3 times (5minutes of accumulation) leads to the decision C1 Thisapproach is more robust with respect to the behavior ofpeople and preferable to Model B

Model A is most universally valid with respect to peoplersquosbehavior Its statistics are so unfavorable that it does not lead toacceptable values of pmd however In all models the numberof RSSI measurement n that are combined before taking anelementary decision is another parameter that can be adaptedLarge values lead to more reliable decisions but also to ahigher number of exchanged messages The rate of messagewill be n middot x0 measurements in 15 minutes

In order to assess pmd we need to know the number oftimes that the distance and attitude condition for C1 betweenA and B are fulfilled This depends on the profession andpersonality of the person It has two components the firstone is determined by the number of people met during oneday Let us assume that this number is k and that it hasprobability pn(k) then the probability pS that a particularfellow A spreads the virus after having been in contact withm isin 1 2 k people who are infectious with probabilitypi under the assumption that i isin 1 2 m of thesecontacts are not detected is given by

pS =

infinsumk=0

pn(k)

ksumm=1

(km

)pmi (1minus pi)kminusm

middotmsumi=1

(mi

)pimd(1minus pmd)mminusi

Since pi and pmd are small numbers the dominant term inthis equation is obtained for m = i = 1

pS Kpipmd (1)

with K =suminfink=0 pn(k)k being the average number of con-

tacts see also the second row in Table II All these contactstake place mostly independently and can thus be treated assuch Each of them is associated with a contact time x isin Nwith a distribution pX(x) The latter is derived from socialmodels and depends on whether people are practicing socialdistancing

4

The accumulation of n measurements leads to a decisionc1 The latter has a probability of missed detection and falsealarm denoted by πmd and πfa respectively In the presentsection both quantities are written without further indices Inlater sections the dependency on n will be made explicitThe combination of x0 such decisions c1 finally leads tothe decision C1 which is associated with a missed detectionprobability

pmd(x) =

x0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd (2)

since the combined missed detection occurs whenever less thanx0 detections succeed Using this in Equation 1 implies thatthe probability that A spreads the disease is

pS KpixMsumx=x0

pX(x)pmd(x)

= Kpi

xMsumx=x0

pX(x)

x0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd

(3)

with xM = 24middot4middotx0 being the number of elementary decisionstaken per day (24 middot 4 quarters of hours times x0) The aboveequation is an approximation since the distribution of contacttimes depends on the people and circumstances of the meetinglike sitting together in the train having a joint lunch and soon If πmd 1 the term m = x0minus1 is dominant in Equation(3)

pS sim KpixMsumx=x0

pX(x)

(x

x0 minus 1

)πmd

x0minus1πxminusx0+1md

= Kpiπx0minus1d

xMminusx0+1sumxprime=1

pX(xprime + x0 minus 1) (4)

middot(xprime + x0 minus 1x0 minus 1

)πxprime

md

le Kpiπx0minus1d

xMminusx0+1sumxprime=1

pX(xprime + x0 minus 1)1

xprime(x0πmd)

xprime

with πd = 1 minus πmd The second line in the equation isobtained by shifting the indices the third one is obtained byexpanding the binomial coefficients and bounding the termsin the numerator Note that the term for xprime = 1 holdswith equality Under the same assumptions used so far theprobability that fellow A is a C1 contact of B after a day is

pC1= Kpi

xMsumx=x0

pX(x) = Npi

xMminusx0+1sumxprime=1

pX(xprime + x0 minus 1)

Thus the comparison of pS ie the probability of spreadingthe virus with tracing and of pC1 ie the correspondingprobability without tracing shows that contact tracing is avery effective option to reduce the spreading whenever

x0πmd (5)

is small This implies that the probability of missed detectionmust be constrained to a value smaller than 1x0 which ispossible to achieve if x0 is small as it is the case in Model

B and Model C and not possible to achieve in Model A evenwith very large values of n Rephrasing this in words may helpdeveloping some intuition since x0 individual c1 decision areneeded for a C1 decision missing any one of them leads toa missed detection Since there are x0 options for that pSbecomes essentially proportional to x0πmd We will use thelatter product as a measure for the reduction in the spreadingof the disease by the tracing app

In order to evaluate pfa we need to additionally knowthe number of times y that a person is close enough for ameasurement to take place The distribution pY (y) does againdepend on social parameters but additionally depends on radiopropagation in the case of Bluetooth measurements and onthe triggering mechanism in the case of audio ranging Thenumber of contacts KY ge K is larger since the presencedetection by Bluetooth signaling is triggered well beyond C1

separation Consider Bluetooth measurements if among they time instances for which a radio contact to one particularfellow B persists and assume that m lt x0 of those contactsare correctly detected as fulfilling the C1 conditions Thenq additional erroneously identified contacts (erroneous c1-decisions) with m+ q ge x0 are needed to cause a false alarmfor that number y of radio contacts to B (see Table III for asummary of the meaning of the variables)

pfa(y) =

xMsumx=0

pX(x)

minxx0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd

yminusxsumq=x0minusm

(y minus xq

)πqfa(1minus πfa)yminusxminusq

(6)

for y ge x0 and pfa(y) = 0 for y lt x0

TABLE IIIVARIABLES USED IN EQUATION III

Variable Meaningy number of radio contactsx number of C1 contactsx0 number of c1-decisions to declare C1

m number of correct c1 estimatesq number of incorrect c1 estimates

Using Equation (6) the expected number of an unnecessaryquarantining of people is approximated by

nQ = KY pi

xMsumy=x0

pY (y)pfa(y)

= KY pi

xMsumy=x0

pY (y)middot

middotxMsumx=0

pX(x)

minxx0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd

middotyminusxsum

q=x0minusm

(y minus xq

)πqfa(1minus πfa)yminusxminusq

(7)

This equation also includes the possibility that users movewith respect to each other which means that the conditions

5

C1 and notC1 alternate as a function of time If C1 is fulfilledπfa = 0 and if notC1 the equation πd = 0 holds At the borderof the C1 domain the two quantities change their role Thisimplies that a small pfa near that border is associated with alarge pmd sim 1 minus pfa on the other side of the border This isuncritical if the distributions are very narrow - concentratedaround a value - as is the case for ranging but becomes ratherproblematic with Bluetooth signal power measurements whichshow a very flat distribution Unless great precautions are takenthe classification becomes unreliable Consider the case thatfellow B is outside of the C1 zone of fellow A ie pX(0) = 1Then x = 0 for these measurements and the equation becomes

nQ = KY pi

xMsumy=x0

pY (y)

ysumx=x0

(yx

)πxfa(1minus πfa)yminusx

Although terms with x gt x0 may be larger the term x = x0

gives us an idea of the scaling Its asymptotic dependencycan be evaluated using Stirlingrsquos formula and limyrarrinfin(y(yminusx0))y = ex0 (

yx0

)πx0

fa simradic

y

2πx0(y minus x0)

((y

x0minus 1

)eπfa

)x0

This means that in the long term it is the duration of theradio contact y which dominates the rate of quarantiningpeople Some target figures for πfa can be obtained for afully occupied train for example In Germanyrsquos 2nd classsetups there are 4 seats in one row on each side of a carriageand around 10 rows in the carriage The range of Bluetoothreaches well beyond the next row forward and backward Thismeans that KY gt 24 of which 4-8 are within the contact zoneand must thus be discounted leading to an effective valueKY = 16 The value y itself is determined by the duration ofthe common journey For commuter trains we choose 15 and30 minutes for inter-city journeys 1 2 and 3 hours whichleads to yx0 = 1 2 4 8 and 12 In such a train a carrierof the disease will send 4 people to quarantine thus it shouldbe tolerable that 2 additional people are sent to quarantine byfalse alarms as well The value of πfa is then obtained bysolving

KY

(yx0

)πx0

fa = 2

Numerical values of πfa are indicated in Table IV They arethe values that can be tolerated leading to a 50 increase inthe quarantining of people riding a German train The situationis rather uncritical on a short commuter train ride πfa lt 093and much more demanding on a longer intercity train journey

TABLE IVROUGH INDICATION FOR ACCEPTABLE VALUES OF πfa THEY ARE

OBTAINED BY CONSIDERING TRAIN RIDES

yx0 1 2 4 8 12πfa 093 025 011 005 003

III BLUETOOTH POWER MEASUREMENTS

The Application Programming Interfaces (API) of Androidand iOS allow to trigger the transmission of Bluetooth Low

Energy (BLE) advertisement messages and to measure the ra-dio signal strength of the received signals The correspondingvalues are provided in the form of a Radio Signal StrengthIndicators (RSSI) which is defined as the received signalpower on a logarithmic scale Bluetooth uses frequenciesfrom a band shared with microwave heating which meansthat Bluetooth signals are strongly absorbed by water As aconsequence any part of a human body obstructing the lineof sight significantly attenuates the signal The wide varietyof options for carrying mobile phones in your hand pocketor bag thus implies an enormous variability in received powerlevels This is further amplified by the directional characteristicof low-cost antennas You might make an experiment yourselfusing a Bluetooth module and a BLE scanner app on yoursmartphone which can be downloaded from the iOS orAndroid stores With the module and phone separated by 15meters I personally found the following RSSI-values -61 to-66 dBm when the module was in my hand and -81 to -89-dB when it was in my pocket Knowing that a 20 dB changecorresponds to a factor 10 in distance exemplifies the difficultyof estimating distances using Bluetooth RSSI values This ledus to propose the rule of carrying smartphones hanging downfrom the neck Note that the smartphone could be replaced bya much smaller device built around a Bluetooth module anInertial Navigation System (INS) and a sonic or ultra-sonicranging system as well

Even if people follow the above recommendation on howto carry their smartphone the situation remains difficult dueto uncertainties in radio propagation which furthermore takesplace on three different carrier frequencies The unknownassociation of carrier frequencies to measurements adds anadditional level of difficulty Gentner et al identified certainpatterns in the use of carriers see [11] which can be usedto reduce the associated uncertainty Traditional models ofpropagation are shortly addressed in the following section andin more details in [8] The section furthermore relates theassociated statistics to the statistics of classification

A Propagation Model

The smartphone is assumed to be worn on the chest see [8]for details of the measurement setup used to obtain numericalresults For each individual carrier the received signal powerPRX is modeled by the equation

PRX =γ

dνPTX + n (8)

with PTX denoting the transmit power γ denoting a stochasticfading coefficient d being the distance between the receiverand the transmitter ν being the exponent of the decay lawwhich is 2 for free space propagation and with n representinga superposition of noise and interference For simplicity thenoise and interference are not further considered here - at lowdistances they are not dominant In this case the receivedpower can be represented on a logarithmic scale which leadsto the definition of the RSSI

RSSI = 10 logPRX = 10 logPTX minus ν middot 10 log d+ η (9)

6

with η = 10 log γ and with logarithms taken to the basis 10The relationship between the reported RSSI value and d isthe basis for distance measurement the measured RSSI iscompared to

Θ = 10 logPTX minus ν middot 10 log dc + 〈η〉

with dc = 2 m being the critical distance Note that Equation(8) defines the units which have to be maintained after takinglogarithms

In order to evaluate the missed detection probability perevent pmd or the false alarm probability per event pfa thestatistics for η or γ need to be known These statistics aredependent on the situation In the case that two fellowsface each other they are in a line of sight situation If thedirect path dominates all other contributions γ is basicallydelta distributed with an average of Γ determined by theantenna pattern In other cases the direct path remains presentbut is superposed by scattered components In this case thedistribution of the amplitude of the received signal is modeledby a Ricean distribution This model is considered to provide afaithful representation of reality whenever the parameters areproperly estimated Presently the model is only considered forcomparative purposes as shall be seen below The receivedpower (or attenuation γ) in this model has a non-central χ2-distribution with two degrees of freedom

pR(γ) =1

2σ2R

eminus(γ+γR)(2σ2R)I0

(radicγγR

σ2R

) (10)

with γR being the non-centrality parameter and σR being thevariance In the case that the decision about C1 is taken onthe basis of a single measurement (n = 1) eg in Model Athe criterion for the decision is

γ ge γc(d

dc

)ν (11)

with γc being given by

γc = 〈γ〉 =

int infin0

dγγpR(γ) (12)

The associated estimate is denoted by c1 and the probabilityof missed detection for the distance d lt dc is given by

πmd(d) =

int γc(ddc)ν

0

dγpR(γ) (13)

If one would add several power measurements ie n gt 1eg in Model B and C this would mean adding n indepen-dent identically distributed variables each of them being χ2-distributed with 2 degrees of freedom The result would thenbe χ2-distributed with 2n degrees of freedom

pRn(γ) =1

2σ2R

(γ

nγR

)nminus12

eminus(γ+nγR)(2σ2R)Inminus1

(radicnγγR

σ2R

)

The Equations (11) and (12) would remain valid and the latterintegral could be computed in closed form for arbitrary nThe value γc is the first moment of the χ2-distribution with2n degrees of freedom and non-centrality parameter nγRσ2

R

γc = n(γR + 2σ2R)

The probability of missed detection (13) in estimating c1 couldthen be computed in closed form using Marcumrsquos Q-functionQn( )

πmdRn(d) = 1minusQn

radicnγRσR

radicγc

(ddc

)ν2σR

(14)

The above distributions are adequate for users A and B inclose proximity of each other as is the case for d le dc It isthe desired result in Model A and shall serve as a benchmarkin the Models B and C The reason for not using this resultdirectly in the latter models is that apps are expected to addthe RSSI values rather than the power values In this case thestatistics cannot be determined in closed form but must ratherbe evaluated numerically Before addressing this case let usconsider the situaiton d gt dc with a line of sight that is oftenobstructed In such cases a lognormal fading distribution isconsidered to be a reasonable model of reality see [12] Thedistribution may either be written in terms of γ

pL(γ) =10 log10(e)radic

2πσLγeminus(10 log γminus10 log γL)2(2σ2

L)

or in terms of η = 10 log γ

pL(η) =1radic

2πσLeminus(ηminusηL)2(2σ2

L) (15)

with ηL = 10 log γL = 〈η〉 Equation (15) makes the Gaussiancharacter and the meaning of ηL and σL obvious In the abovediscussion a decision in the case n = 1 was taken in favorof C1 whenever the power level was above a threshold Onthe logarithmic scale this condition reads RSSI gt Θ iewhenever the difference

RSSI minusΘ = η minus ηL + ν middot 10 logdcd

(16)

is positive or equivalently whenever η gt 〈η〉+ν middot10 log(ddc)Thus a false alert occurs if this condition is fulfilled for d gtdc The probability of a false alarm ie and erroneous decisionfor c1 becomes

πfa(d) =

int infin〈η〉+νmiddot10 log(ddc)

dη pL(η) (17)

= Q

(ν middot 10 log(ddc)

σL

)

with the present Q-function being a scaled version of the errorfunction complement

Q(x) =1

2erfc

(xradic2

)

In the case of n = 1 a closed form of the statistics thus existsfor πmd for d le dc and for πfa for d gt dc In the case n gt 1eg Model B and C the situation changes somewhat sincemeasurements are now combined by adding RSSI-values Thiscorresponds to a geometric average of the received powers

7

In this case the probability of false alarm can be computedeasily

πfan(d) = Q

(radicn middot ν middot 10 log(ddc)

σL

) (18)

for d gt dc This equation is a consequence of the scaling of ηLand σ2

L by n Using the same distribution but with differentparameters for d lt dc is expected to be a worse match toreality but allows to also evaluate the probability of misseddetection in closed from

πmdLn(d) =

int 〈η〉+νmiddot10 log(ddc)

minusinfindη pL(η)

= 1minusQ(radic

n middot ν middot 10 log(ddc)

σL

)= Q

(radicn middot ν middot 10 log(dcd)

σL

)= πfan

(d2c

d

) (19)

It leads to an interesting symmetry between the probabilitiesof missed detection and of false alert

Note that both probabilities πmd and πfa depend on theparameters of the distribution on the true distance d andon the critical distance dc but that they do not depend onthe explicit threshold Θ see Equation (16) and the associatedexplanations The resulting functional dependence can eitherbe used in a simulation of roaming users or can simply beaveraged over the interior of a circle of radius dc for πmd orover its complement or a relevant subset for πfa The closedform of Equation (6) provides the immediate insight thatπfan(dc) = 12 which shows that the models are consistentwith our intuition

B BLE Measurements Results

The companion paper by Dammann et al [8] describes themeasurements and their analysis in more details All thesemeasurements have so far been made using ideal conditionswith no additional people except A and B (in the very initialmeasurements A was a actually a post carrying the receiver)The experimental basis shall be further broadened in thefuture A first result can be derived from the estimated Riceparameters at a distance of 2 meters γR = 247 pW andσR2 = 915 pW as well as for the lognormal distributionat 2 and 4 meters 160 and 197 dBm respectively

This allows plotting the functions from Equation (14) and(17) for πmdRn(d) and for πfan(d2

cd) = πmdLn(d)respectively The values of n determines how many mea-surements are combined into an elementary decision c1 Forn = 1 the values πmdR1(d) and πfa1(d) are the best modelsamong those considered - the use of a decision threshold in theabsolute or logarithmic domain are equivalent The parameterfor 4 meters 197 dBm is used for determining the false alarmrate

If several RSSI values are added (logarithmic domain) thestatistics associated with the more realistic Rice distributionin the near range can not be determined in closed form atleast not today In this case Equation (19) for the lognormal

distribution is used to determine πmdLn(d) with the parame-ter for 2 meters This is used as an approximation of the truedistribution in the exemplary case n = 60 The plots in Figure(2) show two groups of curves The upper group correspondsto n = 1 and the lower group to n = 60 The latter group ofcurves shows the benefit of diversity Within these groups thereare differences between πmdRn(d) (wrong combination) andπmdLn(d) (wrong fading statistics) but they turn out not tobe fundamental

Fig 2 Probability of missed detection as a function of user distance usingBluetooth Radio Signal Strength Indication (RSSI)

TABLE VSELECTED VALUES OF πmdavn UNDER THE ASSUMPTION OF A

LOGNORMAL FADING DISTRIBUTION THESE VALUES DEFINE THENUMBER OF MEASUREMENTS NEEDED TO ACHIEVE THE DESIRED

PROBABILITY OF FURTHER SPREADING THE DISEASE

n πmdavn n πmdavn

1 012 60 00146 0054 120 000715 0034 240 000230 0023 480 00003

In Section III-A the probability of missed detection wasdetermined as a function of distance Since the probability ofdetection is additive in the sense that

πd =

intdS(r) ρ(r)πd(r)

=

int dc

0

2πrdr ρ(r)πd(r) (20)

In this equation πd(r) = 1 minus πmd(r) is the condition prob-ability of detection given that fellow B is at distance r anddS(r) ρ(r) is the probability density for fellow B to be atthat distance Equation 20 thus is the marginalization of πd(r)with respect to r Note that the limitation of the integration isa consequence of πd(r) = 0 whenever r gt dc This allowsto define the average probability of missed detection over thedistribution of users

πmdavn =

int dc0

2πrdr ρ(r)πmdn(r)int dc0

2πrdr ρ(r) (21)

8

The probability distribution of users in Equation (20) and (13)is given by

ρ(r) =dn(r)

dS(r)=dbπr2c2πrdr

In this expression n(r) = bπr2c is the number of peopleat a distance not greater than r in the case of a density ofone person per square meter This corresponds to the densestpacking of people occupying a surface of 1 meter People arecontinuously spread in a symmetric manner around fellow Awhich is a simple way of achieving a densest packing Theldquofunctionrdquo dn(r)dr is mostly zero It jumps at the valuesrm =

radicmπ with

n(rm + ε)minus n(rm minus ε) =

int rm+ε

rmminusε

dn(r)

drdr = 1

which is a distribution in the sense of Schwartz [13] Withthese preparations the integrals become

πmdavn =1

mc

mcsumm=1

πmdn

(radicm

π

) (22)

with mc being the largest integer with such that rmc le dcNote that the density of points rm increases with increasingm which means that the main contribution comes from theborder of the contact zone Using the experimental resultsfrom [8] this integral is evaluated to πmdav1 = 015 forn = 1 for the χ2-distribution and to πmdav1 = 012 for thelognormal distribution which are both not very compatiblewith the need of a small x0πmd Remember that the lattervalue is the reduction factor in the probability of furtherspreading of the disease achieved by contact tracing TableV lists values of πmdavn for different n which can beused to determine the reduction factor Even in the casen = 120 the factor x0πmd = 021 in Model A and it wouldrequire 4 measurement per second It is only with n = 480that factor x0πmd falls below 1 which would require 16measurements per second This would seriously impact thestandby time of the smart phone Assuming Model C and adecision based on 3 minutes intervals ie x0 = 5 means thatwe could achieve a reduction by a factor 007 provided thatn = 60 measurements are performed and aggregated in each 3minutes interval ie that one measurement is performed every3 seconds In the case of a decision every 5 minutes whichassumes a lower dynamics in the relative movement of peoplethe reduction factor is 004 with the same 60 measurementsbut now spread over a 5 minutes interval which correspondsto one measurement every 5 seconds So lower requirementsin the dynamic allow both to improve the suppression of thespreading of the virus and to reduce the measurement rate

Tolerable alarm rates were derived for the train scenarioThis led to the values in Table (IV) The evaluation of πfan(d)is straight forward For d = dc it gives πfan(dc) = 12 as wasalready discussed previously Assuming that people occupy acircular surface of 1 square meter gives them a radius δ =1radicπ Thus the minimum distance to people fully outside of

the critical zone is dc + δ Evaluating Equation (19) yields

pfa1(dc + δ) = 0137 and pfa3(dc + δ) = 0029

respectively This means that n = 1 is compatible with ajourney of 15 minutes before sending more than the two peopleto quarantine For n = 3 long journeys of up to 3 hoursbecome possible with the same consequences The probabilityof false alarm does thus not strongly limit the number n ofmeasurements aggregated to a decision and one might considerthe more demanding homogeneous distribution of users Thisrequires a study of the combination of false alarms Considertwo fellows B and Brsquo there is no alarm if neither B nor Brsquotriggers an alarm ie

1minus πfa = (1minus πfaB)(1minus πfaBprime)

Furthermore let users be at distances dc+δ(k+1) with k isin Z+

being a positive integer and assume that there are

ν(k) = π (dc + 2δ(k + 1))2 minus π (dc + 2δk)

2

users at that distance (they cover an angular shell of thickness2δ) This guarantees a densest packing In that case theprobability of false alarm ie an erroneous decision in favorof C1 becomes

pfan = 1minusinfinprodk=0

(1minus πfan(dc + k))ν(k)

(23)

In this more demanding scenario exemplary values are

pfa3 = 0413 and pfa9 = 0009

which means that n = 9 would be sufficient to reduce theprobability of false alarm to a very small level Table VI showsperformance figures for a number of possible choices for thenumber n of measurements aggregated to an estimate c1 aswell as for the number x0 of estimates c1 that lead to a decisionC1 The product of n and x0 leads to the measurement rateρ = x0n(15 middot 60) The performance figures are the reductionfactor x0πmdn of the spreading achieved by tracing as wellas the probability of unduly sending a person to quarantineThe figures in Table VI all relate to Model C Model A doesnot lead to interesting parameter choices and Model B is toostatic

TABLE VIKEY PERFORMANCE PARAMETERS x0πmdn MEASURES THE REDUCTIONIN SPREADING AND pfan THE PROBABILITY OF UNDUE QUARANTINING

THE PARAMETER ρ IS THE NUMBER OF MEASUREMENTS PER SECOND

x0πmdn pfan ρx0 3 5 - 3 5

n6 016 027 0064 150 130

15 012 017 00002 120 11260 004 007 00000 15 13

A choice with n = 15 and x0 = 3 for example requiresa measurement to be performed every 12 seconds suppressedthe risk of spreading by a factor 012 and does hardly sendanyone unduly to quarantine Performing a measurement everyfive seconds reduces the risk of spreading by a factor 004 Thisassumes that people let their phones hang from their neckand some standard form of environment In reality a numberof additional factors have to be taken into account such asa more complex propagation situation eg due to metallic

9

walls a higher dynamic of user movements eg due topeople entering and exiting commuter trains or unpredictableshadowing due to the userrsquos hands arms or body in the pathof radio signals Thus it is advisable to complement theBluetooth measurement by an alternative Audio ranging isthe option that shall be described in the next section The ideais to use it whenever the situation is not clear

IV AUDIO RANGING

Smartphones have a microphone and a speaker with rathergood transmit and receive conditions if the device is carriedon the chest or held in the hand This can be used for audioranging up to distances of a few meters Signals and theirtransmission can be configured by the API In experiments thatwe performed recently we focused on the use Android phonesThe response of the microphones built into three differentphones is shown in Figure 3 The references were a NT1-A microphone from Rode and an Adagio Infinite Speaker ofA3 on the source side Figure 3 shows the response of threesmartphones from two different brands The curves are verysimilar suggesting that the same microphones are integratedin those phones All microphones show a good sensitivity overall frequencies

