Performance and Cramer–Rao Bounds for DoA/RSS Estimation ... · strength (RSS) estimation, as...

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016 3255

Performance and Cramer–Rao Bounds for DoA/RSSEstimation and Transmitter Localization

Using Sectorized AntennasJanis Werner, Jun Wang, Aki Hakkarainen, Danijela Cabric, Senior Member, IEEE, and

Mikko Valkama, Member, IEEE

Abstract—Using collaborative sensors or other observing de-vices equipped with sectorized antennas provides a practical andlow-cost solution to direction-of-arrival (DoA) and received signalstrength (RSS) estimation, as well as noncooperative transmit-ter (TX) localization. In this paper, we study the performanceand theoretical bounds of DoA/RSS estimation and localizationusing sectorized antennas. We first show that the sector-powermeasurements at an individual sensor form a sufficient statis-tic for DoA/RSS estimation and TX localization. Motivated bythat, we then derive the Cramer–Rao bound (CRB) on DoA/RSSestimation based on sector-powers and study its asymptotic be-havior. Moreover, we derive an analytical expression for themean square error (MSE) of a practical sectorized-antenna-basedDoA estimator, compare its performance to the derived CRB,and study its asymptotic properties. Next, we derive the CRBfor localization based on sector-powers. The resulting CRB is alower bound for a localization system where the DoA/RSS es-timates, obtained from sector-powers at individual sensors, arefused together into a location estimate. Moreover, the CRB alsocovers the more general case of a localization system wherethe sector-powers from individual nodes are directly fused to-gether, without an intermediate DoA/RSS estimation step. Wecompare the obtained CRB to a localization approach employ-ing an intermediate DoA/RSS estimation step and observe thatskipping this intermediate processing step may result in a sub-stantially improved localization performance. Finally, we studythe influence of various important system parameters, such asthe number of sensors, sectors, and measurement samples, on theachievable estimation and localization performance. Overall, thispaper demonstrates and quantifies the achievable DoA/RSS es-timation and localization performance of sectorized antennasand provides comprehensive design guidelines for sector-powerbased low-complexity localization systems.

Manuscript received November 28, 2014; revised April 2, 2015; acceptedJune 5, 2015. Date of publication June 15, 2015; date of current versionMay 12, 2016. This work was supported in part by the Doctoral Program ofthe President of Tampere University of Technology; by the Tuula and YrjöNeuvo Fund; by the Nokia Foundation; by the Finnish Funding Agency forTechnology and Innovation (TEKES) under Project “Reconfigurable Antenna-based Enhancement of Dynamic Spectrum Access Algorithms,” Project “5GNetworks and Device Positioning,” and Project “Future Small-Cell Networksusing Reconfigurable Antennas”); and by the National Science Foundationunder Grant 1117600. The review of this paper was coordinated by Dr. K. Yu.

J. Werner, A. Hakkarainen, and M. Valkama are with the Department of Elec-tronics and Communications Engineering, Tampere University of Technology,33101 Tampere, Finland (e-mail: [email protected]; [email protected];[email protected]).

J. Wang is with Broadcom Corporation, Irvine, CA 92617, USA (e-mail:[email protected]).

D. Cabric is with the Department of Electrical Engineering, University ofCalifornia, Los Angeles, CA 90095 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2015.2445317

Index Terms—Angle of arrival, cognitive radio (CR), Cramer–Rao bounds (CRBs), directional antennas, direction-of-arrival esti-mation, leaky-wave antennas (LWAs), localization, received signalstrength (RSS), reconfigurable antennas, sectorized antennas,single radio-frequency front end.