Fig 3 Frequency response of microphones from three different smartphones

A similar experiment was performed for the speakers witha rather different result In that case only two smartphoneswere analyzed The response on the better device is reducedby roughly 10 dB above 16 kHz as compared to the referenceThe response of the other one is degraded by another 3 dBand the degradation starts 2kHz earlier Covering the speakerby one layer of tissue of a sweater degrades the performanceby another 4 dB If both parties cover their smartphonesthe associated attenuation adds up Thus the use of audioranging requires carrying the devices in an exposed mannereg hanging from onersquos neck see Figure (1) Transmissionat lower less attenuated frequencies is not considered as atrue option since it would be too disturbing The norm ISO2262003 compiles equivalent hearing sensitivity (isophones)which allows to compare the disturbance caused by acousticalsignals on different frequencies

Fig 4 Frequency response of two speakers as well as influence of coveringthe speaker of Smartphone 2 with one layer of a sweater

On the basis of such considerations we propose modulatinga carrier at 18 kHz with a modulation rate of 1 kbaud Thiskeeps the signal in a spectral range that is not too disturbing tomost people A spread spectrum modulation provides a goodrange resolution and allows to operate at a low signal-to-noiseratio at the same time Different options exist and are discussedin [9] Since the velocity of sound in air is cs = 343 ms understandard conditions a chip duration of 1 ms corresponds to alength of 34 cm At a typical signal-to-noise ratio this leadsto a distance resolution of 1 to 3 cm Let us be conservativeand assume a resolution of 5 cm A multipath delay of twometers leads to an offset by 6 chips and is well suppressedby the autocorrelation of the spreading code The length ofthe spreading code is assumed to be around 350 chips Analternative using chirps is also considered The performanceof audio ranging is further developed in Section IV

Audio ranging can be performed in a peer-to-peer or ina networked manner Consider the peer-to-peer situation firstSmartphones do not provide accurate timing control Howeverthe microphone input of a smartphone may be sampled at afixed rate Furthermore smartphones can transmit and receiveat the same time and this is furthermore supported by theAPIs of Android and iOS Let the smartphones thus agreeto start audio ranging via Bluetooth In a first step theyopen their microphone channels and then proceed according toFigure 5 at time tTXA A transmits the ranging signal usingits speaker This transmission is delayed with respect to theAPI by τTXA In parallel to its transmission Arsquos microphonecapture the transmitted signal This signal is delayed by thesum of the local propagation delay τlA and by the internalreceive delay τRXA The delay τlA is determined by thedevice geometry and can be stored in memory A standardvalue of 14 cm should be appropriate for most devices on themarket The time of reception thus is

tprimeRXA = tTXA + τTXA + τlA + τRXA

and is used for calibration purposes The same definition ofdelays applies at B Thus the signal transmitted by A at time

10

tTXA is received at B at the time tRXB

tRXB = tTXA + τTXA + τ + τRXB

with τ being the propagation time from A to B After receptionof the signal from A by B B sends a corresponding signal toA The equations are obtained by changing the roles of A andB

tprimeRXB = tTXB + τTXB + τlB + τRXB

andtRXA = tTXB + τTXB + τ + τRXA

At the end of the reception A sends

∆tA = tRXA minus tprimeRXA + τlA (24)

to B and B sends ∆tB = tRXB minus tprimeRXB + τlB using BLEThus both can compute the propagation time

τ =∆tA + ∆tB

2

and thus the distance d = τcs The property of audio signalswhich is crucial for this self-calibration is the possibility toobserve the own transmitted signal

Fig 5 Signal paths in two-way acoustical ranging with calibration of transmitand receive delays

A Ranging Protocol

The above peer-to-peer protocol can be extended to anetworked protocol In this case the users agree on an orderingof transmissions via Bluetooth All smartphones A1 Akactivate their microphones and one after the other transmittheir audio ranging signals For simplicity the scheduling isprearranged which also works if some of the smartphonecannot acquire all signals In this case all delays are summedup 350 ms for the ranging signal 10 ms (corresponding to4 meters) for propagation and 40 ms for the internal delaysbetween the activation of the transmission command and thestart of transmission (the latter needs to be confirmed by moredata) This allows for a scheduling of a transmission every 400ms After the completion of the cycle and the evaluation of the

reception time tRXAi by terminal A1 this terminal transmitsthe time difference using Bluetooth

∆tA1Ai = tRXAi minus tprimeRXA1+ τlA1

for 2 le i le k

If all terminals see each other they transmit k(k minus 1) suchvalues in total The annoying transmissions of audio signals re-main limited to k however The overall time interval spannedby all transmissions in the networked protocol may be longenough for users to move slightly This is not critical howeverThe snap-shot measurements are simply converted to averagevalues The only instances which require some care are thosein which the audio signals are used to calibrate Bluetoothmeasurements Finally it should be emphasized that audiobeacon transmissions should not be activated if the device isheld to the ear Even if the signals are hardly heard this seemsa reasonable precaution

B Theoretical Performance of Acoustic Range Estimation

The received audio signal is filtered to remove out-of-band interference and noise to the best possible extent Thefiltered signal is used to determine the in-band interference andnoise level N0 and is furthermore correlated using the filteredranging signal For simplicity the further exposition focuses onspread spectrum signals In a first step the I and Q componentsof the correlation C(∆τ) are computed at intervals of Tc2with Tc = 1 ms denoting the chip duration The result issearched for the delay leading to the maximum norm |C(∆τ)|Although the implementations by widely used phones seemnot to require that frequency offsets may be searched as wellThis allows to acquire the signal which may be present or notThus it is sufficient to search for the delay (and frequencyoffsets) leading to the maximum norm from early to late Thelatter ordering is to avoid locking on an echo If the signalto noise ratio is above the expected threshold the signal isassumed present In this case a successive refinement of theresult is performed in a DLL type of processing The powerdiscriminator

DP (∆τ) = |R(∆τ + δ)|2 minus |R(∆τ minus δ)|2

is used to iteratively increasereduce the delay ∆τ dependingon the value of DP (∆τ) ≷ 0 In this equation δ is half thecorrelator spacing and is expressed as a fraction ∆ of thechip duration δ = ∆Tc We will restrict ourselves to ∆ = 1A further optimization is possible see Betz and Kolodziejski[14] [15] The uncertainty of the delay estimate ∆τ due tonoise is given by (see Dierendonck Fenton and Ford [16])

σ2∆τ T 2

c

∆

4EiN0

(1 +

3

(2minus∆)EiN0

) (25)

In this expression Ei is the signal energy accumulated duringthe correlation and N0 is the spectral noise density of theaudio noise and interference The latter quantity is estimatedusing the norm of the filtered I and Q samples of the incomingsignal

N0 =1

BSNTc

Nsumn=1

(s2I + s2

Q

)

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

3

TABLE IC1 CONTACT EVENTS

Event Description ProbabilityC1|C1 C1 is detected and C1 is correct (C1 contact) pdnotC1|C1 C1 is rejected while C1 would have been correct (missed detection) pmd = 1minus pdC1|notC1 C1 is detected but C1 is not correct (false alarm) pfanotC1|notC1 C1 is rejected which is the correct conclusion(no contact) 1minus pfa

TABLE IIC1 CONTACT EVENT PROBABILITIES AND CONSEQUENCES

Event Probability ConsequenceC1|C1 andB pC1pipd sim pC1pi A goes to quarantinenotC1|C1 andB pC1

pipmd A pot spreads the virusC1|notC1 andB (1minus pC1 )pipfa Unnecessary quarantine of A

notB or (notC1|notC1 andB) 1minus pi + (1minus pC1)pi(1minus pfa) none (all other cases)

potential under-performance of an app only becomes apparentin environments with a high number of infectious people

The decision for C1 or notC1 is taken after a substantialnumber of individual measurements They are assumed to beperformed at regular intervals The number of such intervals ina time laps of 15 minutes is denoted by x0 Depending on theassumed behavior of people different methods of analyzingthe measurement data shall be considered

bull Model A People are rather mobile and the environmentis changing quickly - the contact duration is accumulatedover many short intervals Examples of such situationsoccur when people work closely together which is notparticularly critical in terms of classification They occurin underground trains during breaks at conferences atany form of party and the like In these cases a decisionis taken every 15 seconds if x0 = 60 such measurementsindicate that fellow B is in the contact zone of fellowA the contact is classified as being C1 It will turn outthat this model cannot be addressed with the currentcapabilities

bull Model B People come together stay in a given relativepose and then separate again This happens when peopleare seated in a train especially in long-distance inter-city trains in restaurants meeting rooms lecture hallstheaters and the like In this case a single test (x0 = 1)is performed to decide on whether A is in the contactzone of B Specifically in the case of Bluetooth RSSImeasurements a timer is started when the RSSI value ex-ceeds a critical value for the first time From then on thetimes for which the RSSI values are compatible with a C1

contact are accumulated If the time exceeds 15 minutesat the end of the contact a C1 contact is declared Thereare many different options for the implementation of thismodel They will not be further discussed however sincethey assumes a static constellation of people which is notthe most common case

bull Model C Is an intermediate model which allows for slowchanges in the distribution of people In this model theRSSI is accumulated over time like in Model B but onlyover intervals of 3 or 5 minutes Contrary to Model Bthere is no further condition on the accumulation The

accumulated RSSI-values are evaluated against a thresh-old at the end of the interval Exceeding the thresholdx0 = 5 times (3 minutes of accumulation) or 3 times (5minutes of accumulation) leads to the decision C1 Thisapproach is more robust with respect to the behavior ofpeople and preferable to Model B

Model A is most universally valid with respect to peoplersquosbehavior Its statistics are so unfavorable that it does not lead toacceptable values of pmd however In all models the numberof RSSI measurement n that are combined before taking anelementary decision is another parameter that can be adaptedLarge values lead to more reliable decisions but also to ahigher number of exchanged messages The rate of messagewill be n middot x0 measurements in 15 minutes

In order to assess pmd we need to know the number oftimes that the distance and attitude condition for C1 betweenA and B are fulfilled This depends on the profession andpersonality of the person It has two components the firstone is determined by the number of people met during oneday Let us assume that this number is k and that it hasprobability pn(k) then the probability pS that a particularfellow A spreads the virus after having been in contact withm isin 1 2 k people who are infectious with probabilitypi under the assumption that i isin 1 2 m of thesecontacts are not detected is given by

pS =

infinsumk=0

pn(k)

ksumm=1

(km

)pmi (1minus pi)kminusm

middotmsumi=1

(mi

)pimd(1minus pmd)mminusi

Since pi and pmd are small numbers the dominant term inthis equation is obtained for m = i = 1

pS Kpipmd (1)

with K =suminfink=0 pn(k)k being the average number of con-

tacts see also the second row in Table II All these contactstake place mostly independently and can thus be treated assuch Each of them is associated with a contact time x isin Nwith a distribution pX(x) The latter is derived from socialmodels and depends on whether people are practicing socialdistancing

4

The accumulation of n measurements leads to a decisionc1 The latter has a probability of missed detection and falsealarm denoted by πmd and πfa respectively In the presentsection both quantities are written without further indices Inlater sections the dependency on n will be made explicitThe combination of x0 such decisions c1 finally leads tothe decision C1 which is associated with a missed detectionprobability

pmd(x) =

x0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd (2)

since the combined missed detection occurs whenever less thanx0 detections succeed Using this in Equation 1 implies thatthe probability that A spreads the disease is

pS KpixMsumx=x0

pX(x)pmd(x)

= Kpi

xMsumx=x0

pX(x)

x0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd

(3)

with xM = 24middot4middotx0 being the number of elementary decisionstaken per day (24 middot 4 quarters of hours times x0) The aboveequation is an approximation since the distribution of contacttimes depends on the people and circumstances of the meetinglike sitting together in the train having a joint lunch and soon If πmd 1 the term m = x0minus1 is dominant in Equation(3)

pS sim KpixMsumx=x0

pX(x)

(x

x0 minus 1

)πmd

x0minus1πxminusx0+1md

= Kpiπx0minus1d

xMminusx0+1sumxprime=1

pX(xprime + x0 minus 1) (4)

middot(xprime + x0 minus 1x0 minus 1

)πxprime

md

le Kpiπx0minus1d

xMminusx0+1sumxprime=1

pX(xprime + x0 minus 1)1

xprime(x0πmd)

xprime

with πd = 1 minus πmd The second line in the equation isobtained by shifting the indices the third one is obtained byexpanding the binomial coefficients and bounding the termsin the numerator Note that the term for xprime = 1 holdswith equality Under the same assumptions used so far theprobability that fellow A is a C1 contact of B after a day is

pC1= Kpi

xMsumx=x0

pX(x) = Npi

xMminusx0+1sumxprime=1

pX(xprime + x0 minus 1)

Thus the comparison of pS ie the probability of spreadingthe virus with tracing and of pC1 ie the correspondingprobability without tracing shows that contact tracing is avery effective option to reduce the spreading whenever

x0πmd (5)

is small This implies that the probability of missed detectionmust be constrained to a value smaller than 1x0 which ispossible to achieve if x0 is small as it is the case in Model

B and Model C and not possible to achieve in Model A evenwith very large values of n Rephrasing this in words may helpdeveloping some intuition since x0 individual c1 decision areneeded for a C1 decision missing any one of them leads toa missed detection Since there are x0 options for that pSbecomes essentially proportional to x0πmd We will use thelatter product as a measure for the reduction in the spreadingof the disease by the tracing app

In order to evaluate pfa we need to additionally knowthe number of times y that a person is close enough for ameasurement to take place The distribution pY (y) does againdepend on social parameters but additionally depends on radiopropagation in the case of Bluetooth measurements and onthe triggering mechanism in the case of audio ranging Thenumber of contacts KY ge K is larger since the presencedetection by Bluetooth signaling is triggered well beyond C1

separation Consider Bluetooth measurements if among they time instances for which a radio contact to one particularfellow B persists and assume that m lt x0 of those contactsare correctly detected as fulfilling the C1 conditions Thenq additional erroneously identified contacts (erroneous c1-decisions) with m+ q ge x0 are needed to cause a false alarmfor that number y of radio contacts to B (see Table III for asummary of the meaning of the variables)

pfa(y) =

xMsumx=0

pX(x)

minxx0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd

yminusxsumq=x0minusm

(y minus xq

)πqfa(1minus πfa)yminusxminusq

(6)

for y ge x0 and pfa(y) = 0 for y lt x0

TABLE IIIVARIABLES USED IN EQUATION III

Variable Meaningy number of radio contactsx number of C1 contactsx0 number of c1-decisions to declare C1

m number of correct c1 estimatesq number of incorrect c1 estimates

Using Equation (6) the expected number of an unnecessaryquarantining of people is approximated by

nQ = KY pi

xMsumy=x0

pY (y)pfa(y)

= KY pi

xMsumy=x0

pY (y)middot

middotxMsumx=0

pX(x)

minxx0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd

middotyminusxsum

q=x0minusm

(y minus xq

)πqfa(1minus πfa)yminusxminusq

(7)

This equation also includes the possibility that users movewith respect to each other which means that the conditions

5

C1 and notC1 alternate as a function of time If C1 is fulfilledπfa = 0 and if notC1 the equation πd = 0 holds At the borderof the C1 domain the two quantities change their role Thisimplies that a small pfa near that border is associated with alarge pmd sim 1 minus pfa on the other side of the border This isuncritical if the distributions are very narrow - concentratedaround a value - as is the case for ranging but becomes ratherproblematic with Bluetooth signal power measurements whichshow a very flat distribution Unless great precautions are takenthe classification becomes unreliable Consider the case thatfellow B is outside of the C1 zone of fellow A ie pX(0) = 1Then x = 0 for these measurements and the equation becomes

nQ = KY pi

xMsumy=x0

pY (y)

ysumx=x0

(yx

)πxfa(1minus πfa)yminusx

Although terms with x gt x0 may be larger the term x = x0

gives us an idea of the scaling Its asymptotic dependencycan be evaluated using Stirlingrsquos formula and limyrarrinfin(y(yminusx0))y = ex0 (

yx0

)πx0

fa simradic

y

2πx0(y minus x0)

((y

x0minus 1

)eπfa

)x0

This means that in the long term it is the duration of theradio contact y which dominates the rate of quarantiningpeople Some target figures for πfa can be obtained for afully occupied train for example In Germanyrsquos 2nd classsetups there are 4 seats in one row on each side of a carriageand around 10 rows in the carriage The range of Bluetoothreaches well beyond the next row forward and backward Thismeans that KY gt 24 of which 4-8 are within the contact zoneand must thus be discounted leading to an effective valueKY = 16 The value y itself is determined by the duration ofthe common journey For commuter trains we choose 15 and30 minutes for inter-city journeys 1 2 and 3 hours whichleads to yx0 = 1 2 4 8 and 12 In such a train a carrierof the disease will send 4 people to quarantine thus it shouldbe tolerable that 2 additional people are sent to quarantine byfalse alarms as well The value of πfa is then obtained bysolving

KY

(yx0

)πx0

fa = 2

Numerical values of πfa are indicated in Table IV They arethe values that can be tolerated leading to a 50 increase inthe quarantining of people riding a German train The situationis rather uncritical on a short commuter train ride πfa lt 093and much more demanding on a longer intercity train journey

TABLE IVROUGH INDICATION FOR ACCEPTABLE VALUES OF πfa THEY ARE

OBTAINED BY CONSIDERING TRAIN RIDES

yx0 1 2 4 8 12πfa 093 025 011 005 003

III BLUETOOTH POWER MEASUREMENTS

The Application Programming Interfaces (API) of Androidand iOS allow to trigger the transmission of Bluetooth Low

Energy (BLE) advertisement messages and to measure the ra-dio signal strength of the received signals The correspondingvalues are provided in the form of a Radio Signal StrengthIndicators (RSSI) which is defined as the received signalpower on a logarithmic scale Bluetooth uses frequenciesfrom a band shared with microwave heating which meansthat Bluetooth signals are strongly absorbed by water As aconsequence any part of a human body obstructing the lineof sight significantly attenuates the signal The wide varietyof options for carrying mobile phones in your hand pocketor bag thus implies an enormous variability in received powerlevels This is further amplified by the directional characteristicof low-cost antennas You might make an experiment yourselfusing a Bluetooth module and a BLE scanner app on yoursmartphone which can be downloaded from the iOS orAndroid stores With the module and phone separated by 15meters I personally found the following RSSI-values -61 to-66 dBm when the module was in my hand and -81 to -89-dB when it was in my pocket Knowing that a 20 dB changecorresponds to a factor 10 in distance exemplifies the difficultyof estimating distances using Bluetooth RSSI values This ledus to propose the rule of carrying smartphones hanging downfrom the neck Note that the smartphone could be replaced bya much smaller device built around a Bluetooth module anInertial Navigation System (INS) and a sonic or ultra-sonicranging system as well

Even if people follow the above recommendation on howto carry their smartphone the situation remains difficult dueto uncertainties in radio propagation which furthermore takesplace on three different carrier frequencies The unknownassociation of carrier frequencies to measurements adds anadditional level of difficulty Gentner et al identified certainpatterns in the use of carriers see [11] which can be usedto reduce the associated uncertainty Traditional models ofpropagation are shortly addressed in the following section andin more details in [8] The section furthermore relates theassociated statistics to the statistics of classification

A Propagation Model

The smartphone is assumed to be worn on the chest see [8]for details of the measurement setup used to obtain numericalresults For each individual carrier the received signal powerPRX is modeled by the equation

PRX =γ

dνPTX + n (8)

with PTX denoting the transmit power γ denoting a stochasticfading coefficient d being the distance between the receiverand the transmitter ν being the exponent of the decay lawwhich is 2 for free space propagation and with n representinga superposition of noise and interference For simplicity thenoise and interference are not further considered here - at lowdistances they are not dominant In this case the receivedpower can be represented on a logarithmic scale which leadsto the definition of the RSSI

RSSI = 10 logPRX = 10 logPTX minus ν middot 10 log d+ η (9)

6

with η = 10 log γ and with logarithms taken to the basis 10The relationship between the reported RSSI value and d isthe basis for distance measurement the measured RSSI iscompared to

Θ = 10 logPTX minus ν middot 10 log dc + 〈η〉

with dc = 2 m being the critical distance Note that Equation(8) defines the units which have to be maintained after takinglogarithms

In order to evaluate the missed detection probability perevent pmd or the false alarm probability per event pfa thestatistics for η or γ need to be known These statistics aredependent on the situation In the case that two fellowsface each other they are in a line of sight situation If thedirect path dominates all other contributions γ is basicallydelta distributed with an average of Γ determined by theantenna pattern In other cases the direct path remains presentbut is superposed by scattered components In this case thedistribution of the amplitude of the received signal is modeledby a Ricean distribution This model is considered to provide afaithful representation of reality whenever the parameters areproperly estimated Presently the model is only considered forcomparative purposes as shall be seen below The receivedpower (or attenuation γ) in this model has a non-central χ2-distribution with two degrees of freedom

pR(γ) =1

2σ2R

eminus(γ+γR)(2σ2R)I0

(radicγγR

σ2R

) (10)

with γR being the non-centrality parameter and σR being thevariance In the case that the decision about C1 is taken onthe basis of a single measurement (n = 1) eg in Model Athe criterion for the decision is

γ ge γc(d

dc

)ν (11)

with γc being given by

γc = 〈γ〉 =

int infin0

dγγpR(γ) (12)

The associated estimate is denoted by c1 and the probabilityof missed detection for the distance d lt dc is given by

πmd(d) =

int γc(ddc)ν

0

dγpR(γ) (13)

If one would add several power measurements ie n gt 1eg in Model B and C this would mean adding n indepen-dent identically distributed variables each of them being χ2-distributed with 2 degrees of freedom The result would thenbe χ2-distributed with 2n degrees of freedom

pRn(γ) =1

2σ2R

(γ

nγR

)nminus12

eminus(γ+nγR)(2σ2R)Inminus1

(radicnγγR

σ2R

)

The Equations (11) and (12) would remain valid and the latterintegral could be computed in closed form for arbitrary nThe value γc is the first moment of the χ2-distribution with2n degrees of freedom and non-centrality parameter nγRσ2

R

γc = n(γR + 2σ2R)

The probability of missed detection (13) in estimating c1 couldthen be computed in closed form using Marcumrsquos Q-functionQn( )

πmdRn(d) = 1minusQn

radicnγRσR

radicγc

(ddc

)ν2σR

(14)

The above distributions are adequate for users A and B inclose proximity of each other as is the case for d le dc It isthe desired result in Model A and shall serve as a benchmarkin the Models B and C The reason for not using this resultdirectly in the latter models is that apps are expected to addthe RSSI values rather than the power values In this case thestatistics cannot be determined in closed form but must ratherbe evaluated numerically Before addressing this case let usconsider the situaiton d gt dc with a line of sight that is oftenobstructed In such cases a lognormal fading distribution isconsidered to be a reasonable model of reality see [12] Thedistribution may either be written in terms of γ

pL(γ) =10 log10(e)radic

2πσLγeminus(10 log γminus10 log γL)2(2σ2

L)

or in terms of η = 10 log γ

pL(η) =1radic

2πσLeminus(ηminusηL)2(2σ2

L) (15)

with ηL = 10 log γL = 〈η〉 Equation (15) makes the Gaussiancharacter and the meaning of ηL and σL obvious In the abovediscussion a decision in the case n = 1 was taken in favorof C1 whenever the power level was above a threshold Onthe logarithmic scale this condition reads RSSI gt Θ iewhenever the difference

RSSI minusΘ = η minus ηL + ν middot 10 logdcd

(16)

is positive or equivalently whenever η gt 〈η〉+ν middot10 log(ddc)Thus a false alert occurs if this condition is fulfilled for d gtdc The probability of a false alarm ie and erroneous decisionfor c1 becomes

πfa(d) =

int infin〈η〉+νmiddot10 log(ddc)

dη pL(η) (17)

= Q

(ν middot 10 log(ddc)

σL

)

with the present Q-function being a scaled version of the errorfunction complement

Q(x) =1

2erfc

(xradic2

)

In the case of n = 1 a closed form of the statistics thus existsfor πmd for d le dc and for πfa for d gt dc In the case n gt 1eg Model B and C the situation changes somewhat sincemeasurements are now combined by adding RSSI-values Thiscorresponds to a geometric average of the received powers

7

In this case the probability of false alarm can be computedeasily

πfan(d) = Q

(radicn middot ν middot 10 log(ddc)

σL

) (18)

for d gt dc This equation is a consequence of the scaling of ηLand σ2

L by n Using the same distribution but with differentparameters for d lt dc is expected to be a worse match toreality but allows to also evaluate the probability of misseddetection in closed from

πmdLn(d) =

int 〈η〉+νmiddot10 log(ddc)

minusinfindη pL(η)

= 1minusQ(radic

n middot ν middot 10 log(ddc)

σL

)= Q

(radicn middot ν middot 10 log(dcd)

σL

)= πfan

(d2c

d

) (19)

It leads to an interesting symmetry between the probabilitiesof missed detection and of false alert

Note that both probabilities πmd and πfa depend on theparameters of the distribution on the true distance d andon the critical distance dc but that they do not depend onthe explicit threshold Θ see Equation (16) and the associatedexplanations The resulting functional dependence can eitherbe used in a simulation of roaming users or can simply beaveraged over the interior of a circle of radius dc for πmd orover its complement or a relevant subset for πfa The closedform of Equation (6) provides the immediate insight thatπfan(dc) = 12 which shows that the models are consistentwith our intuition

B BLE Measurements Results

The companion paper by Dammann et al [8] describes themeasurements and their analysis in more details All thesemeasurements have so far been made using ideal conditionswith no additional people except A and B (in the very initialmeasurements A was a actually a post carrying the receiver)The experimental basis shall be further broadened in thefuture A first result can be derived from the estimated Riceparameters at a distance of 2 meters γR = 247 pW andσR2 = 915 pW as well as for the lognormal distributionat 2 and 4 meters 160 and 197 dBm respectively

This allows plotting the functions from Equation (14) and(17) for πmdRn(d) and for πfan(d2

cd) = πmdLn(d)respectively The values of n determines how many mea-surements are combined into an elementary decision c1 Forn = 1 the values πmdR1(d) and πfa1(d) are the best modelsamong those considered - the use of a decision threshold in theabsolute or logarithmic domain are equivalent The parameterfor 4 meters 197 dBm is used for determining the false alarmrate

If several RSSI values are added (logarithmic domain) thestatistics associated with the more realistic Rice distributionin the near range can not be determined in closed form atleast not today In this case Equation (19) for the lognormal

distribution is used to determine πmdLn(d) with the parame-ter for 2 meters This is used as an approximation of the truedistribution in the exemplary case n = 60 The plots in Figure(2) show two groups of curves The upper group correspondsto n = 1 and the lower group to n = 60 The latter group ofcurves shows the benefit of diversity Within these groups thereare differences between πmdRn(d) (wrong combination) andπmdLn(d) (wrong fading statistics) but they turn out not tobe fundamental

Fig 2 Probability of missed detection as a function of user distance usingBluetooth Radio Signal Strength Indication (RSSI)

TABLE VSELECTED VALUES OF πmdavn UNDER THE ASSUMPTION OF A

LOGNORMAL FADING DISTRIBUTION THESE VALUES DEFINE THENUMBER OF MEASUREMENTS NEEDED TO ACHIEVE THE DESIRED

PROBABILITY OF FURTHER SPREADING THE DISEASE

n πmdavn n πmdavn

1 012 60 00146 0054 120 000715 0034 240 000230 0023 480 00003

In Section III-A the probability of missed detection wasdetermined as a function of distance Since the probability ofdetection is additive in the sense that

πd =

intdS(r) ρ(r)πd(r)

=

int dc

0

2πrdr ρ(r)πd(r) (20)

In this equation πd(r) = 1 minus πmd(r) is the condition prob-ability of detection given that fellow B is at distance r anddS(r) ρ(r) is the probability density for fellow B to be atthat distance Equation 20 thus is the marginalization of πd(r)with respect to r Note that the limitation of the integration isa consequence of πd(r) = 0 whenever r gt dc This allowsto define the average probability of missed detection over thedistribution of users

πmdavn =

int dc0

2πrdr ρ(r)πmdn(r)int dc0

2πrdr ρ(r) (21)

8

The probability distribution of users in Equation (20) and (13)is given by

ρ(r) =dn(r)

dS(r)=dbπr2c2πrdr

In this expression n(r) = bπr2c is the number of peopleat a distance not greater than r in the case of a density ofone person per square meter This corresponds to the densestpacking of people occupying a surface of 1 meter People arecontinuously spread in a symmetric manner around fellow Awhich is a simple way of achieving a densest packing Theldquofunctionrdquo dn(r)dr is mostly zero It jumps at the valuesrm =

radicmπ with

n(rm + ε)minus n(rm minus ε) =

int rm+ε

rmminusε

dn(r)

drdr = 1

which is a distribution in the sense of Schwartz [13] Withthese preparations the integrals become

πmdavn =1

mc

mcsumm=1

πmdn

(radicm

π

) (22)

with mc being the largest integer with such that rmc le dcNote that the density of points rm increases with increasingm which means that the main contribution comes from theborder of the contact zone Using the experimental resultsfrom [8] this integral is evaluated to πmdav1 = 015 forn = 1 for the χ2-distribution and to πmdav1 = 012 for thelognormal distribution which are both not very compatiblewith the need of a small x0πmd Remember that the lattervalue is the reduction factor in the probability of furtherspreading of the disease achieved by contact tracing TableV lists values of πmdavn for different n which can beused to determine the reduction factor Even in the casen = 120 the factor x0πmd = 021 in Model A and it wouldrequire 4 measurement per second It is only with n = 480that factor x0πmd falls below 1 which would require 16measurements per second This would seriously impact thestandby time of the smart phone Assuming Model C and adecision based on 3 minutes intervals ie x0 = 5 means thatwe could achieve a reduction by a factor 007 provided thatn = 60 measurements are performed and aggregated in each 3minutes interval ie that one measurement is performed every3 seconds In the case of a decision every 5 minutes whichassumes a lower dynamics in the relative movement of peoplethe reduction factor is 004 with the same 60 measurementsbut now spread over a 5 minutes interval which correspondsto one measurement every 5 seconds So lower requirementsin the dynamic allow both to improve the suppression of thespreading of the virus and to reduce the measurement rate

Tolerable alarm rates were derived for the train scenarioThis led to the values in Table (IV) The evaluation of πfan(d)is straight forward For d = dc it gives πfan(dc) = 12 as wasalready discussed previously Assuming that people occupy acircular surface of 1 square meter gives them a radius δ =1radicπ Thus the minimum distance to people fully outside of

the critical zone is dc + δ Evaluating Equation (19) yields

pfa1(dc + δ) = 0137 and pfa3(dc + δ) = 0029

respectively This means that n = 1 is compatible with ajourney of 15 minutes before sending more than the two peopleto quarantine For n = 3 long journeys of up to 3 hoursbecome possible with the same consequences The probabilityof false alarm does thus not strongly limit the number n ofmeasurements aggregated to a decision and one might considerthe more demanding homogeneous distribution of users Thisrequires a study of the combination of false alarms Considertwo fellows B and Brsquo there is no alarm if neither B nor Brsquotriggers an alarm ie

1minus πfa = (1minus πfaB)(1minus πfaBprime)

Furthermore let users be at distances dc+δ(k+1) with k isin Z+

being a positive integer and assume that there are

ν(k) = π (dc + 2δ(k + 1))2 minus π (dc + 2δk)

2

users at that distance (they cover an angular shell of thickness2δ) This guarantees a densest packing In that case theprobability of false alarm ie an erroneous decision in favorof C1 becomes

pfan = 1minusinfinprodk=0

(1minus πfan(dc + k))ν(k)

(23)

In this more demanding scenario exemplary values are

pfa3 = 0413 and pfa9 = 0009

which means that n = 9 would be sufficient to reduce theprobability of false alarm to a very small level Table VI showsperformance figures for a number of possible choices for thenumber n of measurements aggregated to an estimate c1 aswell as for the number x0 of estimates c1 that lead to a decisionC1 The product of n and x0 leads to the measurement rateρ = x0n(15 middot 60) The performance figures are the reductionfactor x0πmdn of the spreading achieved by tracing as wellas the probability of unduly sending a person to quarantineThe figures in Table VI all relate to Model C Model A doesnot lead to interesting parameter choices and Model B is toostatic

TABLE VIKEY PERFORMANCE PARAMETERS x0πmdn MEASURES THE REDUCTIONIN SPREADING AND pfan THE PROBABILITY OF UNDUE QUARANTINING

THE PARAMETER ρ IS THE NUMBER OF MEASUREMENTS PER SECOND

x0πmdn pfan ρx0 3 5 - 3 5

n6 016 027 0064 150 130

15 012 017 00002 120 11260 004 007 00000 15 13

A choice with n = 15 and x0 = 3 for example requiresa measurement to be performed every 12 seconds suppressedthe risk of spreading by a factor 012 and does hardly sendanyone unduly to quarantine Performing a measurement everyfive seconds reduces the risk of spreading by a factor 004 Thisassumes that people let their phones hang from their neckand some standard form of environment In reality a numberof additional factors have to be taken into account such asa more complex propagation situation eg due to metallic

9

walls a higher dynamic of user movements eg due topeople entering and exiting commuter trains or unpredictableshadowing due to the userrsquos hands arms or body in the pathof radio signals Thus it is advisable to complement theBluetooth measurement by an alternative Audio ranging isthe option that shall be described in the next section The ideais to use it whenever the situation is not clear

IV AUDIO RANGING

Smartphones have a microphone and a speaker with rathergood transmit and receive conditions if the device is carriedon the chest or held in the hand This can be used for audioranging up to distances of a few meters Signals and theirtransmission can be configured by the API In experiments thatwe performed recently we focused on the use Android phonesThe response of the microphones built into three differentphones is shown in Figure 3 The references were a NT1-A microphone from Rode and an Adagio Infinite Speaker ofA3 on the source side Figure 3 shows the response of threesmartphones from two different brands The curves are verysimilar suggesting that the same microphones are integratedin those phones All microphones show a good sensitivity overall frequencies

Fig 3 Frequency response of microphones from three different smartphones

A similar experiment was performed for the speakers witha rather different result In that case only two smartphoneswere analyzed The response on the better device is reducedby roughly 10 dB above 16 kHz as compared to the referenceThe response of the other one is degraded by another 3 dBand the degradation starts 2kHz earlier Covering the speakerby one layer of tissue of a sweater degrades the performanceby another 4 dB If both parties cover their smartphonesthe associated attenuation adds up Thus the use of audioranging requires carrying the devices in an exposed mannereg hanging from onersquos neck see Figure (1) Transmissionat lower less attenuated frequencies is not considered as atrue option since it would be too disturbing The norm ISO2262003 compiles equivalent hearing sensitivity (isophones)which allows to compare the disturbance caused by acousticalsignals on different frequencies

Fig 4 Frequency response of two speakers as well as influence of coveringthe speaker of Smartphone 2 with one layer of a sweater

On the basis of such considerations we propose modulatinga carrier at 18 kHz with a modulation rate of 1 kbaud Thiskeeps the signal in a spectral range that is not too disturbing tomost people A spread spectrum modulation provides a goodrange resolution and allows to operate at a low signal-to-noiseratio at the same time Different options exist and are discussedin [9] Since the velocity of sound in air is cs = 343 ms understandard conditions a chip duration of 1 ms corresponds to alength of 34 cm At a typical signal-to-noise ratio this leadsto a distance resolution of 1 to 3 cm Let us be conservativeand assume a resolution of 5 cm A multipath delay of twometers leads to an offset by 6 chips and is well suppressedby the autocorrelation of the spreading code The length ofthe spreading code is assumed to be around 350 chips Analternative using chirps is also considered The performanceof audio ranging is further developed in Section IV

Audio ranging can be performed in a peer-to-peer or ina networked manner Consider the peer-to-peer situation firstSmartphones do not provide accurate timing control Howeverthe microphone input of a smartphone may be sampled at afixed rate Furthermore smartphones can transmit and receiveat the same time and this is furthermore supported by theAPIs of Android and iOS Let the smartphones thus agreeto start audio ranging via Bluetooth In a first step theyopen their microphone channels and then proceed according toFigure 5 at time tTXA A transmits the ranging signal usingits speaker This transmission is delayed with respect to theAPI by τTXA In parallel to its transmission Arsquos microphonecapture the transmitted signal This signal is delayed by thesum of the local propagation delay τlA and by the internalreceive delay τRXA The delay τlA is determined by thedevice geometry and can be stored in memory A standardvalue of 14 cm should be appropriate for most devices on themarket The time of reception thus is

tprimeRXA = tTXA + τTXA + τlA + τRXA

and is used for calibration purposes The same definition ofdelays applies at B Thus the signal transmitted by A at time

10

tTXA is received at B at the time tRXB

tRXB = tTXA + τTXA + τ + τRXB

with τ being the propagation time from A to B After receptionof the signal from A by B B sends a corresponding signal toA The equations are obtained by changing the roles of A andB

tprimeRXB = tTXB + τTXB + τlB + τRXB

andtRXA = tTXB + τTXB + τ + τRXA

At the end of the reception A sends

∆tA = tRXA minus tprimeRXA + τlA (24)

to B and B sends ∆tB = tRXB minus tprimeRXB + τlB using BLEThus both can compute the propagation time

τ =∆tA + ∆tB

2

and thus the distance d = τcs The property of audio signalswhich is crucial for this self-calibration is the possibility toobserve the own transmitted signal

Fig 5 Signal paths in two-way acoustical ranging with calibration of transmitand receive delays

A Ranging Protocol

The above peer-to-peer protocol can be extended to anetworked protocol In this case the users agree on an orderingof transmissions via Bluetooth All smartphones A1 Akactivate their microphones and one after the other transmittheir audio ranging signals For simplicity the scheduling isprearranged which also works if some of the smartphonecannot acquire all signals In this case all delays are summedup 350 ms for the ranging signal 10 ms (corresponding to4 meters) for propagation and 40 ms for the internal delaysbetween the activation of the transmission command and thestart of transmission (the latter needs to be confirmed by moredata) This allows for a scheduling of a transmission every 400ms After the completion of the cycle and the evaluation of the

reception time tRXAi by terminal A1 this terminal transmitsthe time difference using Bluetooth

∆tA1Ai = tRXAi minus tprimeRXA1+ τlA1

for 2 le i le k

If all terminals see each other they transmit k(k minus 1) suchvalues in total The annoying transmissions of audio signals re-main limited to k however The overall time interval spannedby all transmissions in the networked protocol may be longenough for users to move slightly This is not critical howeverThe snap-shot measurements are simply converted to averagevalues The only instances which require some care are thosein which the audio signals are used to calibrate Bluetoothmeasurements Finally it should be emphasized that audiobeacon transmissions should not be activated if the device isheld to the ear Even if the signals are hardly heard this seemsa reasonable precaution

B Theoretical Performance of Acoustic Range Estimation

The received audio signal is filtered to remove out-of-band interference and noise to the best possible extent Thefiltered signal is used to determine the in-band interference andnoise level N0 and is furthermore correlated using the filteredranging signal For simplicity the further exposition focuses onspread spectrum signals In a first step the I and Q componentsof the correlation C(∆τ) are computed at intervals of Tc2with Tc = 1 ms denoting the chip duration The result issearched for the delay leading to the maximum norm |C(∆τ)|Although the implementations by widely used phones seemnot to require that frequency offsets may be searched as wellThis allows to acquire the signal which may be present or notThus it is sufficient to search for the delay (and frequencyoffsets) leading to the maximum norm from early to late Thelatter ordering is to avoid locking on an echo If the signalto noise ratio is above the expected threshold the signal isassumed present In this case a successive refinement of theresult is performed in a DLL type of processing The powerdiscriminator

DP (∆τ) = |R(∆τ + δ)|2 minus |R(∆τ minus δ)|2

is used to iteratively increasereduce the delay ∆τ dependingon the value of DP (∆τ) ≷ 0 In this equation δ is half thecorrelator spacing and is expressed as a fraction ∆ of thechip duration δ = ∆Tc We will restrict ourselves to ∆ = 1A further optimization is possible see Betz and Kolodziejski[14] [15] The uncertainty of the delay estimate ∆τ due tonoise is given by (see Dierendonck Fenton and Ford [16])

σ2∆τ T 2

c

∆

4EiN0

(1 +

3

(2minus∆)EiN0

) (25)

In this expression Ei is the signal energy accumulated duringthe correlation and N0 is the spectral noise density of theaudio noise and interference The latter quantity is estimatedusing the norm of the filtered I and Q samples of the incomingsignal

N0 =1

BSNTc

Nsumn=1

(s2I + s2

Q

)

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

4

The accumulation of n measurements leads to a decisionc1 The latter has a probability of missed detection and falsealarm denoted by πmd and πfa respectively In the presentsection both quantities are written without further indices Inlater sections the dependency on n will be made explicitThe combination of x0 such decisions c1 finally leads tothe decision C1 which is associated with a missed detectionprobability

pmd(x) =

x0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd (2)

since the combined missed detection occurs whenever less thanx0 detections succeed Using this in Equation 1 implies thatthe probability that A spreads the disease is

pS KpixMsumx=x0

pX(x)pmd(x)

= Kpi

xMsumx=x0

pX(x)

x0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd

(3)

with xM = 24middot4middotx0 being the number of elementary decisionstaken per day (24 middot 4 quarters of hours times x0) The aboveequation is an approximation since the distribution of contacttimes depends on the people and circumstances of the meetinglike sitting together in the train having a joint lunch and soon If πmd 1 the term m = x0minus1 is dominant in Equation(3)

pS sim KpixMsumx=x0

pX(x)

(x

x0 minus 1

)πmd

x0minus1πxminusx0+1md

= Kpiπx0minus1d

xMminusx0+1sumxprime=1

pX(xprime + x0 minus 1) (4)

middot(xprime + x0 minus 1x0 minus 1

)πxprime

md

le Kpiπx0minus1d

xMminusx0+1sumxprime=1

pX(xprime + x0 minus 1)1

xprime(x0πmd)

xprime

with πd = 1 minus πmd The second line in the equation isobtained by shifting the indices the third one is obtained byexpanding the binomial coefficients and bounding the termsin the numerator Note that the term for xprime = 1 holdswith equality Under the same assumptions used so far theprobability that fellow A is a C1 contact of B after a day is

pC1= Kpi

xMsumx=x0

pX(x) = Npi

xMminusx0+1sumxprime=1

pX(xprime + x0 minus 1)

Thus the comparison of pS ie the probability of spreadingthe virus with tracing and of pC1 ie the correspondingprobability without tracing shows that contact tracing is avery effective option to reduce the spreading whenever

x0πmd (5)

is small This implies that the probability of missed detectionmust be constrained to a value smaller than 1x0 which ispossible to achieve if x0 is small as it is the case in Model

B and Model C and not possible to achieve in Model A evenwith very large values of n Rephrasing this in words may helpdeveloping some intuition since x0 individual c1 decision areneeded for a C1 decision missing any one of them leads toa missed detection Since there are x0 options for that pSbecomes essentially proportional to x0πmd We will use thelatter product as a measure for the reduction in the spreadingof the disease by the tracing app

In order to evaluate pfa we need to additionally knowthe number of times y that a person is close enough for ameasurement to take place The distribution pY (y) does againdepend on social parameters but additionally depends on radiopropagation in the case of Bluetooth measurements and onthe triggering mechanism in the case of audio ranging Thenumber of contacts KY ge K is larger since the presencedetection by Bluetooth signaling is triggered well beyond C1

separation Consider Bluetooth measurements if among they time instances for which a radio contact to one particularfellow B persists and assume that m lt x0 of those contactsare correctly detected as fulfilling the C1 conditions Thenq additional erroneously identified contacts (erroneous c1-decisions) with m+ q ge x0 are needed to cause a false alarmfor that number y of radio contacts to B (see Table III for asummary of the meaning of the variables)

pfa(y) =

xMsumx=0

pX(x)

minxx0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd

yminusxsumq=x0minusm

(y minus xq

)πqfa(1minus πfa)yminusxminusq

(6)

for y ge x0 and pfa(y) = 0 for y lt x0

TABLE IIIVARIABLES USED IN EQUATION III

Variable Meaningy number of radio contactsx number of C1 contactsx0 number of c1-decisions to declare C1

m number of correct c1 estimatesq number of incorrect c1 estimates

Using Equation (6) the expected number of an unnecessaryquarantining of people is approximated by

nQ = KY pi

xMsumy=x0

pY (y)pfa(y)

= KY pi

xMsumy=x0

pY (y)middot

middotxMsumx=0

pX(x)

minxx0minus1summ=0

(xm

)(1minus πmd)mπxminusmmd

middotyminusxsum

q=x0minusm

(y minus xq

)πqfa(1minus πfa)yminusxminusq

(7)

This equation also includes the possibility that users movewith respect to each other which means that the conditions

5

C1 and notC1 alternate as a function of time If C1 is fulfilledπfa = 0 and if notC1 the equation πd = 0 holds At the borderof the C1 domain the two quantities change their role Thisimplies that a small pfa near that border is associated with alarge pmd sim 1 minus pfa on the other side of the border This isuncritical if the distributions are very narrow - concentratedaround a value - as is the case for ranging but becomes ratherproblematic with Bluetooth signal power measurements whichshow a very flat distribution Unless great precautions are takenthe classification becomes unreliable Consider the case thatfellow B is outside of the C1 zone of fellow A ie pX(0) = 1Then x = 0 for these measurements and the equation becomes

nQ = KY pi

xMsumy=x0

pY (y)

ysumx=x0

(yx

)πxfa(1minus πfa)yminusx

Although terms with x gt x0 may be larger the term x = x0

gives us an idea of the scaling Its asymptotic dependencycan be evaluated using Stirlingrsquos formula and limyrarrinfin(y(yminusx0))y = ex0 (

yx0

)πx0

fa simradic

y

2πx0(y minus x0)

((y

x0minus 1

)eπfa

)x0

This means that in the long term it is the duration of theradio contact y which dominates the rate of quarantiningpeople Some target figures for πfa can be obtained for afully occupied train for example In Germanyrsquos 2nd classsetups there are 4 seats in one row on each side of a carriageand around 10 rows in the carriage The range of Bluetoothreaches well beyond the next row forward and backward Thismeans that KY gt 24 of which 4-8 are within the contact zoneand must thus be discounted leading to an effective valueKY = 16 The value y itself is determined by the duration ofthe common journey For commuter trains we choose 15 and30 minutes for inter-city journeys 1 2 and 3 hours whichleads to yx0 = 1 2 4 8 and 12 In such a train a carrierof the disease will send 4 people to quarantine thus it shouldbe tolerable that 2 additional people are sent to quarantine byfalse alarms as well The value of πfa is then obtained bysolving

KY

(yx0

)πx0

fa = 2

Numerical values of πfa are indicated in Table IV They arethe values that can be tolerated leading to a 50 increase inthe quarantining of people riding a German train The situationis rather uncritical on a short commuter train ride πfa lt 093and much more demanding on a longer intercity train journey

TABLE IVROUGH INDICATION FOR ACCEPTABLE VALUES OF πfa THEY ARE

OBTAINED BY CONSIDERING TRAIN RIDES

yx0 1 2 4 8 12πfa 093 025 011 005 003

III BLUETOOTH POWER MEASUREMENTS

The Application Programming Interfaces (API) of Androidand iOS allow to trigger the transmission of Bluetooth Low

Energy (BLE) advertisement messages and to measure the ra-dio signal strength of the received signals The correspondingvalues are provided in the form of a Radio Signal StrengthIndicators (RSSI) which is defined as the received signalpower on a logarithmic scale Bluetooth uses frequenciesfrom a band shared with microwave heating which meansthat Bluetooth signals are strongly absorbed by water As aconsequence any part of a human body obstructing the lineof sight significantly attenuates the signal The wide varietyof options for carrying mobile phones in your hand pocketor bag thus implies an enormous variability in received powerlevels This is further amplified by the directional characteristicof low-cost antennas You might make an experiment yourselfusing a Bluetooth module and a BLE scanner app on yoursmartphone which can be downloaded from the iOS orAndroid stores With the module and phone separated by 15meters I personally found the following RSSI-values -61 to-66 dBm when the module was in my hand and -81 to -89-dB when it was in my pocket Knowing that a 20 dB changecorresponds to a factor 10 in distance exemplifies the difficultyof estimating distances using Bluetooth RSSI values This ledus to propose the rule of carrying smartphones hanging downfrom the neck Note that the smartphone could be replaced bya much smaller device built around a Bluetooth module anInertial Navigation System (INS) and a sonic or ultra-sonicranging system as well

Even if people follow the above recommendation on howto carry their smartphone the situation remains difficult dueto uncertainties in radio propagation which furthermore takesplace on three different carrier frequencies The unknownassociation of carrier frequencies to measurements adds anadditional level of difficulty Gentner et al identified certainpatterns in the use of carriers see [11] which can be usedto reduce the associated uncertainty Traditional models ofpropagation are shortly addressed in the following section andin more details in [8] The section furthermore relates theassociated statistics to the statistics of classification