I. INTRODUCTION

LOCALIZATION of a noncooperative transmitter (TX),i.e., a TX that does not directly communicate with the lo-

calization network, using multiple cooperating receivers (RXs),is an important task in several application areas [1], [2]. Forexample, knowledge about primary-user location can enableseveral advanced functionalities in cognitive radio (CR) net-works, such as improved spatiotemporal sensing, intelligentlocation-aware power control and routing, and spectrum policyenforcement [2]. However, both the complexity and energy con-straints of the RXs and the wireless communication among theRXs require efficient use of the available resources, namely, theavailable energy of the RX devices [1] and the radio frequency(RF) spectrum [2]. Moreover, it is desirable to distribute thecomputational complexity among the participating RXs [1].As a consequence, it is prohibitive that the RXs communicatetheir measurement samples directly. Instead, every RX shouldpreprocess the samples into compressed measurements that areuseful for localization. These compressed measurements arethen communicated to a fusion center (FC), which may also beone of the sensors, where the compressed measurements fromall RXs are combined into a location estimate [3].

Traditionally, three different types of measurements are usedin the localization of a noncooperative TX, i.e., the receivedsignal strength (RSS), direction of arrival (DoA), and timedifference of arrival (TDoA) [4]. However, TDoA-based algo-rithms require perfect synchronization among the RXs, whichis generally infeasible in many application areas [4]. Therefore,related research has focused on RSS [5], [6] and DoA [7]and joint DoA/RSS-based localization, which has been shownto significantly outperform RSS-only and DoA-only schemes[4], [8]. However, traditional DoA estimation techniques, suchas MUSIC [9], require digitally controlled antenna arrays(DCAAs) with a high number of antenna units and associatedRF chains, and thus may not be suitable for RX devices in manyapplications due to size and cost limits.

Stemming from the above, we have recently proposed to usesectorized antennas for DoA and RSS estimation [10], [11]. Asectorized antenna is an antenna structure that can selectively

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3256 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016

Fig. 1. Required hardware for different directional antennas. (Left) DCAA.(Middle) SBS. (Right) LWA [14].

receive energy from different sectors. A sector denotes a contin-uous range of angles and selectivity means that signals arrivingfrom outside of the activated sector are strongly attenuated. Wefurther assume that only a single sector can be activated at atime. Thus, the class of sectorized antennas encompasses alldirectional antennas with a single RF front end. Examples ofantenna structures that can be used as sectorized antennas areswitched-beam systems (SBSs) [12] and leaky-wave antennas(LWAs) [13]. Using sectorized antennas as opposed to DCAAscan result in a great reduction of hardware complexity asillustrated in Fig. 1. An LWA, for example, is operated with asingle RF front end and one analog-to-digital converter (ADC)only, whereas a DCAA requires an RX chain and a separateADC for each of its antennas [14].

The DoA/RSS estimators for sectorized antennas proposedin [10] and [11] first calculate the power in every sector andsuccessively estimate the DoA and RSS based on these sector-powers. From a practical point of view, this way of processingattenuates the influence of channel fluctuations. In this paper,we will furthermore show that, at an individual RX, the sector-powers are a sufficient statistic for DoA/RSS estimation ifsuccessively received samples are independent. Thus, an RXthat is equipped with a sectorized antenna can estimate theDoA/RSS based on sector-powers without loss in performancecompared with estimation based on raw samples. Therefore,in this paper, we derive the Cramer–Rao Bounds (CRBs) onDoA/RSS estimation based on sector-powers. These DoA/RSSCRBs for sectorized antennas differ significantly from therespective CRBs [15], [16] for DCAAs since different samplesin a sectorized antenna are received sequentially in time insteadof being received in parallel. In [11], we already used theDoA/RSS estimation CRBs as a reference when evaluating theperformance of a practical DoA/RSS estimator. However, botha detailed evaluation and the derivation of the CRBs were notpart of [11]. Moreover, and in contrast to [11], this paper alsopresents an analytical analysis of the asymptotic behavior ofthe CRBs and a novel and highly accurate approximation of theDoA/RSS CRBs that relates the bounds to the RX parametersin a very simple form.

Since the DoA is a function of the TX location, sector-powers at individual RXs are also a sufficient statistic forTX location estimation. However, the TX location cannot beestimated based on the sector-powers from an individual RXalone. Therefore, in this paper, we propose a system wheremultiple collaborating RXs localize a noncooperative TX basedon sector-powers instead of RSSs or DoAs. More specifically,we derive the CRB on sector-power based collaborative local-ization of a noncooperative TX and two approximations of theobtained bound. In related studies, the CRB has been derived

for localization based on RSSs only [5], [6], DoAs only [7]or joint DoA/RSS-based approaches using DCAAs [4], [8].However, to the best of our knowledge, no prior work considerslocalization based on sector-powers nor the localization CRBfor sectorized antennas in general.