A Propagation Model

The smartphone is assumed to be worn on the chest see [8]for details of the measurement setup used to obtain numericalresults For each individual carrier the received signal powerPRX is modeled by the equation

PRX =γ

dνPTX + n (8)

with PTX denoting the transmit power γ denoting a stochasticfading coefficient d being the distance between the receiverand the transmitter ν being the exponent of the decay lawwhich is 2 for free space propagation and with n representinga superposition of noise and interference For simplicity thenoise and interference are not further considered here - at lowdistances they are not dominant In this case the receivedpower can be represented on a logarithmic scale which leadsto the definition of the RSSI

RSSI = 10 logPRX = 10 logPTX minus ν middot 10 log d+ η (9)

6

with η = 10 log γ and with logarithms taken to the basis 10The relationship between the reported RSSI value and d isthe basis for distance measurement the measured RSSI iscompared to

Θ = 10 logPTX minus ν middot 10 log dc + 〈η〉

with dc = 2 m being the critical distance Note that Equation(8) defines the units which have to be maintained after takinglogarithms

In order to evaluate the missed detection probability perevent pmd or the false alarm probability per event pfa thestatistics for η or γ need to be known These statistics aredependent on the situation In the case that two fellowsface each other they are in a line of sight situation If thedirect path dominates all other contributions γ is basicallydelta distributed with an average of Γ determined by theantenna pattern In other cases the direct path remains presentbut is superposed by scattered components In this case thedistribution of the amplitude of the received signal is modeledby a Ricean distribution This model is considered to provide afaithful representation of reality whenever the parameters areproperly estimated Presently the model is only considered forcomparative purposes as shall be seen below The receivedpower (or attenuation γ) in this model has a non-central χ2-distribution with two degrees of freedom

pR(γ) =1

2σ2R

eminus(γ+γR)(2σ2R)I0

(radicγγR

σ2R

) (10)

with γR being the non-centrality parameter and σR being thevariance In the case that the decision about C1 is taken onthe basis of a single measurement (n = 1) eg in Model Athe criterion for the decision is

γ ge γc(d

dc

)ν (11)

with γc being given by

γc = 〈γ〉 =

int infin0

dγγpR(γ) (12)

The associated estimate is denoted by c1 and the probabilityof missed detection for the distance d lt dc is given by

πmd(d) =

int γc(ddc)ν

0

dγpR(γ) (13)

If one would add several power measurements ie n gt 1eg in Model B and C this would mean adding n indepen-dent identically distributed variables each of them being χ2-distributed with 2 degrees of freedom The result would thenbe χ2-distributed with 2n degrees of freedom

pRn(γ) =1

2σ2R

(γ

nγR

)nminus12

eminus(γ+nγR)(2σ2R)Inminus1

(radicnγγR

σ2R

)

The Equations (11) and (12) would remain valid and the latterintegral could be computed in closed form for arbitrary nThe value γc is the first moment of the χ2-distribution with2n degrees of freedom and non-centrality parameter nγRσ2

R

γc = n(γR + 2σ2R)

The probability of missed detection (13) in estimating c1 couldthen be computed in closed form using Marcumrsquos Q-functionQn( )

πmdRn(d) = 1minusQn

radicnγRσR

radicγc

(ddc

)ν2σR

(14)

The above distributions are adequate for users A and B inclose proximity of each other as is the case for d le dc It isthe desired result in Model A and shall serve as a benchmarkin the Models B and C The reason for not using this resultdirectly in the latter models is that apps are expected to addthe RSSI values rather than the power values In this case thestatistics cannot be determined in closed form but must ratherbe evaluated numerically Before addressing this case let usconsider the situaiton d gt dc with a line of sight that is oftenobstructed In such cases a lognormal fading distribution isconsidered to be a reasonable model of reality see [12] Thedistribution may either be written in terms of γ

pL(γ) =10 log10(e)radic

2πσLγeminus(10 log γminus10 log γL)2(2σ2

L)

or in terms of η = 10 log γ

pL(η) =1radic

2πσLeminus(ηminusηL)2(2σ2

L) (15)

with ηL = 10 log γL = 〈η〉 Equation (15) makes the Gaussiancharacter and the meaning of ηL and σL obvious In the abovediscussion a decision in the case n = 1 was taken in favorof C1 whenever the power level was above a threshold Onthe logarithmic scale this condition reads RSSI gt Θ iewhenever the difference

RSSI minusΘ = η minus ηL + ν middot 10 logdcd

(16)

is positive or equivalently whenever η gt 〈η〉+ν middot10 log(ddc)Thus a false alert occurs if this condition is fulfilled for d gtdc The probability of a false alarm ie and erroneous decisionfor c1 becomes

πfa(d) =

int infin〈η〉+νmiddot10 log(ddc)

dη pL(η) (17)

= Q

(ν middot 10 log(ddc)

σL

)

with the present Q-function being a scaled version of the errorfunction complement

Q(x) =1

2erfc

(xradic2

)

In the case of n = 1 a closed form of the statistics thus existsfor πmd for d le dc and for πfa for d gt dc In the case n gt 1eg Model B and C the situation changes somewhat sincemeasurements are now combined by adding RSSI-values Thiscorresponds to a geometric average of the received powers

7

In this case the probability of false alarm can be computedeasily

πfan(d) = Q

(radicn middot ν middot 10 log(ddc)

σL

) (18)

for d gt dc This equation is a consequence of the scaling of ηLand σ2

L by n Using the same distribution but with differentparameters for d lt dc is expected to be a worse match toreality but allows to also evaluate the probability of misseddetection in closed from

πmdLn(d) =

int 〈η〉+νmiddot10 log(ddc)

minusinfindη pL(η)

= 1minusQ(radic

n middot ν middot 10 log(ddc)

σL

)= Q

(radicn middot ν middot 10 log(dcd)

σL

)= πfan

(d2c

d

) (19)

It leads to an interesting symmetry between the probabilitiesof missed detection and of false alert

Note that both probabilities πmd and πfa depend on theparameters of the distribution on the true distance d andon the critical distance dc but that they do not depend onthe explicit threshold Θ see Equation (16) and the associatedexplanations The resulting functional dependence can eitherbe used in a simulation of roaming users or can simply beaveraged over the interior of a circle of radius dc for πmd orover its complement or a relevant subset for πfa The closedform of Equation (6) provides the immediate insight thatπfan(dc) = 12 which shows that the models are consistentwith our intuition

B BLE Measurements Results

The companion paper by Dammann et al [8] describes themeasurements and their analysis in more details All thesemeasurements have so far been made using ideal conditionswith no additional people except A and B (in the very initialmeasurements A was a actually a post carrying the receiver)The experimental basis shall be further broadened in thefuture A first result can be derived from the estimated Riceparameters at a distance of 2 meters γR = 247 pW andσR2 = 915 pW as well as for the lognormal distributionat 2 and 4 meters 160 and 197 dBm respectively

This allows plotting the functions from Equation (14) and(17) for πmdRn(d) and for πfan(d2

cd) = πmdLn(d)respectively The values of n determines how many mea-surements are combined into an elementary decision c1 Forn = 1 the values πmdR1(d) and πfa1(d) are the best modelsamong those considered - the use of a decision threshold in theabsolute or logarithmic domain are equivalent The parameterfor 4 meters 197 dBm is used for determining the false alarmrate

If several RSSI values are added (logarithmic domain) thestatistics associated with the more realistic Rice distributionin the near range can not be determined in closed form atleast not today In this case Equation (19) for the lognormal

distribution is used to determine πmdLn(d) with the parame-ter for 2 meters This is used as an approximation of the truedistribution in the exemplary case n = 60 The plots in Figure(2) show two groups of curves The upper group correspondsto n = 1 and the lower group to n = 60 The latter group ofcurves shows the benefit of diversity Within these groups thereare differences between πmdRn(d) (wrong combination) andπmdLn(d) (wrong fading statistics) but they turn out not tobe fundamental

Fig 2 Probability of missed detection as a function of user distance usingBluetooth Radio Signal Strength Indication (RSSI)

TABLE VSELECTED VALUES OF πmdavn UNDER THE ASSUMPTION OF A

LOGNORMAL FADING DISTRIBUTION THESE VALUES DEFINE THENUMBER OF MEASUREMENTS NEEDED TO ACHIEVE THE DESIRED

PROBABILITY OF FURTHER SPREADING THE DISEASE

n πmdavn n πmdavn

1 012 60 00146 0054 120 000715 0034 240 000230 0023 480 00003

In Section III-A the probability of missed detection wasdetermined as a function of distance Since the probability ofdetection is additive in the sense that

πd =

intdS(r) ρ(r)πd(r)

=

int dc

0

2πrdr ρ(r)πd(r) (20)

In this equation πd(r) = 1 minus πmd(r) is the condition prob-ability of detection given that fellow B is at distance r anddS(r) ρ(r) is the probability density for fellow B to be atthat distance Equation 20 thus is the marginalization of πd(r)with respect to r Note that the limitation of the integration isa consequence of πd(r) = 0 whenever r gt dc This allowsto define the average probability of missed detection over thedistribution of users

πmdavn =

int dc0

2πrdr ρ(r)πmdn(r)int dc0

2πrdr ρ(r) (21)

8

The probability distribution of users in Equation (20) and (13)is given by

ρ(r) =dn(r)

dS(r)=dbπr2c2πrdr

In this expression n(r) = bπr2c is the number of peopleat a distance not greater than r in the case of a density ofone person per square meter This corresponds to the densestpacking of people occupying a surface of 1 meter People arecontinuously spread in a symmetric manner around fellow Awhich is a simple way of achieving a densest packing Theldquofunctionrdquo dn(r)dr is mostly zero It jumps at the valuesrm =

radicmπ with

n(rm + ε)minus n(rm minus ε) =

int rm+ε

rmminusε

dn(r)

drdr = 1

which is a distribution in the sense of Schwartz [13] Withthese preparations the integrals become

πmdavn =1

mc

mcsumm=1

πmdn

(radicm

π

) (22)

with mc being the largest integer with such that rmc le dcNote that the density of points rm increases with increasingm which means that the main contribution comes from theborder of the contact zone Using the experimental resultsfrom [8] this integral is evaluated to πmdav1 = 015 forn = 1 for the χ2-distribution and to πmdav1 = 012 for thelognormal distribution which are both not very compatiblewith the need of a small x0πmd Remember that the lattervalue is the reduction factor in the probability of furtherspreading of the disease achieved by contact tracing TableV lists values of πmdavn for different n which can beused to determine the reduction factor Even in the casen = 120 the factor x0πmd = 021 in Model A and it wouldrequire 4 measurement per second It is only with n = 480that factor x0πmd falls below 1 which would require 16measurements per second This would seriously impact thestandby time of the smart phone Assuming Model C and adecision based on 3 minutes intervals ie x0 = 5 means thatwe could achieve a reduction by a factor 007 provided thatn = 60 measurements are performed and aggregated in each 3minutes interval ie that one measurement is performed every3 seconds In the case of a decision every 5 minutes whichassumes a lower dynamics in the relative movement of peoplethe reduction factor is 004 with the same 60 measurementsbut now spread over a 5 minutes interval which correspondsto one measurement every 5 seconds So lower requirementsin the dynamic allow both to improve the suppression of thespreading of the virus and to reduce the measurement rate

Tolerable alarm rates were derived for the train scenarioThis led to the values in Table (IV) The evaluation of πfan(d)is straight forward For d = dc it gives πfan(dc) = 12 as wasalready discussed previously Assuming that people occupy acircular surface of 1 square meter gives them a radius δ =1radicπ Thus the minimum distance to people fully outside of

the critical zone is dc + δ Evaluating Equation (19) yields

pfa1(dc + δ) = 0137 and pfa3(dc + δ) = 0029

respectively This means that n = 1 is compatible with ajourney of 15 minutes before sending more than the two peopleto quarantine For n = 3 long journeys of up to 3 hoursbecome possible with the same consequences The probabilityof false alarm does thus not strongly limit the number n ofmeasurements aggregated to a decision and one might considerthe more demanding homogeneous distribution of users Thisrequires a study of the combination of false alarms Considertwo fellows B and Brsquo there is no alarm if neither B nor Brsquotriggers an alarm ie

1minus πfa = (1minus πfaB)(1minus πfaBprime)

Furthermore let users be at distances dc+δ(k+1) with k isin Z+

being a positive integer and assume that there are

ν(k) = π (dc + 2δ(k + 1))2 minus π (dc + 2δk)

2

users at that distance (they cover an angular shell of thickness2δ) This guarantees a densest packing In that case theprobability of false alarm ie an erroneous decision in favorof C1 becomes

pfan = 1minusinfinprodk=0

(1minus πfan(dc + k))ν(k)

(23)

In this more demanding scenario exemplary values are

pfa3 = 0413 and pfa9 = 0009

which means that n = 9 would be sufficient to reduce theprobability of false alarm to a very small level Table VI showsperformance figures for a number of possible choices for thenumber n of measurements aggregated to an estimate c1 aswell as for the number x0 of estimates c1 that lead to a decisionC1 The product of n and x0 leads to the measurement rateρ = x0n(15 middot 60) The performance figures are the reductionfactor x0πmdn of the spreading achieved by tracing as wellas the probability of unduly sending a person to quarantineThe figures in Table VI all relate to Model C Model A doesnot lead to interesting parameter choices and Model B is toostatic

TABLE VIKEY PERFORMANCE PARAMETERS x0πmdn MEASURES THE REDUCTIONIN SPREADING AND pfan THE PROBABILITY OF UNDUE QUARANTINING

THE PARAMETER ρ IS THE NUMBER OF MEASUREMENTS PER SECOND

x0πmdn pfan ρx0 3 5 - 3 5

n6 016 027 0064 150 130

15 012 017 00002 120 11260 004 007 00000 15 13

A choice with n = 15 and x0 = 3 for example requiresa measurement to be performed every 12 seconds suppressedthe risk of spreading by a factor 012 and does hardly sendanyone unduly to quarantine Performing a measurement everyfive seconds reduces the risk of spreading by a factor 004 Thisassumes that people let their phones hang from their neckand some standard form of environment In reality a numberof additional factors have to be taken into account such asa more complex propagation situation eg due to metallic

9

walls a higher dynamic of user movements eg due topeople entering and exiting commuter trains or unpredictableshadowing due to the userrsquos hands arms or body in the pathof radio signals Thus it is advisable to complement theBluetooth measurement by an alternative Audio ranging isthe option that shall be described in the next section The ideais to use it whenever the situation is not clear

IV AUDIO RANGING

Smartphones have a microphone and a speaker with rathergood transmit and receive conditions if the device is carriedon the chest or held in the hand This can be used for audioranging up to distances of a few meters Signals and theirtransmission can be configured by the API In experiments thatwe performed recently we focused on the use Android phonesThe response of the microphones built into three differentphones is shown in Figure 3 The references were a NT1-A microphone from Rode and an Adagio Infinite Speaker ofA3 on the source side Figure 3 shows the response of threesmartphones from two different brands The curves are verysimilar suggesting that the same microphones are integratedin those phones All microphones show a good sensitivity overall frequencies

Fig 3 Frequency response of microphones from three different smartphones

A similar experiment was performed for the speakers witha rather different result In that case only two smartphoneswere analyzed The response on the better device is reducedby roughly 10 dB above 16 kHz as compared to the referenceThe response of the other one is degraded by another 3 dBand the degradation starts 2kHz earlier Covering the speakerby one layer of tissue of a sweater degrades the performanceby another 4 dB If both parties cover their smartphonesthe associated attenuation adds up Thus the use of audioranging requires carrying the devices in an exposed mannereg hanging from onersquos neck see Figure (1) Transmissionat lower less attenuated frequencies is not considered as atrue option since it would be too disturbing The norm ISO2262003 compiles equivalent hearing sensitivity (isophones)which allows to compare the disturbance caused by acousticalsignals on different frequencies

Fig 4 Frequency response of two speakers as well as influence of coveringthe speaker of Smartphone 2 with one layer of a sweater

On the basis of such considerations we propose modulatinga carrier at 18 kHz with a modulation rate of 1 kbaud Thiskeeps the signal in a spectral range that is not too disturbing tomost people A spread spectrum modulation provides a goodrange resolution and allows to operate at a low signal-to-noiseratio at the same time Different options exist and are discussedin [9] Since the velocity of sound in air is cs = 343 ms understandard conditions a chip duration of 1 ms corresponds to alength of 34 cm At a typical signal-to-noise ratio this leadsto a distance resolution of 1 to 3 cm Let us be conservativeand assume a resolution of 5 cm A multipath delay of twometers leads to an offset by 6 chips and is well suppressedby the autocorrelation of the spreading code The length ofthe spreading code is assumed to be around 350 chips Analternative using chirps is also considered The performanceof audio ranging is further developed in Section IV

Audio ranging can be performed in a peer-to-peer or ina networked manner Consider the peer-to-peer situation firstSmartphones do not provide accurate timing control Howeverthe microphone input of a smartphone may be sampled at afixed rate Furthermore smartphones can transmit and receiveat the same time and this is furthermore supported by theAPIs of Android and iOS Let the smartphones thus agreeto start audio ranging via Bluetooth In a first step theyopen their microphone channels and then proceed according toFigure 5 at time tTXA A transmits the ranging signal usingits speaker This transmission is delayed with respect to theAPI by τTXA In parallel to its transmission Arsquos microphonecapture the transmitted signal This signal is delayed by thesum of the local propagation delay τlA and by the internalreceive delay τRXA The delay τlA is determined by thedevice geometry and can be stored in memory A standardvalue of 14 cm should be appropriate for most devices on themarket The time of reception thus is

tprimeRXA = tTXA + τTXA + τlA + τRXA

and is used for calibration purposes The same definition ofdelays applies at B Thus the signal transmitted by A at time

10

tTXA is received at B at the time tRXB

tRXB = tTXA + τTXA + τ + τRXB

with τ being the propagation time from A to B After receptionof the signal from A by B B sends a corresponding signal toA The equations are obtained by changing the roles of A andB

tprimeRXB = tTXB + τTXB + τlB + τRXB

andtRXA = tTXB + τTXB + τ + τRXA

At the end of the reception A sends

∆tA = tRXA minus tprimeRXA + τlA (24)

to B and B sends ∆tB = tRXB minus tprimeRXB + τlB using BLEThus both can compute the propagation time

τ =∆tA + ∆tB

2

and thus the distance d = τcs The property of audio signalswhich is crucial for this self-calibration is the possibility toobserve the own transmitted signal

Fig 5 Signal paths in two-way acoustical ranging with calibration of transmitand receive delays

A Ranging Protocol

The above peer-to-peer protocol can be extended to anetworked protocol In this case the users agree on an orderingof transmissions via Bluetooth All smartphones A1 Akactivate their microphones and one after the other transmittheir audio ranging signals For simplicity the scheduling isprearranged which also works if some of the smartphonecannot acquire all signals In this case all delays are summedup 350 ms for the ranging signal 10 ms (corresponding to4 meters) for propagation and 40 ms for the internal delaysbetween the activation of the transmission command and thestart of transmission (the latter needs to be confirmed by moredata) This allows for a scheduling of a transmission every 400ms After the completion of the cycle and the evaluation of the

reception time tRXAi by terminal A1 this terminal transmitsthe time difference using Bluetooth

∆tA1Ai = tRXAi minus tprimeRXA1+ τlA1

for 2 le i le k

If all terminals see each other they transmit k(k minus 1) suchvalues in total The annoying transmissions of audio signals re-main limited to k however The overall time interval spannedby all transmissions in the networked protocol may be longenough for users to move slightly This is not critical howeverThe snap-shot measurements are simply converted to averagevalues The only instances which require some care are thosein which the audio signals are used to calibrate Bluetoothmeasurements Finally it should be emphasized that audiobeacon transmissions should not be activated if the device isheld to the ear Even if the signals are hardly heard this seemsa reasonable precaution

B Theoretical Performance of Acoustic Range Estimation

The received audio signal is filtered to remove out-of-band interference and noise to the best possible extent Thefiltered signal is used to determine the in-band interference andnoise level N0 and is furthermore correlated using the filteredranging signal For simplicity the further exposition focuses onspread spectrum signals In a first step the I and Q componentsof the correlation C(∆τ) are computed at intervals of Tc2with Tc = 1 ms denoting the chip duration The result issearched for the delay leading to the maximum norm |C(∆τ)|Although the implementations by widely used phones seemnot to require that frequency offsets may be searched as wellThis allows to acquire the signal which may be present or notThus it is sufficient to search for the delay (and frequencyoffsets) leading to the maximum norm from early to late Thelatter ordering is to avoid locking on an echo If the signalto noise ratio is above the expected threshold the signal isassumed present In this case a successive refinement of theresult is performed in a DLL type of processing The powerdiscriminator

DP (∆τ) = |R(∆τ + δ)|2 minus |R(∆τ minus δ)|2

is used to iteratively increasereduce the delay ∆τ dependingon the value of DP (∆τ) ≷ 0 In this equation δ is half thecorrelator spacing and is expressed as a fraction ∆ of thechip duration δ = ∆Tc We will restrict ourselves to ∆ = 1A further optimization is possible see Betz and Kolodziejski[14] [15] The uncertainty of the delay estimate ∆τ due tonoise is given by (see Dierendonck Fenton and Ford [16])

σ2∆τ T 2

c

∆

4EiN0

(1 +

3

(2minus∆)EiN0

) (25)

In this expression Ei is the signal energy accumulated duringthe correlation and N0 is the spectral noise density of theaudio noise and interference The latter quantity is estimatedusing the norm of the filtered I and Q samples of the incomingsignal

N0 =1

BSNTc

Nsumn=1

(s2I + s2

Q

)

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

5

C1 and notC1 alternate as a function of time If C1 is fulfilledπfa = 0 and if notC1 the equation πd = 0 holds At the borderof the C1 domain the two quantities change their role Thisimplies that a small pfa near that border is associated with alarge pmd sim 1 minus pfa on the other side of the border This isuncritical if the distributions are very narrow - concentratedaround a value - as is the case for ranging but becomes ratherproblematic with Bluetooth signal power measurements whichshow a very flat distribution Unless great precautions are takenthe classification becomes unreliable Consider the case thatfellow B is outside of the C1 zone of fellow A ie pX(0) = 1Then x = 0 for these measurements and the equation becomes

nQ = KY pi

xMsumy=x0

pY (y)

ysumx=x0

(yx

)πxfa(1minus πfa)yminusx

Although terms with x gt x0 may be larger the term x = x0

gives us an idea of the scaling Its asymptotic dependencycan be evaluated using Stirlingrsquos formula and limyrarrinfin(y(yminusx0))y = ex0 (

yx0

)πx0

fa simradic

y

2πx0(y minus x0)

((y

x0minus 1

)eπfa

)x0

This means that in the long term it is the duration of theradio contact y which dominates the rate of quarantiningpeople Some target figures for πfa can be obtained for afully occupied train for example In Germanyrsquos 2nd classsetups there are 4 seats in one row on each side of a carriageand around 10 rows in the carriage The range of Bluetoothreaches well beyond the next row forward and backward Thismeans that KY gt 24 of which 4-8 are within the contact zoneand must thus be discounted leading to an effective valueKY = 16 The value y itself is determined by the duration ofthe common journey For commuter trains we choose 15 and30 minutes for inter-city journeys 1 2 and 3 hours whichleads to yx0 = 1 2 4 8 and 12 In such a train a carrierof the disease will send 4 people to quarantine thus it shouldbe tolerable that 2 additional people are sent to quarantine byfalse alarms as well The value of πfa is then obtained bysolving

KY

(yx0

)πx0

fa = 2

Numerical values of πfa are indicated in Table IV They arethe values that can be tolerated leading to a 50 increase inthe quarantining of people riding a German train The situationis rather uncritical on a short commuter train ride πfa lt 093and much more demanding on a longer intercity train journey

TABLE IVROUGH INDICATION FOR ACCEPTABLE VALUES OF πfa THEY ARE

OBTAINED BY CONSIDERING TRAIN RIDES

yx0 1 2 4 8 12πfa 093 025 011 005 003

III BLUETOOTH POWER MEASUREMENTS

The Application Programming Interfaces (API) of Androidand iOS allow to trigger the transmission of Bluetooth Low

Energy (BLE) advertisement messages and to measure the ra-dio signal strength of the received signals The correspondingvalues are provided in the form of a Radio Signal StrengthIndicators (RSSI) which is defined as the received signalpower on a logarithmic scale Bluetooth uses frequenciesfrom a band shared with microwave heating which meansthat Bluetooth signals are strongly absorbed by water As aconsequence any part of a human body obstructing the lineof sight significantly attenuates the signal The wide varietyof options for carrying mobile phones in your hand pocketor bag thus implies an enormous variability in received powerlevels This is further amplified by the directional characteristicof low-cost antennas You might make an experiment yourselfusing a Bluetooth module and a BLE scanner app on yoursmartphone which can be downloaded from the iOS orAndroid stores With the module and phone separated by 15meters I personally found the following RSSI-values -61 to-66 dBm when the module was in my hand and -81 to -89-dB when it was in my pocket Knowing that a 20 dB changecorresponds to a factor 10 in distance exemplifies the difficultyof estimating distances using Bluetooth RSSI values This ledus to propose the rule of carrying smartphones hanging downfrom the neck Note that the smartphone could be replaced bya much smaller device built around a Bluetooth module anInertial Navigation System (INS) and a sonic or ultra-sonicranging system as well

Even if people follow the above recommendation on howto carry their smartphone the situation remains difficult dueto uncertainties in radio propagation which furthermore takesplace on three different carrier frequencies The unknownassociation of carrier frequencies to measurements adds anadditional level of difficulty Gentner et al identified certainpatterns in the use of carriers see [11] which can be usedto reduce the associated uncertainty Traditional models ofpropagation are shortly addressed in the following section andin more details in [8] The section furthermore relates theassociated statistics to the statistics of classification

A Propagation Model

The smartphone is assumed to be worn on the chest see [8]for details of the measurement setup used to obtain numericalresults For each individual carrier the received signal powerPRX is modeled by the equation

PRX =γ

dνPTX + n (8)

with PTX denoting the transmit power γ denoting a stochasticfading coefficient d being the distance between the receiverand the transmitter ν being the exponent of the decay lawwhich is 2 for free space propagation and with n representinga superposition of noise and interference For simplicity thenoise and interference are not further considered here - at lowdistances they are not dominant In this case the receivedpower can be represented on a logarithmic scale which leadsto the definition of the RSSI

RSSI = 10 logPRX = 10 logPTX minus ν middot 10 log d+ η (9)

6

with η = 10 log γ and with logarithms taken to the basis 10The relationship between the reported RSSI value and d isthe basis for distance measurement the measured RSSI iscompared to

Θ = 10 logPTX minus ν middot 10 log dc + 〈η〉

with dc = 2 m being the critical distance Note that Equation(8) defines the units which have to be maintained after takinglogarithms

In order to evaluate the missed detection probability perevent pmd or the false alarm probability per event pfa thestatistics for η or γ need to be known These statistics aredependent on the situation In the case that two fellowsface each other they are in a line of sight situation If thedirect path dominates all other contributions γ is basicallydelta distributed with an average of Γ determined by theantenna pattern In other cases the direct path remains presentbut is superposed by scattered components In this case thedistribution of the amplitude of the received signal is modeledby a Ricean distribution This model is considered to provide afaithful representation of reality whenever the parameters areproperly estimated Presently the model is only considered forcomparative purposes as shall be seen below The receivedpower (or attenuation γ) in this model has a non-central χ2-distribution with two degrees of freedom

pR(γ) =1

2σ2R

eminus(γ+γR)(2σ2R)I0

(radicγγR

σ2R

) (10)

with γR being the non-centrality parameter and σR being thevariance In the case that the decision about C1 is taken onthe basis of a single measurement (n = 1) eg in Model Athe criterion for the decision is

γ ge γc(d

dc

)ν (11)

with γc being given by

γc = 〈γ〉 =

int infin0

dγγpR(γ) (12)

The associated estimate is denoted by c1 and the probabilityof missed detection for the distance d lt dc is given by

πmd(d) =

int γc(ddc)ν

0

dγpR(γ) (13)

If one would add several power measurements ie n gt 1eg in Model B and C this would mean adding n indepen-dent identically distributed variables each of them being χ2-distributed with 2 degrees of freedom The result would thenbe χ2-distributed with 2n degrees of freedom

pRn(γ) =1

2σ2R

(γ

nγR

)nminus12

eminus(γ+nγR)(2σ2R)Inminus1

(radicnγγR

σ2R

)

The Equations (11) and (12) would remain valid and the latterintegral could be computed in closed form for arbitrary nThe value γc is the first moment of the χ2-distribution with2n degrees of freedom and non-centrality parameter nγRσ2

R

γc = n(γR + 2σ2R)

The probability of missed detection (13) in estimating c1 couldthen be computed in closed form using Marcumrsquos Q-functionQn( )

πmdRn(d) = 1minusQn

radicnγRσR

radicγc

(ddc

)ν2σR

(14)

The above distributions are adequate for users A and B inclose proximity of each other as is the case for d le dc It isthe desired result in Model A and shall serve as a benchmarkin the Models B and C The reason for not using this resultdirectly in the latter models is that apps are expected to addthe RSSI values rather than the power values In this case thestatistics cannot be determined in closed form but must ratherbe evaluated numerically Before addressing this case let usconsider the situaiton d gt dc with a line of sight that is oftenobstructed In such cases a lognormal fading distribution isconsidered to be a reasonable model of reality see [12] Thedistribution may either be written in terms of γ

pL(γ) =10 log10(e)radic

2πσLγeminus(10 log γminus10 log γL)2(2σ2

L)

or in terms of η = 10 log γ

pL(η) =1radic

2πσLeminus(ηminusηL)2(2σ2

L) (15)

with ηL = 10 log γL = 〈η〉 Equation (15) makes the Gaussiancharacter and the meaning of ηL and σL obvious In the abovediscussion a decision in the case n = 1 was taken in favorof C1 whenever the power level was above a threshold Onthe logarithmic scale this condition reads RSSI gt Θ iewhenever the difference

RSSI minusΘ = η minus ηL + ν middot 10 logdcd

(16)

is positive or equivalently whenever η gt 〈η〉+ν middot10 log(ddc)Thus a false alert occurs if this condition is fulfilled for d gtdc The probability of a false alarm ie and erroneous decisionfor c1 becomes

πfa(d) =

int infin〈η〉+νmiddot10 log(ddc)

dη pL(η) (17)

= Q

(ν middot 10 log(ddc)

σL

)

with the present Q-function being a scaled version of the errorfunction complement

Q(x) =1

2erfc

(xradic2

)

In the case of n = 1 a closed form of the statistics thus existsfor πmd for d le dc and for πfa for d gt dc In the case n gt 1eg Model B and C the situation changes somewhat sincemeasurements are now combined by adding RSSI-values Thiscorresponds to a geometric average of the received powers

7

In this case the probability of false alarm can be computedeasily

πfan(d) = Q

(radicn middot ν middot 10 log(ddc)

σL

) (18)

for d gt dc This equation is a consequence of the scaling of ηLand σ2

L by n Using the same distribution but with differentparameters for d lt dc is expected to be a worse match toreality but allows to also evaluate the probability of misseddetection in closed from

πmdLn(d) =

int 〈η〉+νmiddot10 log(ddc)

minusinfindη pL(η)

= 1minusQ(radic

n middot ν middot 10 log(ddc)

σL

)= Q

(radicn middot ν middot 10 log(dcd)

σL

)= πfan

(d2c

d

) (19)

It leads to an interesting symmetry between the probabilitiesof missed detection and of false alert

Note that both probabilities πmd and πfa depend on theparameters of the distribution on the true distance d andon the critical distance dc but that they do not depend onthe explicit threshold Θ see Equation (16) and the associatedexplanations The resulting functional dependence can eitherbe used in a simulation of roaming users or can simply beaveraged over the interior of a circle of radius dc for πmd orover its complement or a relevant subset for πfa The closedform of Equation (6) provides the immediate insight thatπfan(dc) = 12 which shows that the models are consistentwith our intuition

B BLE Measurements Results

The companion paper by Dammann et al [8] describes themeasurements and their analysis in more details All thesemeasurements have so far been made using ideal conditionswith no additional people except A and B (in the very initialmeasurements A was a actually a post carrying the receiver)The experimental basis shall be further broadened in thefuture A first result can be derived from the estimated Riceparameters at a distance of 2 meters γR = 247 pW andσR2 = 915 pW as well as for the lognormal distributionat 2 and 4 meters 160 and 197 dBm respectively

This allows plotting the functions from Equation (14) and(17) for πmdRn(d) and for πfan(d2

cd) = πmdLn(d)respectively The values of n determines how many mea-surements are combined into an elementary decision c1 Forn = 1 the values πmdR1(d) and πfa1(d) are the best modelsamong those considered - the use of a decision threshold in theabsolute or logarithmic domain are equivalent The parameterfor 4 meters 197 dBm is used for determining the false alarmrate

If several RSSI values are added (logarithmic domain) thestatistics associated with the more realistic Rice distributionin the near range can not be determined in closed form atleast not today In this case Equation (19) for the lognormal

distribution is used to determine πmdLn(d) with the parame-ter for 2 meters This is used as an approximation of the truedistribution in the exemplary case n = 60 The plots in Figure(2) show two groups of curves The upper group correspondsto n = 1 and the lower group to n = 60 The latter group ofcurves shows the benefit of diversity Within these groups thereare differences between πmdRn(d) (wrong combination) andπmdLn(d) (wrong fading statistics) but they turn out not tobe fundamental

Fig 2 Probability of missed detection as a function of user distance usingBluetooth Radio Signal Strength Indication (RSSI)

TABLE VSELECTED VALUES OF πmdavn UNDER THE ASSUMPTION OF A

LOGNORMAL FADING DISTRIBUTION THESE VALUES DEFINE THENUMBER OF MEASUREMENTS NEEDED TO ACHIEVE THE DESIRED

PROBABILITY OF FURTHER SPREADING THE DISEASE

n πmdavn n πmdavn

1 012 60 00146 0054 120 000715 0034 240 000230 0023 480 00003

In Section III-A the probability of missed detection wasdetermined as a function of distance Since the probability ofdetection is additive in the sense that

πd =

intdS(r) ρ(r)πd(r)

=

int dc

0

2πrdr ρ(r)πd(r) (20)

In this equation πd(r) = 1 minus πmd(r) is the condition prob-ability of detection given that fellow B is at distance r anddS(r) ρ(r) is the probability density for fellow B to be atthat distance Equation 20 thus is the marginalization of πd(r)with respect to r Note that the limitation of the integration isa consequence of πd(r) = 0 whenever r gt dc This allowsto define the average probability of missed detection over thedistribution of users

πmdavn =

int dc0

2πrdr ρ(r)πmdn(r)int dc0

2πrdr ρ(r) (21)

8

The probability distribution of users in Equation (20) and (13)is given by

ρ(r) =dn(r)

dS(r)=dbπr2c2πrdr

In this expression n(r) = bπr2c is the number of peopleat a distance not greater than r in the case of a density ofone person per square meter This corresponds to the densestpacking of people occupying a surface of 1 meter People arecontinuously spread in a symmetric manner around fellow Awhich is a simple way of achieving a densest packing Theldquofunctionrdquo dn(r)dr is mostly zero It jumps at the valuesrm =

radicmπ with

n(rm + ε)minus n(rm minus ε) =

int rm+ε

rmminusε

dn(r)

drdr = 1

which is a distribution in the sense of Schwartz [13] Withthese preparations the integrals become

πmdavn =1

mc

mcsumm=1

πmdn

(radicm

π

) (22)

with mc being the largest integer with such that rmc le dcNote that the density of points rm increases with increasingm which means that the main contribution comes from theborder of the contact zone Using the experimental resultsfrom [8] this integral is evaluated to πmdav1 = 015 forn = 1 for the χ2-distribution and to πmdav1 = 012 for thelognormal distribution which are both not very compatiblewith the need of a small x0πmd Remember that the lattervalue is the reduction factor in the probability of furtherspreading of the disease achieved by contact tracing TableV lists values of πmdavn for different n which can beused to determine the reduction factor Even in the casen = 120 the factor x0πmd = 021 in Model A and it wouldrequire 4 measurement per second It is only with n = 480that factor x0πmd falls below 1 which would require 16measurements per second This would seriously impact thestandby time of the smart phone Assuming Model C and adecision based on 3 minutes intervals ie x0 = 5 means thatwe could achieve a reduction by a factor 007 provided thatn = 60 measurements are performed and aggregated in each 3minutes interval ie that one measurement is performed every3 seconds In the case of a decision every 5 minutes whichassumes a lower dynamics in the relative movement of peoplethe reduction factor is 004 with the same 60 measurementsbut now spread over a 5 minutes interval which correspondsto one measurement every 5 seconds So lower requirementsin the dynamic allow both to improve the suppression of thespreading of the virus and to reduce the measurement rate

Tolerable alarm rates were derived for the train scenarioThis led to the values in Table (IV) The evaluation of πfan(d)is straight forward For d = dc it gives πfan(dc) = 12 as wasalready discussed previously Assuming that people occupy acircular surface of 1 square meter gives them a radius δ =1radicπ Thus the minimum distance to people fully outside of

the critical zone is dc + δ Evaluating Equation (19) yields

pfa1(dc + δ) = 0137 and pfa3(dc + δ) = 0029

respectively This means that n = 1 is compatible with ajourney of 15 minutes before sending more than the two peopleto quarantine For n = 3 long journeys of up to 3 hoursbecome possible with the same consequences The probabilityof false alarm does thus not strongly limit the number n ofmeasurements aggregated to a decision and one might considerthe more demanding homogeneous distribution of users Thisrequires a study of the combination of false alarms Considertwo fellows B and Brsquo there is no alarm if neither B nor Brsquotriggers an alarm ie

1minus πfa = (1minus πfaB)(1minus πfaBprime)

Furthermore let users be at distances dc+δ(k+1) with k isin Z+

being a positive integer and assume that there are

ν(k) = π (dc + 2δ(k + 1))2 minus π (dc + 2δk)

2

users at that distance (they cover an angular shell of thickness2δ) This guarantees a densest packing In that case theprobability of false alarm ie an erroneous decision in favorof C1 becomes

pfan = 1minusinfinprodk=0

(1minus πfan(dc + k))ν(k)

(23)

In this more demanding scenario exemplary values are

pfa3 = 0413 and pfa9 = 0009

which means that n = 9 would be sufficient to reduce theprobability of false alarm to a very small level Table VI showsperformance figures for a number of possible choices for thenumber n of measurements aggregated to an estimate c1 aswell as for the number x0 of estimates c1 that lead to a decisionC1 The product of n and x0 leads to the measurement rateρ = x0n(15 middot 60) The performance figures are the reductionfactor x0πmdn of the spreading achieved by tracing as wellas the probability of unduly sending a person to quarantineThe figures in Table VI all relate to Model C Model A doesnot lead to interesting parameter choices and Model B is toostatic

TABLE VIKEY PERFORMANCE PARAMETERS x0πmdn MEASURES THE REDUCTIONIN SPREADING AND pfan THE PROBABILITY OF UNDUE QUARANTINING

THE PARAMETER ρ IS THE NUMBER OF MEASUREMENTS PER SECOND

x0πmdn pfan ρx0 3 5 - 3 5

n6 016 027 0064 150 130

15 012 017 00002 120 11260 004 007 00000 15 13

A choice with n = 15 and x0 = 3 for example requiresa measurement to be performed every 12 seconds suppressedthe risk of spreading by a factor 012 and does hardly sendanyone unduly to quarantine Performing a measurement everyfive seconds reduces the risk of spreading by a factor 004 Thisassumes that people let their phones hang from their neckand some standard form of environment In reality a numberof additional factors have to be taken into account such asa more complex propagation situation eg due to metallic

9

walls a higher dynamic of user movements eg due topeople entering and exiting commuter trains or unpredictableshadowing due to the userrsquos hands arms or body in the pathof radio signals Thus it is advisable to complement theBluetooth measurement by an alternative Audio ranging isthe option that shall be described in the next section The ideais to use it whenever the situation is not clear

IV AUDIO RANGING

Smartphones have a microphone and a speaker with rathergood transmit and receive conditions if the device is carriedon the chest or held in the hand This can be used for audioranging up to distances of a few meters Signals and theirtransmission can be configured by the API In experiments thatwe performed recently we focused on the use Android phonesThe response of the microphones built into three differentphones is shown in Figure 3 The references were a NT1-A microphone from Rode and an Adagio Infinite Speaker ofA3 on the source side Figure 3 shows the response of threesmartphones from two different brands The curves are verysimilar suggesting that the same microphones are integratedin those phones All microphones show a good sensitivity overall frequencies

Fig 3 Frequency response of microphones from three different smartphones

A similar experiment was performed for the speakers witha rather different result In that case only two smartphoneswere analyzed The response on the better device is reducedby roughly 10 dB above 16 kHz as compared to the referenceThe response of the other one is degraded by another 3 dBand the degradation starts 2kHz earlier Covering the speakerby one layer of tissue of a sweater degrades the performanceby another 4 dB If both parties cover their smartphonesthe associated attenuation adds up Thus the use of audioranging requires carrying the devices in an exposed mannereg hanging from onersquos neck see Figure (1) Transmissionat lower less attenuated frequencies is not considered as atrue option since it would be too disturbing The norm ISO2262003 compiles equivalent hearing sensitivity (isophones)which allows to compare the disturbance caused by acousticalsignals on different frequencies

Fig 4 Frequency response of two speakers as well as influence of coveringthe speaker of Smartphone 2 with one layer of a sweater

On the basis of such considerations we propose modulatinga carrier at 18 kHz with a modulation rate of 1 kbaud Thiskeeps the signal in a spectral range that is not too disturbing tomost people A spread spectrum modulation provides a goodrange resolution and allows to operate at a low signal-to-noiseratio at the same time Different options exist and are discussedin [9] Since the velocity of sound in air is cs = 343 ms understandard conditions a chip duration of 1 ms corresponds to alength of 34 cm At a typical signal-to-noise ratio this leadsto a distance resolution of 1 to 3 cm Let us be conservativeand assume a resolution of 5 cm A multipath delay of twometers leads to an offset by 6 chips and is well suppressedby the autocorrelation of the spreading code The length ofthe spreading code is assumed to be around 350 chips Analternative using chirps is also considered The performanceof audio ranging is further developed in Section IV

Audio ranging can be performed in a peer-to-peer or ina networked manner Consider the peer-to-peer situation firstSmartphones do not provide accurate timing control Howeverthe microphone input of a smartphone may be sampled at afixed rate Furthermore smartphones can transmit and receiveat the same time and this is furthermore supported by theAPIs of Android and iOS Let the smartphones thus agreeto start audio ranging via Bluetooth In a first step theyopen their microphone channels and then proceed according toFigure 5 at time tTXA A transmits the ranging signal usingits speaker This transmission is delayed with respect to theAPI by τTXA In parallel to its transmission Arsquos microphonecapture the transmitted signal This signal is delayed by thesum of the local propagation delay τlA and by the internalreceive delay τRXA The delay τlA is determined by thedevice geometry and can be stored in memory A standardvalue of 14 cm should be appropriate for most devices on themarket The time of reception thus is

tprimeRXA = tTXA + τTXA + τlA + τRXA

and is used for calibration purposes The same definition ofdelays applies at B Thus the signal transmitted by A at time

10

tTXA is received at B at the time tRXB

tRXB = tTXA + τTXA + τ + τRXB

with τ being the propagation time from A to B After receptionof the signal from A by B B sends a corresponding signal toA The equations are obtained by changing the roles of A andB

tprimeRXB = tTXB + τTXB + τlB + τRXB

andtRXA = tTXB + τTXB + τ + τRXA

At the end of the reception A sends

∆tA = tRXA minus tprimeRXA + τlA (24)

to B and B sends ∆tB = tRXB minus tprimeRXB + τlB using BLEThus both can compute the propagation time

τ =∆tA + ∆tB

2

and thus the distance d = τcs The property of audio signalswhich is crucial for this self-calibration is the possibility toobserve the own transmitted signal

Fig 5 Signal paths in two-way acoustical ranging with calibration of transmitand receive delays

A Ranging Protocol

The above peer-to-peer protocol can be extended to anetworked protocol In this case the users agree on an orderingof transmissions via Bluetooth All smartphones A1 Akactivate their microphones and one after the other transmittheir audio ranging signals For simplicity the scheduling isprearranged which also works if some of the smartphonecannot acquire all signals In this case all delays are summedup 350 ms for the ranging signal 10 ms (corresponding to4 meters) for propagation and 40 ms for the internal delaysbetween the activation of the transmission command and thestart of transmission (the latter needs to be confirmed by moredata) This allows for a scheduling of a transmission every 400ms After the completion of the cycle and the evaluation of the

reception time tRXAi by terminal A1 this terminal transmitsthe time difference using Bluetooth

∆tA1Ai = tRXAi minus tprimeRXA1+ τlA1

for 2 le i le k

If all terminals see each other they transmit k(k minus 1) suchvalues in total The annoying transmissions of audio signals re-main limited to k however The overall time interval spannedby all transmissions in the networked protocol may be longenough for users to move slightly This is not critical howeverThe snap-shot measurements are simply converted to averagevalues The only instances which require some care are thosein which the audio signals are used to calibrate Bluetoothmeasurements Finally it should be emphasized that audiobeacon transmissions should not be activated if the device isheld to the ear Even if the signals are hardly heard this seemsa reasonable precaution

B Theoretical Performance of Acoustic Range Estimation

The received audio signal is filtered to remove out-of-band interference and noise to the best possible extent Thefiltered signal is used to determine the in-band interference andnoise level N0 and is furthermore correlated using the filteredranging signal For simplicity the further exposition focuses onspread spectrum signals In a first step the I and Q componentsof the correlation C(∆τ) are computed at intervals of Tc2with Tc = 1 ms denoting the chip duration The result issearched for the delay leading to the maximum norm |C(∆τ)|Although the implementations by widely used phones seemnot to require that frequency offsets may be searched as wellThis allows to acquire the signal which may be present or notThus it is sufficient to search for the delay (and frequencyoffsets) leading to the maximum norm from early to late Thelatter ordering is to avoid locking on an echo If the signalto noise ratio is above the expected threshold the signal isassumed present In this case a successive refinement of theresult is performed in a DLL type of processing The powerdiscriminator

DP (∆τ) = |R(∆τ + δ)|2 minus |R(∆τ minus δ)|2

is used to iteratively increasereduce the delay ∆τ dependingon the value of DP (∆τ) ≷ 0 In this equation δ is half thecorrelator spacing and is expressed as a fraction ∆ of thechip duration δ = ∆Tc We will restrict ourselves to ∆ = 1A further optimization is possible see Betz and Kolodziejski[14] [15] The uncertainty of the delay estimate ∆τ due tonoise is given by (see Dierendonck Fenton and Ford [16])

σ2∆τ T 2

c

∆

4EiN0

(1 +

3

(2minus∆)EiN0

) (25)

In this expression Ei is the signal energy accumulated duringthe correlation and N0 is the spectral noise density of theaudio noise and interference The latter quantity is estimatedusing the norm of the filtered I and Q samples of the incomingsignal

N0 =1

BSNTc

Nsumn=1

(s2I + s2

Q

)

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

6

with η = 10 log γ and with logarithms taken to the basis 10The relationship between the reported RSSI value and d isthe basis for distance measurement the measured RSSI iscompared to

Θ = 10 logPTX minus ν middot 10 log dc + 〈η〉

with dc = 2 m being the critical distance Note that Equation(8) defines the units which have to be maintained after takinglogarithms

In order to evaluate the missed detection probability perevent pmd or the false alarm probability per event pfa thestatistics for η or γ need to be known These statistics aredependent on the situation In the case that two fellowsface each other they are in a line of sight situation If thedirect path dominates all other contributions γ is basicallydelta distributed with an average of Γ determined by theantenna pattern In other cases the direct path remains presentbut is superposed by scattered components In this case thedistribution of the amplitude of the received signal is modeledby a Ricean distribution This model is considered to provide afaithful representation of reality whenever the parameters areproperly estimated Presently the model is only considered forcomparative purposes as shall be seen below The receivedpower (or attenuation γ) in this model has a non-central χ2-distribution with two degrees of freedom

pR(γ) =1

2σ2R

eminus(γ+γR)(2σ2R)I0

(radicγγR

σ2R

) (10)

with γR being the non-centrality parameter and σR being thevariance In the case that the decision about C1 is taken onthe basis of a single measurement (n = 1) eg in Model Athe criterion for the decision is

γ ge γc(d

dc

)ν (11)

with γc being given by

γc = 〈γ〉 =

int infin0

dγγpR(γ) (12)

The associated estimate is denoted by c1 and the probabilityof missed detection for the distance d lt dc is given by

πmd(d) =

int γc(ddc)ν

0

dγpR(γ) (13)