The derived DoA/RSS CRBs are compared with the perfor-mance of the simplified least squares (SLS) algorithm, whichis one of the earlier proposed [10] practical sector-power basedDoA/RSS estimators. Toward this end, we derive an accurateanalytical expression for the root mean square error (RMSE) ofSLS DoA estimation, which is valid also for low SNR regions incontrast to the asymptotic high SNR expression in [17]. Overall,the comparison between SLS and the derived DoA/RSS CRBsgives valuable insights into performance and mechanisms ofthe algorithm that are useful for further improvements. Since,currently, no algorithm exists that calculates TX location esti-mates directly from sector-powers, we compare the localizationCRB to a combination of SLS DoA/RSS estimation and amodified version of the Stansfield DoA fusion algorithm [3],[10]. Although this combined algorithm, referred to as SLS-ST, computes DoA estimates from the sector-powers in anintermediate step, it is overall lower bounded by the localizationCRB, as will be discussed in Section III-A. Moreover, SLSperforms fairly close to the CRB on DoA/RSS estimation, andthe Stansfield algorithm is known to very efficiently computelocation estimates from DoA/RSS estimates [4]. Therefore, thecomparison of SLS-ST to the sector-power localization CRB isa good indication for the potential benefit of sector-power basedlocalization without an intermediate DoA/RSS estimation step.In summary, the contributions of this paper are the following:

• introduction of a sector-power based localization systemand its statistical model;

• analysis showing that sector-powers at individual RXsare a sufficient statistic for DoA/RSS estimation and TXlocation;

• derivation of CRBs for DoA/RSS estimation based onsector-powers;

• analytical study of the asymptotic behavior of the DoA/RSS CRBs;

• derivation of an accurate analytical expression for theRMSE of SLS DoA estimation;

• derivation of the CRB for localization based on sector-powers;

• providing simple and accurate approximations of the de-rived CRBs;

• extensive evaluation of the derived CRBs with respect toimportant system parameters;

• comparison of the CRBs to the performance of exist-ing algorithms and identification of weaknesses in thealgorithms.

This paper is organized as follows. Section II introducesthe localization system under consideration. A more detaileddescription of sector-powers and the motivation for using themin the localization system is presented in Section III. TheCRBs for both DoA/RSS estimation and TX localization arederived in Section IV. In Section V, we briefly summarize the

WERNER et al.: PERFORMANCE AND CRB FOR DoA/RSS ESTIMATION AND TRANSMITTER LOCALIZATION 3257

TABLE IMOST COMMONLY USED SYMBOLS

algorithms that we compare with the CRBs. We then derive ananalytical expression for the error of the selected DoA estima-tion algorithm in Section VI. A comparison of the algorithmsto the CRBs along with a detailed evaluation and analysis canbe found in Section VII. Finally, the work is concluded inSection VIII. Details of derivations and proofs are reported inAppendixes A–C.

Notation: Throughout this paper, vectors and matrices arewritten as boldface letters. We will be handling signals witha continuous time nature, as well as the samples of the samesignals. We indicate a signal with continuous time as x(t), t ∈R, and the respective signal samples as x(n), n ∈ N, withthe convention that x(nTs) = x(n), where Ts is the samplingperiod. For z ∈ C, �(z) and �(z) denote the real and imaginaryparts of z, and z∗ is its complex conjugate. E[X ] and var[X ]denote expected value and variance of the random variable X ,respectively, whereas bias[y] = E[y]− y denotes the bias ofestimator y for a deterministic quantity y. mody(x) expressesthe remainder of the division x/y. For a vector x and amatrix M, we denote an individual element as xi and Mij oralternatively (x)i and (M)ij . The transpose of a matrix M iswritten as MT and its conjugate transpose as MH. |M| denotesthe determinant of matrix M, whereas M = diag(x) denotesthe diagonal matrix that is composed of all the elements ofvector x, i.e., [diag(M)]ii = xi. A ◦B denotes the Hadamardproduct of two M ×N matrices A and B. Derivatives arewritten in short as (df/dx) = [f ]x, and matrix derivatives arealways elementwise, i.e., ([M]x)ij = [Mij ]x.