If one would add several power measurements ie n gt 1eg in Model B and C this would mean adding n indepen-dent identically distributed variables each of them being χ2-distributed with 2 degrees of freedom The result would thenbe χ2-distributed with 2n degrees of freedom

pRn(γ) =1

2σ2R

(γ

nγR

)nminus12

eminus(γ+nγR)(2σ2R)Inminus1

(radicnγγR

σ2R

)

The Equations (11) and (12) would remain valid and the latterintegral could be computed in closed form for arbitrary nThe value γc is the first moment of the χ2-distribution with2n degrees of freedom and non-centrality parameter nγRσ2

R

γc = n(γR + 2σ2R)

The probability of missed detection (13) in estimating c1 couldthen be computed in closed form using Marcumrsquos Q-functionQn( )

πmdRn(d) = 1minusQn

radicnγRσR

radicγc

(ddc

)ν2σR

(14)

The above distributions are adequate for users A and B inclose proximity of each other as is the case for d le dc It isthe desired result in Model A and shall serve as a benchmarkin the Models B and C The reason for not using this resultdirectly in the latter models is that apps are expected to addthe RSSI values rather than the power values In this case thestatistics cannot be determined in closed form but must ratherbe evaluated numerically Before addressing this case let usconsider the situaiton d gt dc with a line of sight that is oftenobstructed In such cases a lognormal fading distribution isconsidered to be a reasonable model of reality see [12] Thedistribution may either be written in terms of γ

pL(γ) =10 log10(e)radic

2πσLγeminus(10 log γminus10 log γL)2(2σ2

L)

or in terms of η = 10 log γ

pL(η) =1radic

2πσLeminus(ηminusηL)2(2σ2

L) (15)

with ηL = 10 log γL = 〈η〉 Equation (15) makes the Gaussiancharacter and the meaning of ηL and σL obvious In the abovediscussion a decision in the case n = 1 was taken in favorof C1 whenever the power level was above a threshold Onthe logarithmic scale this condition reads RSSI gt Θ iewhenever the difference

RSSI minusΘ = η minus ηL + ν middot 10 logdcd

(16)

is positive or equivalently whenever η gt 〈η〉+ν middot10 log(ddc)Thus a false alert occurs if this condition is fulfilled for d gtdc The probability of a false alarm ie and erroneous decisionfor c1 becomes

πfa(d) =

int infin〈η〉+νmiddot10 log(ddc)

dη pL(η) (17)

= Q

(ν middot 10 log(ddc)

σL

)

with the present Q-function being a scaled version of the errorfunction complement

Q(x) =1

2erfc

(xradic2

)

In the case of n = 1 a closed form of the statistics thus existsfor πmd for d le dc and for πfa for d gt dc In the case n gt 1eg Model B and C the situation changes somewhat sincemeasurements are now combined by adding RSSI-values Thiscorresponds to a geometric average of the received powers

7

In this case the probability of false alarm can be computedeasily

πfan(d) = Q

(radicn middot ν middot 10 log(ddc)

σL

) (18)

for d gt dc This equation is a consequence of the scaling of ηLand σ2

L by n Using the same distribution but with differentparameters for d lt dc is expected to be a worse match toreality but allows to also evaluate the probability of misseddetection in closed from

πmdLn(d) =

int 〈η〉+νmiddot10 log(ddc)

minusinfindη pL(η)

= 1minusQ(radic

n middot ν middot 10 log(ddc)

σL

)= Q

(radicn middot ν middot 10 log(dcd)

σL

)= πfan

(d2c

d

) (19)

It leads to an interesting symmetry between the probabilitiesof missed detection and of false alert

Note that both probabilities πmd and πfa depend on theparameters of the distribution on the true distance d andon the critical distance dc but that they do not depend onthe explicit threshold Θ see Equation (16) and the associatedexplanations The resulting functional dependence can eitherbe used in a simulation of roaming users or can simply beaveraged over the interior of a circle of radius dc for πmd orover its complement or a relevant subset for πfa The closedform of Equation (6) provides the immediate insight thatπfan(dc) = 12 which shows that the models are consistentwith our intuition

B BLE Measurements Results

The companion paper by Dammann et al [8] describes themeasurements and their analysis in more details All thesemeasurements have so far been made using ideal conditionswith no additional people except A and B (in the very initialmeasurements A was a actually a post carrying the receiver)The experimental basis shall be further broadened in thefuture A first result can be derived from the estimated Riceparameters at a distance of 2 meters γR = 247 pW andσR2 = 915 pW as well as for the lognormal distributionat 2 and 4 meters 160 and 197 dBm respectively

This allows plotting the functions from Equation (14) and(17) for πmdRn(d) and for πfan(d2

cd) = πmdLn(d)respectively The values of n determines how many mea-surements are combined into an elementary decision c1 Forn = 1 the values πmdR1(d) and πfa1(d) are the best modelsamong those considered - the use of a decision threshold in theabsolute or logarithmic domain are equivalent The parameterfor 4 meters 197 dBm is used for determining the false alarmrate

If several RSSI values are added (logarithmic domain) thestatistics associated with the more realistic Rice distributionin the near range can not be determined in closed form atleast not today In this case Equation (19) for the lognormal

distribution is used to determine πmdLn(d) with the parame-ter for 2 meters This is used as an approximation of the truedistribution in the exemplary case n = 60 The plots in Figure(2) show two groups of curves The upper group correspondsto n = 1 and the lower group to n = 60 The latter group ofcurves shows the benefit of diversity Within these groups thereare differences between πmdRn(d) (wrong combination) andπmdLn(d) (wrong fading statistics) but they turn out not tobe fundamental

Fig 2 Probability of missed detection as a function of user distance usingBluetooth Radio Signal Strength Indication (RSSI)

TABLE VSELECTED VALUES OF πmdavn UNDER THE ASSUMPTION OF A

LOGNORMAL FADING DISTRIBUTION THESE VALUES DEFINE THENUMBER OF MEASUREMENTS NEEDED TO ACHIEVE THE DESIRED

PROBABILITY OF FURTHER SPREADING THE DISEASE

n πmdavn n πmdavn

1 012 60 00146 0054 120 000715 0034 240 000230 0023 480 00003

In Section III-A the probability of missed detection wasdetermined as a function of distance Since the probability ofdetection is additive in the sense that

πd =

intdS(r) ρ(r)πd(r)

=

int dc

0

2πrdr ρ(r)πd(r) (20)

In this equation πd(r) = 1 minus πmd(r) is the condition prob-ability of detection given that fellow B is at distance r anddS(r) ρ(r) is the probability density for fellow B to be atthat distance Equation 20 thus is the marginalization of πd(r)with respect to r Note that the limitation of the integration isa consequence of πd(r) = 0 whenever r gt dc This allowsto define the average probability of missed detection over thedistribution of users

πmdavn =

int dc0

2πrdr ρ(r)πmdn(r)int dc0

2πrdr ρ(r) (21)

8

The probability distribution of users in Equation (20) and (13)is given by

ρ(r) =dn(r)

dS(r)=dbπr2c2πrdr

In this expression n(r) = bπr2c is the number of peopleat a distance not greater than r in the case of a density ofone person per square meter This corresponds to the densestpacking of people occupying a surface of 1 meter People arecontinuously spread in a symmetric manner around fellow Awhich is a simple way of achieving a densest packing Theldquofunctionrdquo dn(r)dr is mostly zero It jumps at the valuesrm =

radicmπ with

n(rm + ε)minus n(rm minus ε) =

int rm+ε

rmminusε

dn(r)

drdr = 1

which is a distribution in the sense of Schwartz [13] Withthese preparations the integrals become

πmdavn =1

mc

mcsumm=1

πmdn

(radicm

π

) (22)

with mc being the largest integer with such that rmc le dcNote that the density of points rm increases with increasingm which means that the main contribution comes from theborder of the contact zone Using the experimental resultsfrom [8] this integral is evaluated to πmdav1 = 015 forn = 1 for the χ2-distribution and to πmdav1 = 012 for thelognormal distribution which are both not very compatiblewith the need of a small x0πmd Remember that the lattervalue is the reduction factor in the probability of furtherspreading of the disease achieved by contact tracing TableV lists values of πmdavn for different n which can beused to determine the reduction factor Even in the casen = 120 the factor x0πmd = 021 in Model A and it wouldrequire 4 measurement per second It is only with n = 480that factor x0πmd falls below 1 which would require 16measurements per second This would seriously impact thestandby time of the smart phone Assuming Model C and adecision based on 3 minutes intervals ie x0 = 5 means thatwe could achieve a reduction by a factor 007 provided thatn = 60 measurements are performed and aggregated in each 3minutes interval ie that one measurement is performed every3 seconds In the case of a decision every 5 minutes whichassumes a lower dynamics in the relative movement of peoplethe reduction factor is 004 with the same 60 measurementsbut now spread over a 5 minutes interval which correspondsto one measurement every 5 seconds So lower requirementsin the dynamic allow both to improve the suppression of thespreading of the virus and to reduce the measurement rate

Tolerable alarm rates were derived for the train scenarioThis led to the values in Table (IV) The evaluation of πfan(d)is straight forward For d = dc it gives πfan(dc) = 12 as wasalready discussed previously Assuming that people occupy acircular surface of 1 square meter gives them a radius δ =1radicπ Thus the minimum distance to people fully outside of

the critical zone is dc + δ Evaluating Equation (19) yields

pfa1(dc + δ) = 0137 and pfa3(dc + δ) = 0029

respectively This means that n = 1 is compatible with ajourney of 15 minutes before sending more than the two peopleto quarantine For n = 3 long journeys of up to 3 hoursbecome possible with the same consequences The probabilityof false alarm does thus not strongly limit the number n ofmeasurements aggregated to a decision and one might considerthe more demanding homogeneous distribution of users Thisrequires a study of the combination of false alarms Considertwo fellows B and Brsquo there is no alarm if neither B nor Brsquotriggers an alarm ie

1minus πfa = (1minus πfaB)(1minus πfaBprime)

Furthermore let users be at distances dc+δ(k+1) with k isin Z+

being a positive integer and assume that there are

ν(k) = π (dc + 2δ(k + 1))2 minus π (dc + 2δk)

2

users at that distance (they cover an angular shell of thickness2δ) This guarantees a densest packing In that case theprobability of false alarm ie an erroneous decision in favorof C1 becomes

pfan = 1minusinfinprodk=0

(1minus πfan(dc + k))ν(k)

(23)

In this more demanding scenario exemplary values are

pfa3 = 0413 and pfa9 = 0009

which means that n = 9 would be sufficient to reduce theprobability of false alarm to a very small level Table VI showsperformance figures for a number of possible choices for thenumber n of measurements aggregated to an estimate c1 aswell as for the number x0 of estimates c1 that lead to a decisionC1 The product of n and x0 leads to the measurement rateρ = x0n(15 middot 60) The performance figures are the reductionfactor x0πmdn of the spreading achieved by tracing as wellas the probability of unduly sending a person to quarantineThe figures in Table VI all relate to Model C Model A doesnot lead to interesting parameter choices and Model B is toostatic

TABLE VIKEY PERFORMANCE PARAMETERS x0πmdn MEASURES THE REDUCTIONIN SPREADING AND pfan THE PROBABILITY OF UNDUE QUARANTINING

THE PARAMETER ρ IS THE NUMBER OF MEASUREMENTS PER SECOND

x0πmdn pfan ρx0 3 5 - 3 5

n6 016 027 0064 150 130

15 012 017 00002 120 11260 004 007 00000 15 13

A choice with n = 15 and x0 = 3 for example requiresa measurement to be performed every 12 seconds suppressedthe risk of spreading by a factor 012 and does hardly sendanyone unduly to quarantine Performing a measurement everyfive seconds reduces the risk of spreading by a factor 004 Thisassumes that people let their phones hang from their neckand some standard form of environment In reality a numberof additional factors have to be taken into account such asa more complex propagation situation eg due to metallic

9

walls a higher dynamic of user movements eg due topeople entering and exiting commuter trains or unpredictableshadowing due to the userrsquos hands arms or body in the pathof radio signals Thus it is advisable to complement theBluetooth measurement by an alternative Audio ranging isthe option that shall be described in the next section The ideais to use it whenever the situation is not clear

IV AUDIO RANGING

Smartphones have a microphone and a speaker with rathergood transmit and receive conditions if the device is carriedon the chest or held in the hand This can be used for audioranging up to distances of a few meters Signals and theirtransmission can be configured by the API In experiments thatwe performed recently we focused on the use Android phonesThe response of the microphones built into three differentphones is shown in Figure 3 The references were a NT1-A microphone from Rode and an Adagio Infinite Speaker ofA3 on the source side Figure 3 shows the response of threesmartphones from two different brands The curves are verysimilar suggesting that the same microphones are integratedin those phones All microphones show a good sensitivity overall frequencies

Fig 3 Frequency response of microphones from three different smartphones

A similar experiment was performed for the speakers witha rather different result In that case only two smartphoneswere analyzed The response on the better device is reducedby roughly 10 dB above 16 kHz as compared to the referenceThe response of the other one is degraded by another 3 dBand the degradation starts 2kHz earlier Covering the speakerby one layer of tissue of a sweater degrades the performanceby another 4 dB If both parties cover their smartphonesthe associated attenuation adds up Thus the use of audioranging requires carrying the devices in an exposed mannereg hanging from onersquos neck see Figure (1) Transmissionat lower less attenuated frequencies is not considered as atrue option since it would be too disturbing The norm ISO2262003 compiles equivalent hearing sensitivity (isophones)which allows to compare the disturbance caused by acousticalsignals on different frequencies

Fig 4 Frequency response of two speakers as well as influence of coveringthe speaker of Smartphone 2 with one layer of a sweater

On the basis of such considerations we propose modulatinga carrier at 18 kHz with a modulation rate of 1 kbaud Thiskeeps the signal in a spectral range that is not too disturbing tomost people A spread spectrum modulation provides a goodrange resolution and allows to operate at a low signal-to-noiseratio at the same time Different options exist and are discussedin [9] Since the velocity of sound in air is cs = 343 ms understandard conditions a chip duration of 1 ms corresponds to alength of 34 cm At a typical signal-to-noise ratio this leadsto a distance resolution of 1 to 3 cm Let us be conservativeand assume a resolution of 5 cm A multipath delay of twometers leads to an offset by 6 chips and is well suppressedby the autocorrelation of the spreading code The length ofthe spreading code is assumed to be around 350 chips Analternative using chirps is also considered The performanceof audio ranging is further developed in Section IV

Audio ranging can be performed in a peer-to-peer or ina networked manner Consider the peer-to-peer situation firstSmartphones do not provide accurate timing control Howeverthe microphone input of a smartphone may be sampled at afixed rate Furthermore smartphones can transmit and receiveat the same time and this is furthermore supported by theAPIs of Android and iOS Let the smartphones thus agreeto start audio ranging via Bluetooth In a first step theyopen their microphone channels and then proceed according toFigure 5 at time tTXA A transmits the ranging signal usingits speaker This transmission is delayed with respect to theAPI by τTXA In parallel to its transmission Arsquos microphonecapture the transmitted signal This signal is delayed by thesum of the local propagation delay τlA and by the internalreceive delay τRXA The delay τlA is determined by thedevice geometry and can be stored in memory A standardvalue of 14 cm should be appropriate for most devices on themarket The time of reception thus is

tprimeRXA = tTXA + τTXA + τlA + τRXA

and is used for calibration purposes The same definition ofdelays applies at B Thus the signal transmitted by A at time

10

tTXA is received at B at the time tRXB

tRXB = tTXA + τTXA + τ + τRXB

with τ being the propagation time from A to B After receptionof the signal from A by B B sends a corresponding signal toA The equations are obtained by changing the roles of A andB

tprimeRXB = tTXB + τTXB + τlB + τRXB

andtRXA = tTXB + τTXB + τ + τRXA

At the end of the reception A sends

∆tA = tRXA minus tprimeRXA + τlA (24)

to B and B sends ∆tB = tRXB minus tprimeRXB + τlB using BLEThus both can compute the propagation time

τ =∆tA + ∆tB

2

and thus the distance d = τcs The property of audio signalswhich is crucial for this self-calibration is the possibility toobserve the own transmitted signal

Fig 5 Signal paths in two-way acoustical ranging with calibration of transmitand receive delays

A Ranging Protocol

The above peer-to-peer protocol can be extended to anetworked protocol In this case the users agree on an orderingof transmissions via Bluetooth All smartphones A1 Akactivate their microphones and one after the other transmittheir audio ranging signals For simplicity the scheduling isprearranged which also works if some of the smartphonecannot acquire all signals In this case all delays are summedup 350 ms for the ranging signal 10 ms (corresponding to4 meters) for propagation and 40 ms for the internal delaysbetween the activation of the transmission command and thestart of transmission (the latter needs to be confirmed by moredata) This allows for a scheduling of a transmission every 400ms After the completion of the cycle and the evaluation of the

reception time tRXAi by terminal A1 this terminal transmitsthe time difference using Bluetooth

∆tA1Ai = tRXAi minus tprimeRXA1+ τlA1

for 2 le i le k

If all terminals see each other they transmit k(k minus 1) suchvalues in total The annoying transmissions of audio signals re-main limited to k however The overall time interval spannedby all transmissions in the networked protocol may be longenough for users to move slightly This is not critical howeverThe snap-shot measurements are simply converted to averagevalues The only instances which require some care are thosein which the audio signals are used to calibrate Bluetoothmeasurements Finally it should be emphasized that audiobeacon transmissions should not be activated if the device isheld to the ear Even if the signals are hardly heard this seemsa reasonable precaution

B Theoretical Performance of Acoustic Range Estimation

The received audio signal is filtered to remove out-of-band interference and noise to the best possible extent Thefiltered signal is used to determine the in-band interference andnoise level N0 and is furthermore correlated using the filteredranging signal For simplicity the further exposition focuses onspread spectrum signals In a first step the I and Q componentsof the correlation C(∆τ) are computed at intervals of Tc2with Tc = 1 ms denoting the chip duration The result issearched for the delay leading to the maximum norm |C(∆τ)|Although the implementations by widely used phones seemnot to require that frequency offsets may be searched as wellThis allows to acquire the signal which may be present or notThus it is sufficient to search for the delay (and frequencyoffsets) leading to the maximum norm from early to late Thelatter ordering is to avoid locking on an echo If the signalto noise ratio is above the expected threshold the signal isassumed present In this case a successive refinement of theresult is performed in a DLL type of processing The powerdiscriminator

DP (∆τ) = |R(∆τ + δ)|2 minus |R(∆τ minus δ)|2

is used to iteratively increasereduce the delay ∆τ dependingon the value of DP (∆τ) ≷ 0 In this equation δ is half thecorrelator spacing and is expressed as a fraction ∆ of thechip duration δ = ∆Tc We will restrict ourselves to ∆ = 1A further optimization is possible see Betz and Kolodziejski[14] [15] The uncertainty of the delay estimate ∆τ due tonoise is given by (see Dierendonck Fenton and Ford [16])

σ2∆τ T 2

c

∆

4EiN0

(1 +

3

(2minus∆)EiN0

) (25)

In this expression Ei is the signal energy accumulated duringthe correlation and N0 is the spectral noise density of theaudio noise and interference The latter quantity is estimatedusing the norm of the filtered I and Q samples of the incomingsignal

N0 =1

BSNTc

Nsumn=1

(s2I + s2

Q

)

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

7

In this case the probability of false alarm can be computedeasily

πfan(d) = Q

(radicn middot ν middot 10 log(ddc)

σL

) (18)

for d gt dc This equation is a consequence of the scaling of ηLand σ2

L by n Using the same distribution but with differentparameters for d lt dc is expected to be a worse match toreality but allows to also evaluate the probability of misseddetection in closed from

πmdLn(d) =

int 〈η〉+νmiddot10 log(ddc)

minusinfindη pL(η)

= 1minusQ(radic

n middot ν middot 10 log(ddc)

σL

)= Q

(radicn middot ν middot 10 log(dcd)

σL

)= πfan

(d2c

d

) (19)

It leads to an interesting symmetry between the probabilitiesof missed detection and of false alert

Note that both probabilities πmd and πfa depend on theparameters of the distribution on the true distance d andon the critical distance dc but that they do not depend onthe explicit threshold Θ see Equation (16) and the associatedexplanations The resulting functional dependence can eitherbe used in a simulation of roaming users or can simply beaveraged over the interior of a circle of radius dc for πmd orover its complement or a relevant subset for πfa The closedform of Equation (6) provides the immediate insight thatπfan(dc) = 12 which shows that the models are consistentwith our intuition

B BLE Measurements Results

The companion paper by Dammann et al [8] describes themeasurements and their analysis in more details All thesemeasurements have so far been made using ideal conditionswith no additional people except A and B (in the very initialmeasurements A was a actually a post carrying the receiver)The experimental basis shall be further broadened in thefuture A first result can be derived from the estimated Riceparameters at a distance of 2 meters γR = 247 pW andσR2 = 915 pW as well as for the lognormal distributionat 2 and 4 meters 160 and 197 dBm respectively

This allows plotting the functions from Equation (14) and(17) for πmdRn(d) and for πfan(d2

cd) = πmdLn(d)respectively The values of n determines how many mea-surements are combined into an elementary decision c1 Forn = 1 the values πmdR1(d) and πfa1(d) are the best modelsamong those considered - the use of a decision threshold in theabsolute or logarithmic domain are equivalent The parameterfor 4 meters 197 dBm is used for determining the false alarmrate

If several RSSI values are added (logarithmic domain) thestatistics associated with the more realistic Rice distributionin the near range can not be determined in closed form atleast not today In this case Equation (19) for the lognormal

distribution is used to determine πmdLn(d) with the parame-ter for 2 meters This is used as an approximation of the truedistribution in the exemplary case n = 60 The plots in Figure(2) show two groups of curves The upper group correspondsto n = 1 and the lower group to n = 60 The latter group ofcurves shows the benefit of diversity Within these groups thereare differences between πmdRn(d) (wrong combination) andπmdLn(d) (wrong fading statistics) but they turn out not tobe fundamental

Fig 2 Probability of missed detection as a function of user distance usingBluetooth Radio Signal Strength Indication (RSSI)

TABLE VSELECTED VALUES OF πmdavn UNDER THE ASSUMPTION OF A

LOGNORMAL FADING DISTRIBUTION THESE VALUES DEFINE THENUMBER OF MEASUREMENTS NEEDED TO ACHIEVE THE DESIRED

PROBABILITY OF FURTHER SPREADING THE DISEASE

n πmdavn n πmdavn

1 012 60 00146 0054 120 000715 0034 240 000230 0023 480 00003

In Section III-A the probability of missed detection wasdetermined as a function of distance Since the probability ofdetection is additive in the sense that

πd =

intdS(r) ρ(r)πd(r)

=

int dc

0

2πrdr ρ(r)πd(r) (20)

In this equation πd(r) = 1 minus πmd(r) is the condition prob-ability of detection given that fellow B is at distance r anddS(r) ρ(r) is the probability density for fellow B to be atthat distance Equation 20 thus is the marginalization of πd(r)with respect to r Note that the limitation of the integration isa consequence of πd(r) = 0 whenever r gt dc This allowsto define the average probability of missed detection over thedistribution of users

πmdavn =

int dc0

2πrdr ρ(r)πmdn(r)int dc0

2πrdr ρ(r) (21)

8

The probability distribution of users in Equation (20) and (13)is given by

ρ(r) =dn(r)

dS(r)=dbπr2c2πrdr

In this expression n(r) = bπr2c is the number of peopleat a distance not greater than r in the case of a density ofone person per square meter This corresponds to the densestpacking of people occupying a surface of 1 meter People arecontinuously spread in a symmetric manner around fellow Awhich is a simple way of achieving a densest packing Theldquofunctionrdquo dn(r)dr is mostly zero It jumps at the valuesrm =

radicmπ with

n(rm + ε)minus n(rm minus ε) =

int rm+ε

rmminusε

dn(r)

drdr = 1

which is a distribution in the sense of Schwartz [13] Withthese preparations the integrals become

πmdavn =1

mc

mcsumm=1

πmdn

(radicm

π

) (22)

with mc being the largest integer with such that rmc le dcNote that the density of points rm increases with increasingm which means that the main contribution comes from theborder of the contact zone Using the experimental resultsfrom [8] this integral is evaluated to πmdav1 = 015 forn = 1 for the χ2-distribution and to πmdav1 = 012 for thelognormal distribution which are both not very compatiblewith the need of a small x0πmd Remember that the lattervalue is the reduction factor in the probability of furtherspreading of the disease achieved by contact tracing TableV lists values of πmdavn for different n which can beused to determine the reduction factor Even in the casen = 120 the factor x0πmd = 021 in Model A and it wouldrequire 4 measurement per second It is only with n = 480that factor x0πmd falls below 1 which would require 16measurements per second This would seriously impact thestandby time of the smart phone Assuming Model C and adecision based on 3 minutes intervals ie x0 = 5 means thatwe could achieve a reduction by a factor 007 provided thatn = 60 measurements are performed and aggregated in each 3minutes interval ie that one measurement is performed every3 seconds In the case of a decision every 5 minutes whichassumes a lower dynamics in the relative movement of peoplethe reduction factor is 004 with the same 60 measurementsbut now spread over a 5 minutes interval which correspondsto one measurement every 5 seconds So lower requirementsin the dynamic allow both to improve the suppression of thespreading of the virus and to reduce the measurement rate