For the readers’ convenience, we have collected the mostcommonly used symbols in Table I.

II. PRELIMINARIES AND SYSTEM MODEL

In this paper, we consider a localization system, which isshown in Fig. 2. In this system, K RXs with given locations�k = [xk, yk]

T, k = 1, . . . ,K collaborate to estimate the loca-tion �P = [xP, yP]

T of a noncooperative TX. Toward this end,

Fig. 2. Studied localization system. Every RX is equipped with a sectorizedantenna and preprocesses the received samples into a compressed measure-ment Ψk that is then send to the FC for the final location estimation. Thederived CRBs are for compressed measurements that contain the sector-powers,whereas the algorithm that we use as a comparison employs an intermediatestep that turns sector-powers values into DoA/RSS estimates and could, conse-quently, communicate DoA/RSSs instead of sector-powers.

each of the RXs is equipped with a sectorized antenna, as willbe further discussed in Section II-A. During the localization,RX k receives N samples of the incoming TX signal sk,m(t) atantenna sector m,m = 1, . . . ,M . The respective transmissionmodel is described in detail in Section II-B. Since we aretargeting low network overhead, it is prohibitive that the RXscommunicate raw samples to the FC. Instead, the raw samplesare preprocessed into a compressed measurement Ψk at eachRX k, which is then communicated to the FC. Using these com-pressed measurement from all RXs, the FC finally estimatesthe TX location. In addition to limiting the network overhead,RX preprocessing has a further advantage that computations aredistributed and fewer data have to be handled within the FC.This also makes it possible to use one of the RXs as the FC,even if using resource-constrained devices.

For the derivation of the localization CRB, we assume com-pressed measurements that contain sector-powers, as will bemotivated in Section III. However, so far, no algorithm existsthat turns sector-powers directly into TX location estimates.Therefore, we compare the CRB to a localization algorithmthat obtains the location estimate via an intermediate step (seeSection V). In this step, the DoAs are estimated based onthe sector-powers, and only the DoAs and RSSs are finallyused in TX location estimation. Consequently, if the DoAestimation takes place in every individual RX, the compressedmeasurements can consist of the RSSs and DoAs only.

A. Sectorized Antenna Model

A sector is characterized by its radiation pattern, whichdescribes the attenuation of the received signal as a functionof its DoA. In this paper, we first develop the CRBs in a generalform such that they are applicable to any sectorized antenna.However, to identify some general trends, we then assumethat the radiation pattern ζk,m = ζ(ϕk − ϑm) depends onlyon the angular distance between the incoming signal’s DoA,


Fig. 3. Example for the approximation (dotted line) of an antenna’s mainbeam through a Gaussian radiation pattern (1). (a) Measured radiation pattern ofa sector with ϑ = 90◦ from a LWA [13] and (b) theoretical radiation pattern ofa sector with ϑ = 0◦ from a switched-beam system (SBS) with an underlyinguniform circular antenna array consisting of eight antenna elements.

ϕk = arctan((yP − yk)/(xP − xk)), at RX k and orientationϑm of sector m, m = 1, . . . ,M . Thereby, we assume equalsector orientations for all RXs for simplicity and without lossof generality. In addition, for analysis purposes, we furthersimplify the model of the radiation pattern by approximatingonly the antenna’s main beam using a Gaussian-like shape as in[10] and [11]. Then, the radiation pattern can be expressed as

ζ(ϕ) = exp

(−[M(ϕ)]2

β2

)(1)

with M(ϕ) = mod2π(ϕ+ π)− π and beamwidth β. This ap-proximation is very accurate for, e.g., the measured radiationpattern of the LWA [13], [18] or the radiation pattern of an SBSbased on a circular array, as shown in Fig. 3. In this paper, weassume a spacing between the sectors that is Δϑ = (2π/M),resulting in an orientation of sector m equal to