Tolerable alarm rates were derived for the train scenarioThis led to the values in Table (IV) The evaluation of πfan(d)is straight forward For d = dc it gives πfan(dc) = 12 as wasalready discussed previously Assuming that people occupy acircular surface of 1 square meter gives them a radius δ =1radicπ Thus the minimum distance to people fully outside of

the critical zone is dc + δ Evaluating Equation (19) yields

pfa1(dc + δ) = 0137 and pfa3(dc + δ) = 0029

respectively This means that n = 1 is compatible with ajourney of 15 minutes before sending more than the two peopleto quarantine For n = 3 long journeys of up to 3 hoursbecome possible with the same consequences The probabilityof false alarm does thus not strongly limit the number n ofmeasurements aggregated to a decision and one might considerthe more demanding homogeneous distribution of users Thisrequires a study of the combination of false alarms Considertwo fellows B and Brsquo there is no alarm if neither B nor Brsquotriggers an alarm ie

1minus πfa = (1minus πfaB)(1minus πfaBprime)

Furthermore let users be at distances dc+δ(k+1) with k isin Z+

being a positive integer and assume that there are

ν(k) = π (dc + 2δ(k + 1))2 minus π (dc + 2δk)

2

users at that distance (they cover an angular shell of thickness2δ) This guarantees a densest packing In that case theprobability of false alarm ie an erroneous decision in favorof C1 becomes

pfan = 1minusinfinprodk=0

(1minus πfan(dc + k))ν(k)

(23)

In this more demanding scenario exemplary values are

pfa3 = 0413 and pfa9 = 0009

which means that n = 9 would be sufficient to reduce theprobability of false alarm to a very small level Table VI showsperformance figures for a number of possible choices for thenumber n of measurements aggregated to an estimate c1 aswell as for the number x0 of estimates c1 that lead to a decisionC1 The product of n and x0 leads to the measurement rateρ = x0n(15 middot 60) The performance figures are the reductionfactor x0πmdn of the spreading achieved by tracing as wellas the probability of unduly sending a person to quarantineThe figures in Table VI all relate to Model C Model A doesnot lead to interesting parameter choices and Model B is toostatic

TABLE VIKEY PERFORMANCE PARAMETERS x0πmdn MEASURES THE REDUCTIONIN SPREADING AND pfan THE PROBABILITY OF UNDUE QUARANTINING

THE PARAMETER ρ IS THE NUMBER OF MEASUREMENTS PER SECOND

x0πmdn pfan ρx0 3 5 - 3 5

n6 016 027 0064 150 130

15 012 017 00002 120 11260 004 007 00000 15 13

A choice with n = 15 and x0 = 3 for example requiresa measurement to be performed every 12 seconds suppressedthe risk of spreading by a factor 012 and does hardly sendanyone unduly to quarantine Performing a measurement everyfive seconds reduces the risk of spreading by a factor 004 Thisassumes that people let their phones hang from their neckand some standard form of environment In reality a numberof additional factors have to be taken into account such asa more complex propagation situation eg due to metallic

9

walls a higher dynamic of user movements eg due topeople entering and exiting commuter trains or unpredictableshadowing due to the userrsquos hands arms or body in the pathof radio signals Thus it is advisable to complement theBluetooth measurement by an alternative Audio ranging isthe option that shall be described in the next section The ideais to use it whenever the situation is not clear

IV AUDIO RANGING

Smartphones have a microphone and a speaker with rathergood transmit and receive conditions if the device is carriedon the chest or held in the hand This can be used for audioranging up to distances of a few meters Signals and theirtransmission can be configured by the API In experiments thatwe performed recently we focused on the use Android phonesThe response of the microphones built into three differentphones is shown in Figure 3 The references were a NT1-A microphone from Rode and an Adagio Infinite Speaker ofA3 on the source side Figure 3 shows the response of threesmartphones from two different brands The curves are verysimilar suggesting that the same microphones are integratedin those phones All microphones show a good sensitivity overall frequencies

Fig 3 Frequency response of microphones from three different smartphones

A similar experiment was performed for the speakers witha rather different result In that case only two smartphoneswere analyzed The response on the better device is reducedby roughly 10 dB above 16 kHz as compared to the referenceThe response of the other one is degraded by another 3 dBand the degradation starts 2kHz earlier Covering the speakerby one layer of tissue of a sweater degrades the performanceby another 4 dB If both parties cover their smartphonesthe associated attenuation adds up Thus the use of audioranging requires carrying the devices in an exposed mannereg hanging from onersquos neck see Figure (1) Transmissionat lower less attenuated frequencies is not considered as atrue option since it would be too disturbing The norm ISO2262003 compiles equivalent hearing sensitivity (isophones)which allows to compare the disturbance caused by acousticalsignals on different frequencies

Fig 4 Frequency response of two speakers as well as influence of coveringthe speaker of Smartphone 2 with one layer of a sweater

On the basis of such considerations we propose modulatinga carrier at 18 kHz with a modulation rate of 1 kbaud Thiskeeps the signal in a spectral range that is not too disturbing tomost people A spread spectrum modulation provides a goodrange resolution and allows to operate at a low signal-to-noiseratio at the same time Different options exist and are discussedin [9] Since the velocity of sound in air is cs = 343 ms understandard conditions a chip duration of 1 ms corresponds to alength of 34 cm At a typical signal-to-noise ratio this leadsto a distance resolution of 1 to 3 cm Let us be conservativeand assume a resolution of 5 cm A multipath delay of twometers leads to an offset by 6 chips and is well suppressedby the autocorrelation of the spreading code The length ofthe spreading code is assumed to be around 350 chips Analternative using chirps is also considered The performanceof audio ranging is further developed in Section IV

Audio ranging can be performed in a peer-to-peer or ina networked manner Consider the peer-to-peer situation firstSmartphones do not provide accurate timing control Howeverthe microphone input of a smartphone may be sampled at afixed rate Furthermore smartphones can transmit and receiveat the same time and this is furthermore supported by theAPIs of Android and iOS Let the smartphones thus agreeto start audio ranging via Bluetooth In a first step theyopen their microphone channels and then proceed according toFigure 5 at time tTXA A transmits the ranging signal usingits speaker This transmission is delayed with respect to theAPI by τTXA In parallel to its transmission Arsquos microphonecapture the transmitted signal This signal is delayed by thesum of the local propagation delay τlA and by the internalreceive delay τRXA The delay τlA is determined by thedevice geometry and can be stored in memory A standardvalue of 14 cm should be appropriate for most devices on themarket The time of reception thus is

tprimeRXA = tTXA + τTXA + τlA + τRXA

and is used for calibration purposes The same definition ofdelays applies at B Thus the signal transmitted by A at time

10

tTXA is received at B at the time tRXB

tRXB = tTXA + τTXA + τ + τRXB

with τ being the propagation time from A to B After receptionof the signal from A by B B sends a corresponding signal toA The equations are obtained by changing the roles of A andB

tprimeRXB = tTXB + τTXB + τlB + τRXB

andtRXA = tTXB + τTXB + τ + τRXA

At the end of the reception A sends

∆tA = tRXA minus tprimeRXA + τlA (24)

to B and B sends ∆tB = tRXB minus tprimeRXB + τlB using BLEThus both can compute the propagation time

τ =∆tA + ∆tB

2

and thus the distance d = τcs The property of audio signalswhich is crucial for this self-calibration is the possibility toobserve the own transmitted signal

Fig 5 Signal paths in two-way acoustical ranging with calibration of transmitand receive delays

A Ranging Protocol

The above peer-to-peer protocol can be extended to anetworked protocol In this case the users agree on an orderingof transmissions via Bluetooth All smartphones A1 Akactivate their microphones and one after the other transmittheir audio ranging signals For simplicity the scheduling isprearranged which also works if some of the smartphonecannot acquire all signals In this case all delays are summedup 350 ms for the ranging signal 10 ms (corresponding to4 meters) for propagation and 40 ms for the internal delaysbetween the activation of the transmission command and thestart of transmission (the latter needs to be confirmed by moredata) This allows for a scheduling of a transmission every 400ms After the completion of the cycle and the evaluation of the

reception time tRXAi by terminal A1 this terminal transmitsthe time difference using Bluetooth

∆tA1Ai = tRXAi minus tprimeRXA1+ τlA1

for 2 le i le k

If all terminals see each other they transmit k(k minus 1) suchvalues in total The annoying transmissions of audio signals re-main limited to k however The overall time interval spannedby all transmissions in the networked protocol may be longenough for users to move slightly This is not critical howeverThe snap-shot measurements are simply converted to averagevalues The only instances which require some care are thosein which the audio signals are used to calibrate Bluetoothmeasurements Finally it should be emphasized that audiobeacon transmissions should not be activated if the device isheld to the ear Even if the signals are hardly heard this seemsa reasonable precaution

B Theoretical Performance of Acoustic Range Estimation

The received audio signal is filtered to remove out-of-band interference and noise to the best possible extent Thefiltered signal is used to determine the in-band interference andnoise level N0 and is furthermore correlated using the filteredranging signal For simplicity the further exposition focuses onspread spectrum signals In a first step the I and Q componentsof the correlation C(∆τ) are computed at intervals of Tc2with Tc = 1 ms denoting the chip duration The result issearched for the delay leading to the maximum norm |C(∆τ)|Although the implementations by widely used phones seemnot to require that frequency offsets may be searched as wellThis allows to acquire the signal which may be present or notThus it is sufficient to search for the delay (and frequencyoffsets) leading to the maximum norm from early to late Thelatter ordering is to avoid locking on an echo If the signalto noise ratio is above the expected threshold the signal isassumed present In this case a successive refinement of theresult is performed in a DLL type of processing The powerdiscriminator

DP (∆τ) = |R(∆τ + δ)|2 minus |R(∆τ minus δ)|2

is used to iteratively increasereduce the delay ∆τ dependingon the value of DP (∆τ) ≷ 0 In this equation δ is half thecorrelator spacing and is expressed as a fraction ∆ of thechip duration δ = ∆Tc We will restrict ourselves to ∆ = 1A further optimization is possible see Betz and Kolodziejski[14] [15] The uncertainty of the delay estimate ∆τ due tonoise is given by (see Dierendonck Fenton and Ford [16])

σ2∆τ T 2

c

∆

4EiN0

(1 +

3

(2minus∆)EiN0

) (25)

In this expression Ei is the signal energy accumulated duringthe correlation and N0 is the spectral noise density of theaudio noise and interference The latter quantity is estimatedusing the norm of the filtered I and Q samples of the incomingsignal

N0 =1

BSNTc

Nsumn=1

(s2I + s2

Q

)

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

8

The probability distribution of users in Equation (20) and (13)is given by

ρ(r) =dn(r)

dS(r)=dbπr2c2πrdr

In this expression n(r) = bπr2c is the number of peopleat a distance not greater than r in the case of a density ofone person per square meter This corresponds to the densestpacking of people occupying a surface of 1 meter People arecontinuously spread in a symmetric manner around fellow Awhich is a simple way of achieving a densest packing Theldquofunctionrdquo dn(r)dr is mostly zero It jumps at the valuesrm =

radicmπ with

n(rm + ε)minus n(rm minus ε) =

int rm+ε

rmminusε

dn(r)

drdr = 1

which is a distribution in the sense of Schwartz [13] Withthese preparations the integrals become

πmdavn =1

mc

mcsumm=1

πmdn

(radicm

π

) (22)

with mc being the largest integer with such that rmc le dcNote that the density of points rm increases with increasingm which means that the main contribution comes from theborder of the contact zone Using the experimental resultsfrom [8] this integral is evaluated to πmdav1 = 015 forn = 1 for the χ2-distribution and to πmdav1 = 012 for thelognormal distribution which are both not very compatiblewith the need of a small x0πmd Remember that the lattervalue is the reduction factor in the probability of furtherspreading of the disease achieved by contact tracing TableV lists values of πmdavn for different n which can beused to determine the reduction factor Even in the casen = 120 the factor x0πmd = 021 in Model A and it wouldrequire 4 measurement per second It is only with n = 480that factor x0πmd falls below 1 which would require 16measurements per second This would seriously impact thestandby time of the smart phone Assuming Model C and adecision based on 3 minutes intervals ie x0 = 5 means thatwe could achieve a reduction by a factor 007 provided thatn = 60 measurements are performed and aggregated in each 3minutes interval ie that one measurement is performed every3 seconds In the case of a decision every 5 minutes whichassumes a lower dynamics in the relative movement of peoplethe reduction factor is 004 with the same 60 measurementsbut now spread over a 5 minutes interval which correspondsto one measurement every 5 seconds So lower requirementsin the dynamic allow both to improve the suppression of thespreading of the virus and to reduce the measurement rate

Tolerable alarm rates were derived for the train scenarioThis led to the values in Table (IV) The evaluation of πfan(d)is straight forward For d = dc it gives πfan(dc) = 12 as wasalready discussed previously Assuming that people occupy acircular surface of 1 square meter gives them a radius δ =1radicπ Thus the minimum distance to people fully outside of

the critical zone is dc + δ Evaluating Equation (19) yields

pfa1(dc + δ) = 0137 and pfa3(dc + δ) = 0029

respectively This means that n = 1 is compatible with ajourney of 15 minutes before sending more than the two peopleto quarantine For n = 3 long journeys of up to 3 hoursbecome possible with the same consequences The probabilityof false alarm does thus not strongly limit the number n ofmeasurements aggregated to a decision and one might considerthe more demanding homogeneous distribution of users Thisrequires a study of the combination of false alarms Considertwo fellows B and Brsquo there is no alarm if neither B nor Brsquotriggers an alarm ie

1minus πfa = (1minus πfaB)(1minus πfaBprime)

Furthermore let users be at distances dc+δ(k+1) with k isin Z+

being a positive integer and assume that there are

ν(k) = π (dc + 2δ(k + 1))2 minus π (dc + 2δk)

2

users at that distance (they cover an angular shell of thickness2δ) This guarantees a densest packing In that case theprobability of false alarm ie an erroneous decision in favorof C1 becomes

pfan = 1minusinfinprodk=0

(1minus πfan(dc + k))ν(k)

(23)

In this more demanding scenario exemplary values are

pfa3 = 0413 and pfa9 = 0009

which means that n = 9 would be sufficient to reduce theprobability of false alarm to a very small level Table VI showsperformance figures for a number of possible choices for thenumber n of measurements aggregated to an estimate c1 aswell as for the number x0 of estimates c1 that lead to a decisionC1 The product of n and x0 leads to the measurement rateρ = x0n(15 middot 60) The performance figures are the reductionfactor x0πmdn of the spreading achieved by tracing as wellas the probability of unduly sending a person to quarantineThe figures in Table VI all relate to Model C Model A doesnot lead to interesting parameter choices and Model B is toostatic

TABLE VIKEY PERFORMANCE PARAMETERS x0πmdn MEASURES THE REDUCTIONIN SPREADING AND pfan THE PROBABILITY OF UNDUE QUARANTINING

THE PARAMETER ρ IS THE NUMBER OF MEASUREMENTS PER SECOND

x0πmdn pfan ρx0 3 5 - 3 5

n6 016 027 0064 150 130

15 012 017 00002 120 11260 004 007 00000 15 13

A choice with n = 15 and x0 = 3 for example requiresa measurement to be performed every 12 seconds suppressedthe risk of spreading by a factor 012 and does hardly sendanyone unduly to quarantine Performing a measurement everyfive seconds reduces the risk of spreading by a factor 004 Thisassumes that people let their phones hang from their neckand some standard form of environment In reality a numberof additional factors have to be taken into account such asa more complex propagation situation eg due to metallic

9

walls a higher dynamic of user movements eg due topeople entering and exiting commuter trains or unpredictableshadowing due to the userrsquos hands arms or body in the pathof radio signals Thus it is advisable to complement theBluetooth measurement by an alternative Audio ranging isthe option that shall be described in the next section The ideais to use it whenever the situation is not clear

IV AUDIO RANGING

Smartphones have a microphone and a speaker with rathergood transmit and receive conditions if the device is carriedon the chest or held in the hand This can be used for audioranging up to distances of a few meters Signals and theirtransmission can be configured by the API In experiments thatwe performed recently we focused on the use Android phonesThe response of the microphones built into three differentphones is shown in Figure 3 The references were a NT1-A microphone from Rode and an Adagio Infinite Speaker ofA3 on the source side Figure 3 shows the response of threesmartphones from two different brands The curves are verysimilar suggesting that the same microphones are integratedin those phones All microphones show a good sensitivity overall frequencies

Fig 3 Frequency response of microphones from three different smartphones

A similar experiment was performed for the speakers witha rather different result In that case only two smartphoneswere analyzed The response on the better device is reducedby roughly 10 dB above 16 kHz as compared to the referenceThe response of the other one is degraded by another 3 dBand the degradation starts 2kHz earlier Covering the speakerby one layer of tissue of a sweater degrades the performanceby another 4 dB If both parties cover their smartphonesthe associated attenuation adds up Thus the use of audioranging requires carrying the devices in an exposed mannereg hanging from onersquos neck see Figure (1) Transmissionat lower less attenuated frequencies is not considered as atrue option since it would be too disturbing The norm ISO2262003 compiles equivalent hearing sensitivity (isophones)which allows to compare the disturbance caused by acousticalsignals on different frequencies

Fig 4 Frequency response of two speakers as well as influence of coveringthe speaker of Smartphone 2 with one layer of a sweater

On the basis of such considerations we propose modulatinga carrier at 18 kHz with a modulation rate of 1 kbaud Thiskeeps the signal in a spectral range that is not too disturbing tomost people A spread spectrum modulation provides a goodrange resolution and allows to operate at a low signal-to-noiseratio at the same time Different options exist and are discussedin [9] Since the velocity of sound in air is cs = 343 ms understandard conditions a chip duration of 1 ms corresponds to alength of 34 cm At a typical signal-to-noise ratio this leadsto a distance resolution of 1 to 3 cm Let us be conservativeand assume a resolution of 5 cm A multipath delay of twometers leads to an offset by 6 chips and is well suppressedby the autocorrelation of the spreading code The length ofthe spreading code is assumed to be around 350 chips Analternative using chirps is also considered The performanceof audio ranging is further developed in Section IV

Audio ranging can be performed in a peer-to-peer or ina networked manner Consider the peer-to-peer situation firstSmartphones do not provide accurate timing control Howeverthe microphone input of a smartphone may be sampled at afixed rate Furthermore smartphones can transmit and receiveat the same time and this is furthermore supported by theAPIs of Android and iOS Let the smartphones thus agreeto start audio ranging via Bluetooth In a first step theyopen their microphone channels and then proceed according toFigure 5 at time tTXA A transmits the ranging signal usingits speaker This transmission is delayed with respect to theAPI by τTXA In parallel to its transmission Arsquos microphonecapture the transmitted signal This signal is delayed by thesum of the local propagation delay τlA and by the internalreceive delay τRXA The delay τlA is determined by thedevice geometry and can be stored in memory A standardvalue of 14 cm should be appropriate for most devices on themarket The time of reception thus is

tprimeRXA = tTXA + τTXA + τlA + τRXA

and is used for calibration purposes The same definition ofdelays applies at B Thus the signal transmitted by A at time

10

tTXA is received at B at the time tRXB

tRXB = tTXA + τTXA + τ + τRXB

with τ being the propagation time from A to B After receptionof the signal from A by B B sends a corresponding signal toA The equations are obtained by changing the roles of A andB

tprimeRXB = tTXB + τTXB + τlB + τRXB

andtRXA = tTXB + τTXB + τ + τRXA

At the end of the reception A sends

∆tA = tRXA minus tprimeRXA + τlA (24)

to B and B sends ∆tB = tRXB minus tprimeRXB + τlB using BLEThus both can compute the propagation time

τ =∆tA + ∆tB

2

and thus the distance d = τcs The property of audio signalswhich is crucial for this self-calibration is the possibility toobserve the own transmitted signal

Fig 5 Signal paths in two-way acoustical ranging with calibration of transmitand receive delays

A Ranging Protocol

The above peer-to-peer protocol can be extended to anetworked protocol In this case the users agree on an orderingof transmissions via Bluetooth All smartphones A1 Akactivate their microphones and one after the other transmittheir audio ranging signals For simplicity the scheduling isprearranged which also works if some of the smartphonecannot acquire all signals In this case all delays are summedup 350 ms for the ranging signal 10 ms (corresponding to4 meters) for propagation and 40 ms for the internal delaysbetween the activation of the transmission command and thestart of transmission (the latter needs to be confirmed by moredata) This allows for a scheduling of a transmission every 400ms After the completion of the cycle and the evaluation of the

reception time tRXAi by terminal A1 this terminal transmitsthe time difference using Bluetooth

∆tA1Ai = tRXAi minus tprimeRXA1+ τlA1

for 2 le i le k

If all terminals see each other they transmit k(k minus 1) suchvalues in total The annoying transmissions of audio signals re-main limited to k however The overall time interval spannedby all transmissions in the networked protocol may be longenough for users to move slightly This is not critical howeverThe snap-shot measurements are simply converted to averagevalues The only instances which require some care are thosein which the audio signals are used to calibrate Bluetoothmeasurements Finally it should be emphasized that audiobeacon transmissions should not be activated if the device isheld to the ear Even if the signals are hardly heard this seemsa reasonable precaution

B Theoretical Performance of Acoustic Range Estimation

The received audio signal is filtered to remove out-of-band interference and noise to the best possible extent Thefiltered signal is used to determine the in-band interference andnoise level N0 and is furthermore correlated using the filteredranging signal For simplicity the further exposition focuses onspread spectrum signals In a first step the I and Q componentsof the correlation C(∆τ) are computed at intervals of Tc2with Tc = 1 ms denoting the chip duration The result issearched for the delay leading to the maximum norm |C(∆τ)|Although the implementations by widely used phones seemnot to require that frequency offsets may be searched as wellThis allows to acquire the signal which may be present or notThus it is sufficient to search for the delay (and frequencyoffsets) leading to the maximum norm from early to late Thelatter ordering is to avoid locking on an echo If the signalto noise ratio is above the expected threshold the signal isassumed present In this case a successive refinement of theresult is performed in a DLL type of processing The powerdiscriminator

DP (∆τ) = |R(∆τ + δ)|2 minus |R(∆τ minus δ)|2

is used to iteratively increasereduce the delay ∆τ dependingon the value of DP (∆τ) ≷ 0 In this equation δ is half thecorrelator spacing and is expressed as a fraction ∆ of thechip duration δ = ∆Tc We will restrict ourselves to ∆ = 1A further optimization is possible see Betz and Kolodziejski[14] [15] The uncertainty of the delay estimate ∆τ due tonoise is given by (see Dierendonck Fenton and Ford [16])

σ2∆τ T 2

c

∆

4EiN0

(1 +

3

(2minus∆)EiN0

) (25)

In this expression Ei is the signal energy accumulated duringthe correlation and N0 is the spectral noise density of theaudio noise and interference The latter quantity is estimatedusing the norm of the filtered I and Q samples of the incomingsignal

N0 =1

BSNTc

Nsumn=1

(s2I + s2

Q

)

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

9

walls a higher dynamic of user movements eg due topeople entering and exiting commuter trains or unpredictableshadowing due to the userrsquos hands arms or body in the pathof radio signals Thus it is advisable to complement theBluetooth measurement by an alternative Audio ranging isthe option that shall be described in the next section The ideais to use it whenever the situation is not clear

IV AUDIO RANGING

Smartphones have a microphone and a speaker with rathergood transmit and receive conditions if the device is carriedon the chest or held in the hand This can be used for audioranging up to distances of a few meters Signals and theirtransmission can be configured by the API In experiments thatwe performed recently we focused on the use Android phonesThe response of the microphones built into three differentphones is shown in Figure 3 The references were a NT1-A microphone from Rode and an Adagio Infinite Speaker ofA3 on the source side Figure 3 shows the response of threesmartphones from two different brands The curves are verysimilar suggesting that the same microphones are integratedin those phones All microphones show a good sensitivity overall frequencies

Fig 3 Frequency response of microphones from three different smartphones

A similar experiment was performed for the speakers witha rather different result In that case only two smartphoneswere analyzed The response on the better device is reducedby roughly 10 dB above 16 kHz as compared to the referenceThe response of the other one is degraded by another 3 dBand the degradation starts 2kHz earlier Covering the speakerby one layer of tissue of a sweater degrades the performanceby another 4 dB If both parties cover their smartphonesthe associated attenuation adds up Thus the use of audioranging requires carrying the devices in an exposed mannereg hanging from onersquos neck see Figure (1) Transmissionat lower less attenuated frequencies is not considered as atrue option since it would be too disturbing The norm ISO2262003 compiles equivalent hearing sensitivity (isophones)which allows to compare the disturbance caused by acousticalsignals on different frequencies

Fig 4 Frequency response of two speakers as well as influence of coveringthe speaker of Smartphone 2 with one layer of a sweater

On the basis of such considerations we propose modulatinga carrier at 18 kHz with a modulation rate of 1 kbaud Thiskeeps the signal in a spectral range that is not too disturbing tomost people A spread spectrum modulation provides a goodrange resolution and allows to operate at a low signal-to-noiseratio at the same time Different options exist and are discussedin [9] Since the velocity of sound in air is cs = 343 ms understandard conditions a chip duration of 1 ms corresponds to alength of 34 cm At a typical signal-to-noise ratio this leadsto a distance resolution of 1 to 3 cm Let us be conservativeand assume a resolution of 5 cm A multipath delay of twometers leads to an offset by 6 chips and is well suppressedby the autocorrelation of the spreading code The length ofthe spreading code is assumed to be around 350 chips Analternative using chirps is also considered The performanceof audio ranging is further developed in Section IV

Audio ranging can be performed in a peer-to-peer or ina networked manner Consider the peer-to-peer situation firstSmartphones do not provide accurate timing control Howeverthe microphone input of a smartphone may be sampled at afixed rate Furthermore smartphones can transmit and receiveat the same time and this is furthermore supported by theAPIs of Android and iOS Let the smartphones thus agreeto start audio ranging via Bluetooth In a first step theyopen their microphone channels and then proceed according toFigure 5 at time tTXA A transmits the ranging signal usingits speaker This transmission is delayed with respect to theAPI by τTXA In parallel to its transmission Arsquos microphonecapture the transmitted signal This signal is delayed by thesum of the local propagation delay τlA and by the internalreceive delay τRXA The delay τlA is determined by thedevice geometry and can be stored in memory A standardvalue of 14 cm should be appropriate for most devices on themarket The time of reception thus is

tprimeRXA = tTXA + τTXA + τlA + τRXA

and is used for calibration purposes The same definition ofdelays applies at B Thus the signal transmitted by A at time

10

tTXA is received at B at the time tRXB

tRXB = tTXA + τTXA + τ + τRXB

with τ being the propagation time from A to B After receptionof the signal from A by B B sends a corresponding signal toA The equations are obtained by changing the roles of A andB

tprimeRXB = tTXB + τTXB + τlB + τRXB

andtRXA = tTXB + τTXB + τ + τRXA

At the end of the reception A sends

∆tA = tRXA minus tprimeRXA + τlA (24)

to B and B sends ∆tB = tRXB minus tprimeRXB + τlB using BLEThus both can compute the propagation time

τ =∆tA + ∆tB

2

and thus the distance d = τcs The property of audio signalswhich is crucial for this self-calibration is the possibility toobserve the own transmitted signal

Fig 5 Signal paths in two-way acoustical ranging with calibration of transmitand receive delays

A Ranging Protocol

The above peer-to-peer protocol can be extended to anetworked protocol In this case the users agree on an orderingof transmissions via Bluetooth All smartphones A1 Akactivate their microphones and one after the other transmittheir audio ranging signals For simplicity the scheduling isprearranged which also works if some of the smartphonecannot acquire all signals In this case all delays are summedup 350 ms for the ranging signal 10 ms (corresponding to4 meters) for propagation and 40 ms for the internal delaysbetween the activation of the transmission command and thestart of transmission (the latter needs to be confirmed by moredata) This allows for a scheduling of a transmission every 400ms After the completion of the cycle and the evaluation of the

reception time tRXAi by terminal A1 this terminal transmitsthe time difference using Bluetooth

∆tA1Ai = tRXAi minus tprimeRXA1+ τlA1

for 2 le i le k

If all terminals see each other they transmit k(k minus 1) suchvalues in total The annoying transmissions of audio signals re-main limited to k however The overall time interval spannedby all transmissions in the networked protocol may be longenough for users to move slightly This is not critical howeverThe snap-shot measurements are simply converted to averagevalues The only instances which require some care are thosein which the audio signals are used to calibrate Bluetoothmeasurements Finally it should be emphasized that audiobeacon transmissions should not be activated if the device isheld to the ear Even if the signals are hardly heard this seemsa reasonable precaution

B Theoretical Performance of Acoustic Range Estimation

The received audio signal is filtered to remove out-of-band interference and noise to the best possible extent Thefiltered signal is used to determine the in-band interference andnoise level N0 and is furthermore correlated using the filteredranging signal For simplicity the further exposition focuses onspread spectrum signals In a first step the I and Q componentsof the correlation C(∆τ) are computed at intervals of Tc2with Tc = 1 ms denoting the chip duration The result issearched for the delay leading to the maximum norm |C(∆τ)|Although the implementations by widely used phones seemnot to require that frequency offsets may be searched as wellThis allows to acquire the signal which may be present or notThus it is sufficient to search for the delay (and frequencyoffsets) leading to the maximum norm from early to late Thelatter ordering is to avoid locking on an echo If the signalto noise ratio is above the expected threshold the signal isassumed present In this case a successive refinement of theresult is performed in a DLL type of processing The powerdiscriminator

DP (∆τ) = |R(∆τ + δ)|2 minus |R(∆τ minus δ)|2

is used to iteratively increasereduce the delay ∆τ dependingon the value of DP (∆τ) ≷ 0 In this equation δ is half thecorrelator spacing and is expressed as a fraction ∆ of thechip duration δ = ∆Tc We will restrict ourselves to ∆ = 1A further optimization is possible see Betz and Kolodziejski[14] [15] The uncertainty of the delay estimate ∆τ due tonoise is given by (see Dierendonck Fenton and Ford [16])

σ2∆τ T 2

c

∆

4EiN0

(1 +

3

(2minus∆)EiN0

) (25)

In this expression Ei is the signal energy accumulated duringthe correlation and N0 is the spectral noise density of theaudio noise and interference The latter quantity is estimatedusing the norm of the filtered I and Q samples of the incomingsignal

N0 =1

BSNTc

Nsumn=1

(s2I + s2

Q

)

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

10

tTXA is received at B at the time tRXB

tRXB = tTXA + τTXA + τ + τRXB

with τ being the propagation time from A to B After receptionof the signal from A by B B sends a corresponding signal toA The equations are obtained by changing the roles of A andB

tprimeRXB = tTXB + τTXB + τlB + τRXB

andtRXA = tTXB + τTXB + τ + τRXA

At the end of the reception A sends

∆tA = tRXA minus tprimeRXA + τlA (24)

to B and B sends ∆tB = tRXB minus tprimeRXB + τlB using BLEThus both can compute the propagation time

τ =∆tA + ∆tB

2

and thus the distance d = τcs The property of audio signalswhich is crucial for this self-calibration is the possibility toobserve the own transmitted signal

Fig 5 Signal paths in two-way acoustical ranging with calibration of transmitand receive delays

A Ranging Protocol

The above peer-to-peer protocol can be extended to anetworked protocol In this case the users agree on an orderingof transmissions via Bluetooth All smartphones A1 Akactivate their microphones and one after the other transmittheir audio ranging signals For simplicity the scheduling isprearranged which also works if some of the smartphonecannot acquire all signals In this case all delays are summedup 350 ms for the ranging signal 10 ms (corresponding to4 meters) for propagation and 40 ms for the internal delaysbetween the activation of the transmission command and thestart of transmission (the latter needs to be confirmed by moredata) This allows for a scheduling of a transmission every 400ms After the completion of the cycle and the evaluation of the

reception time tRXAi by terminal A1 this terminal transmitsthe time difference using Bluetooth

∆tA1Ai = tRXAi minus tprimeRXA1+ τlA1

for 2 le i le k

If all terminals see each other they transmit k(k minus 1) suchvalues in total The annoying transmissions of audio signals re-main limited to k however The overall time interval spannedby all transmissions in the networked protocol may be longenough for users to move slightly This is not critical howeverThe snap-shot measurements are simply converted to averagevalues The only instances which require some care are thosein which the audio signals are used to calibrate Bluetoothmeasurements Finally it should be emphasized that audiobeacon transmissions should not be activated if the device isheld to the ear Even if the signals are hardly heard this seemsa reasonable precaution

B Theoretical Performance of Acoustic Range Estimation

The received audio signal is filtered to remove out-of-band interference and noise to the best possible extent Thefiltered signal is used to determine the in-band interference andnoise level N0 and is furthermore correlated using the filteredranging signal For simplicity the further exposition focuses onspread spectrum signals In a first step the I and Q componentsof the correlation C(∆τ) are computed at intervals of Tc2with Tc = 1 ms denoting the chip duration The result issearched for the delay leading to the maximum norm |C(∆τ)|Although the implementations by widely used phones seemnot to require that frequency offsets may be searched as wellThis allows to acquire the signal which may be present or notThus it is sufficient to search for the delay (and frequencyoffsets) leading to the maximum norm from early to late Thelatter ordering is to avoid locking on an echo If the signalto noise ratio is above the expected threshold the signal isassumed present In this case a successive refinement of theresult is performed in a DLL type of processing The powerdiscriminator

DP (∆τ) = |R(∆τ + δ)|2 minus |R(∆τ minus δ)|2

is used to iteratively increasereduce the delay ∆τ dependingon the value of DP (∆τ) ≷ 0 In this equation δ is half thecorrelator spacing and is expressed as a fraction ∆ of thechip duration δ = ∆Tc We will restrict ourselves to ∆ = 1A further optimization is possible see Betz and Kolodziejski[14] [15] The uncertainty of the delay estimate ∆τ due tonoise is given by (see Dierendonck Fenton and Ford [16])

σ2∆τ T 2

c

∆

4EiN0

(1 +

3

(2minus∆)EiN0

) (25)

In this expression Ei is the signal energy accumulated duringthe correlation and N0 is the spectral noise density of theaudio noise and interference The latter quantity is estimatedusing the norm of the filtered I and Q samples of the incomingsignal

N0 =1

BSNTc

Nsumn=1

(s2I + s2

Q

)

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

11

with N denoting the number of samples and with BS de-noting the bandwidth of the passband filter This estimate isperformed ahead of time and is used for setting the volumeof the transmission such that EiN0= 6 dB at 4 meters Atthis level the signal can be acquired and Equation (25) impliesthat σ∆τ Tc4which corresponds to 9 cm At 2 meters thisis half that value ie 45 cm The calibration of the transmitpower may be performed by listening to the own beacon Thisallows detecting whether the user is inadvertently covering themicrophone or the speaker which should trigger a request tothe user to remove the blockage The distribution of audioranging measurements is Gaussian with a standard deviationgiven by Equation (25) This allows computing πmd ie theprobability of deciding against c1 as a function of the distanced le dc

πmd(d) =

int infindc

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= Q

(dc minus dσ∆τ

) (26)

and πfa ie the probability of wrongly deciding in favor ofc1 for distances d gt dc

πfa(d) =

int dc

0

dx1radic

2πσ∆τ

eminus(xminusd)2(2σ∆τ )

= 1minusQ(dc minus dσ∆τ

) (27)

Note that the symmetry of lognormal fading between πmd(d)and πfa(d2

cd) is lost The plot for audio ranging correspond-ing to σ∆τ = 5 cm is shown in Figure 6

Fig 6 Probabilities of missed detection as a function of d and of false alarmas a function of d2cd for audio ranging

Again one might evaluate the average rate of misseddetection and of false alarm as in Equation (22) In thiscase the averaged probability of missed detection becomesπmdav = 0016 In the present case the number of mea-surements is primarily limited by the acoustical disturbancesassociated with the transmission of the beacon The numberof measurements n used for taking a decision is always 1Furthermore the number of measurements x0 per 15 minutes

must also be small for the same reason With x0 = 3 thereduction of the spreading rate of disease is x0πmdav lt 005which is a low figure The probability of false alarm describedby Equation (27) decays so quickly that it is insignificantat d = dc + δ ie πfa(dc + δ) 0 The same appliesfor the integration over a two-dimensional plane according toEquation (23)

The present discussion was about the contributions of un-certainty due to signaling Additionally the relative geometryof the microphones and speakers may add some bias whichmay lead to a shift of the border to a contact zone bya few centimeters This is rather uncritical however Theimportant conclusion is that audio ranging provides sharpresults This form of ranging might thus be activated wheneverthe information gained by Bluetooth measurements may leadto a wrong conclusion

V ATTITUDE SENSING

This section is more a reference to options that may beconsidered The benefits will become visible by the qualitativediscussion of Section VI Earth gravity in the minus~ez directionie towards the center of the earth and the magnetic field inthe direction of ~emN ie towards magnetic North providetwo directions that enable attitude determination Both areseriously disturbed in ways that depend on the environmentA number of authors have investigated the quality of attitudesensing both using algorithms built into smartphones and usingown estimation algorithms Michel and co-authors summarizea number of findings [17] They report an accuracy of 6

with a sampling rate of 40Hz whenever the smartphone iskept in a relatively calm position (front pocket texting orphoning) These results apply to their own algorithms ldquoMich-elObsFrdquo and ldquoMichelEkfFrdquo They did not study the behaviorin a train which is a particularity difficult environment withmany sources of acceleration due to the track geometry dueto passing switches or simply due to irregularities in thetracks themselves Similarly the magnetic field in trains ismodulated by electrical motors permanent magnets and largecurrents On the other hand people sitting or standing nextto each others are likely to be affected in a similar mannerExploiting the latter property however requires the use ofcommon standardized algorithm and precise time stamping ofmeasurements

Carrying the smartphone by letting it hang down onersquosneck leads to two stable orientation one with the displayfacing the chest and one with the display facing aheadThe resolution of the associated ambiguity is rather straight-forward at least as long as people do not predominantlywalk backward Alternatively the cameras could be used fordetermining the orientation since the brightness of the picturesis very different Pitch angles are suppressed by gravity aslong as people do not bend backwards which is unnaturalRoll angles may occur if one strap is shorter than the otherone They are compensated by sensing earth gravity In ouropinion the context of COVID-tracing is quite favorable tothe use of relative attitude estimation which would provide aninteresting complement to Bluetooth sensing andor acousticranging This needs to be developed however

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

12

VI CLASSIFICATION

The definition of a Category 1 contact by the Robert KochInstitute [18] includes three elements

bull an accumulated duration of 15 minutes which can easilybe metered

bull a distance of less than 2 meters which is more difficultto establish

bull and the concept of being face-to-face discussed below

From the previous sections specially Section II and IIIwe learned that under idealized conditions Bluetooth RSSImeasurements provide an adequate estimation of the distancebetween two fellows or more exactly an estimate on whetherB is in the critical zone of A The probability of misseddetection was found to the be a critical performance measureAudio ranging was found to be an interesting complementto Bluetooth measurements in particular if the latter measure-ments are disturbed by shadowing or multipath They provide acomparatively sharp answer and may be used to calibrate pastand future Bluetooth RSSI measurement Audio measurementsmay be audible and thus annoying for younger people aswell as for dogs and other animals As a consequence it isbeneficial to keep them sparse In Section V we very shortlyaddressed the use of attitude sensing

In this section we shall superficially address the potentialof combining these measurement types For this discussionit is meaningful to differentiate different poses as shown inFigure 7 A selection of essential poses of two fellows inclose proximity is shown in a top view Fellow B is infectedand exhales air charged with microscopic droplets carrying thevirus Fellow A inhales the droplets Pose (a) in Figure 7 iswhat everyone would agree to call a face-to-face situation It isthe type of situation which occurs during a meeting lunch orin public transportation for people sitting or standing oppositeto each other It might also occur when desks are facing eachother and in some other special situations Pose (b) occursin public transportation in queues as well as in lecture hallsconcert halls cinemas or the like It also appears dangerousalthough Fellow B needs to be closer for that but this mightoften be the case However unless B stands and is much tallerthan A the air flow will only partially reach Arsquos nose andmouth A further specification by medical authorities wouldbe helpful in this case Pose (c) occurs in similar situationsas Pose (b) Pose (d) (e) and (f) occur during meetings bothwhile standing and sitting in public transportation and someother situations Pose (c) and (d) do not appear too criticalalthough B is likely to turn his head from time to time whichis not detected by the sensors considered Pose (d) (e) and(f) are difficult to differentiate even using perfect ranging andorientation

Assuming that there is no specific direction in the air-flowdue to wind or draft and that the different poses can bedifferentiated medical requirements would probably choose

bull Pose (a) (d) and (e) to be Category 1 ie criticalbull Pose (b) would be critical for a lower distance which

might depend on the height differencesbull Pose (c) and (f) would be essentially uncritical

The possibility to discriminate the cases depends on the type ofsensing as described so far and is discussed in the followingthree sections

Fig 7 Different potential poses of a COVID-19 carrier A and of a nearbyperson B The bubble in front of A shows the area into which A exhales aircarrying droplets with the virus

A Bluetooth-only Measurements

BLE RSSI measurements will return similar results forthe Poses (a) (c) (d) and (e) The distance d between thefellows might appear larger in Pose (f) than it actually isThis is uncritical however In Pose (b) the received powerwill be associated with a larger distance than the actual oneas well Depending on how Pose (b) is classified this leadsto a missed detection A similar situation may also occur inPose (e) whenever Fellow A obstructs the line of sight withhis left harm eg by holding himself on a bar in publictransportation All missed detection events are critical sincethey leave close encounters undetected Finally the poses(c) and (d) will typically generate false alarms which sendspeople to quarantine and testing This sort of differentiationhas not been considered so far at least to our knowledge

B Bluetooth Attitude Sensing

The addition of a attitude sensing allows to separate thecases of ldquoPose (b) with a small distancerdquo from ldquoPose (a) witha large distancerdquo Thus it might use a lower threshold in thecase of an aligned attitude and thus avoid the missed detectionevents in Pose (b) With a lower threshold however fellowsin Pose (c) will be identified as C1 up to a rather large relativedistance potentially generating many false alarms

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile

13

C Bluetooth Attitude Sensing and Audio RangingAn extensive use of audio ranging would eliminate false

alarms mostly It would implement the conditions of Category1 without the alleviation due to the the condition of beingface-to-face When combined with the other measurementsaudio measurements provide additional discrimination andallow reducing the rate of missed detection and false alarmsIn reality acoustical signals are subject to multipath whichmight be critical if the direct path is strongly attenuated Sincethe receiver searches from early to late it is unlikely to beinduced in error however as long as the direct path can stillbe detected

VII CONCLUSIONS

Difficulties in Bluetooth RSSI-based ranging are mentionedby a number of scientists orally The significant attenuationby the human body and other influencing factors such askeys coins metallic pens business card holders and the likemake the power levels very unpredictable We thus propose tostandardize the wearing of smartphones or alternative deviceson the chest when not held in the hand or used for makingphone calls This provides an environment that is much betterdefined for Bluetooth RSSI-based ranging audio ranging andattitude determination Currently we donrsquot see an alternativesetting to the present one that allows for an analysis ofthe tracing performance in terms of identifying Category 1contacts and avoiding unduly frequent alerts for contacts thatare not Category 1 The analysis shows that the accumulatedstatistics require low figures for the per event missed detectionrate This can be achieved with measurements every fewseconds aggregated into decisions every few minutes whichis adequate for stable distributions of people such as in ameeting at lunch in a train and the like The false alarmrate is a lesser problem as soon as a few measurements areaggregated The analysis presented in the paper is a prelimi-nary one Much more experimental data should be generatedto refine the findings In Germany the current probability ofencountering an infected person is rather low In such a contextthe performance does not matter too much There are manyregions in the world where this is not the case howeverIt would thus be quite beneficial if this work was taken upand further developed in particular with respect to attitudesensing Some individuals may reject the idea of carryingtheir smartphone around their neck This could be addressedby producing decorative gadgets which are less obstructive towear Beyond that the carrying of a device around the neckalso enables the use of the camera This would allow to furtherrefine the evaluation of the risk but would drain the batteriesmuch more and would raise concerns about privacy Thusthe use of the sensors addressed in the present papers seemto remain most promising In the future Bluetooth rangingshould be considered as well The complete analysis of thepaper and its validity rely on the current model of infectionof the Robert-Koch Institute

ACKNOWLEDGMENT

The authors would like to thank Dr Armin Dammannfrom the German Aerospace Center (DLR) for comments on

Section III-A and for providing us with early results from theevaluation of the experiments

VIII REFERENCES

REFERENCES

[1] K Kupferschmidt and J Cohen ldquoCan Chinarsquos COVID-19 Strategy WorkElsewhererdquo Science vol 367 no 6482 pp 1061ndash1062 2020 [Online]Available httpssciencesciencemagorgcontent36764821061

[2] ldquoCorona-Warn-Apprdquo 2020 Robert Koch-Institut Berlin [Online]Available httpsdewikipediaorgwikiCorona-Warn-App

[3] S von Arx I Becker-Mayer D Blank J Colligan R FenwickM Hittle M Ingle O Nash V Nguyen J Petrie J SchwaberZ Szabo A Veeraghanta M Voloshin T White and H XueldquoSlowing the Spread of Infectious Diseases Using Crowdsourced Datardquo2020 [Online] Available httpswwwcovid-watchorgarticle

[4] C Guumlnther M Guumlnther and D Guumlnther ldquoTracing Contactsto Control the COVID-19 Pandemicrdquo 2020 [Online] Availablehttpsarxivorgabs200400517

[5] M Nanni G Andrienko C Boldrini F Bonchi C CattutoF Chiaromonte G Comandeacute M Conti M Coteacute F Dignumet al ldquoGive more Data Awareness and Control to Individual Citi-zens and They Will Help COVID-19 Containmentrdquo arXiv preprintarXiv200405222 2020

[6] R Raskar I Schunemann R Barbar K Vilcans J GrayP Vepakomma S Kapa A Nuzzo R Gupta A Berke et al ldquoAppsGone Rogue Maintaining Personal Privacy in an Epidemicrdquo arXivpreprint arXiv200308567 2020

[7] P OrsquoNeill T Ryan-Mosley and B Johnson ldquoA flood of coronavirusapps are tracking us now itrsquos time to keep track of themrdquo 2020

[8] A Dammann C Gentner and D Guumlnther ldquoOn BLE ProximityDetection Performance for COVID-19 Contact Tracingrdquo 2020 underpreparation

[9] L Kurz C Guumlnther and D Guumlnther ldquoAudio Ranging for COVID-19Contact Tracingrdquo 2020 under preparation

[10] P OrsquoNeill ldquoBluetooth contact tracing needs bigger better datardquo MITTechnology Review 2020

[11] C Gentner D Guumlnther and P Kindt ldquoIdentifying the BLE AdvertisingChannel for Reliable Distance Estimationrdquo 2020 [Online] Availablehttpsarxivorgabs200609099

[12] H Hashemi ldquoThe Indoor Radio Propagation Channelrdquo Proceedings ofthe IEEE vol 81 no 7 pp 943ndash968 1993

[13] L Schwartz Theacuteorie des Distributions Hermann Paris 1966[14] J W Betz and K R Kolodziejski ldquoGeneralized Theory of Code

Tracking with an Early-Late Discriminator Part I Lower Bound andCoherent Processingrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 4 pp 1538ndash1556 2009

[15] mdashmdash ldquoGeneralized Theory of Code Tracking with an Early-Late Dis-criminator Part II Noncoherent Processing and Numerical ResultsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 45 no 4pp 1557ndash1564 2009

[16] A Van Dierendonck P Fenton and T Ford ldquoTheory and Performanceof Narrow Correlator Spacing in a GPS Receiverrdquo Navigation vol 39no 3 pp 265ndash283 1992

[17] T Michel P Geneves H Fourati and N Layaiumlda ldquoOn AttitudeEstimation with Smartphonesrdquo in 2017 IEEE International Conferenceon Pervasive Computing and Communications (PerCom) IEEE 2017pp 267ndash275

[18] ldquoKontaktpersonennachverfolgung bei Respiratorischen Erkrankun-gen durch das Coronavirus SARS-CoV-2rdquo 2020Robert Koch Institut Berlin [Online] Avail-able httpswwwrkideDEContentInfAZNNeuartiges_CoronavirusKontaktpersonManagement_Downloadpdf__blob=publicationFile