ϑm =

{M

[(m− 1

2

)Δϑ

], if M is even

M [(m− 1)Δϑ] , if M is odd(2)

where we define the angles such that the sector orientationsare symmetric around 0◦ for convenience and without loss ofgenerality. We parameterize the radiation pattern via the side-sector suppression as. The side-sector suppression is definedas the attenuation that a signal, arriving at the orientationϑm of the mth sector, experiences in the neighboring sectorsm− 1 and m+ 1, i.e., as = ζ(Δϑ) [10]. Therefore, the side-sector suppression determines the amount of overlap betweenneighboring sectors, independent of the number of sectors.The beamwidth can be derived from the above assumptionsas β = 2π/[M

√ln(1/as)]. An illustration of the side-sector

suppression can be found in Fig. 4.

B. Channel and Receiver Model

We denote the complex baseband equivalent signal emittedby the noncooperative TX as s(t) and make the common

Fig. 4. Side-sector suppression of a sectorized antenna with Gaussian radia-tion pattern, M = 3 sectors, and as = 0.4.

assumption that its distribution can be well approximated ascircular symmetric complex Gaussian s(t) ∼ CN (0, σ2

s ), i.e.,an approximation that fits well for a range of practical relevantsignals, such as orthogonal frequency-division multiplexingwith a reasonable number of subcarriers. In our derivations,we do not assume any specific autocorrelation function (ACF)of the TX signal us(τ) = E[s(t)s∗(t− τ)]. For simplicity, weonly require the ACF to satisfy us(nTs) = 0, where Ts is thesampling period at the RXs, and n = ±1,±2, . . .. However, inour evaluations of the localization CRBs, we need to assumea specific ACF. In Section VII-B, we therefore model the TXsignal as bandlimited to the frequencies −B/2 < f < B/2,where B denotes the physical bandwidth of the actual RFsignal. Then, the TX signal has an ACF [19, pp. 416–418]

us(τ) = E [s(t)s∗(t− τ)] = σ2s sinc(Bτ). (3)

For the ACF (3), our earlier assumption uns(Ts) = 0 holds ifTs = 1/B, which means in practice we do not oversample atthe RXs. In general, we assume that the ACF is unknown to thelocalizing network since the TX is noncooperative.

The incoming signal in sector m at RX k is next written as

sk,m(t) = hksk (t− (m− 1)Ta) (4)

= hks (t− ηk − (m− 1)Ta) (5)

where the delay at RX k is dependent on ηk = dk/c, i.e., thefraction of RX–TX distance dk = ‖�k − �P‖ and the speedof light c and on the sector-switching period Ta = NTs. Theincoming signal is furthermore experiencing propagation lossby the TX–RX channel modeled here with the channel co-efficients hk. In this paper, we do not consider fast-varyingchannels explicitly. Instead, we generally assume that thechannel coefficients are constant during the relatively shortlocalization phase TL = TaM (e.g., M = 6 sectors, N = 100samples, fs = 1/Ts = 20 MHz: TL = 30 μs). For simplicity,we moreover consider scenarios where the energy of the line ofsight (LoS) path is dominating. At the RX, the incoming signal


is then attenuated by the DoA-dependent antenna attenuationand corrupted by additive circular symmetric complex whiteGaussian noise wk,m(t) ∼ CN (0, σ2

w), so that the receivedsignal in sector m at RX k becomes

rk,m(t) = ζk,msk,m(t) + wk,m(t) (6)

or

rk,m(n) = ζk,msk,m(n) + wk,m(n) (7)

in sample notation. Hence, at every RX k, we obtain a sam-ple vector rk = [rTk,1, r

Tk,2, . . . , r

Tk,M ]T ∈ CMN×1 composed

of the N samples rk,m = [rk,m(0), rk,m(1), . . . , rk,m(N −1)]T ∈ CN×1 by the M sectors.

The CRBs for DoA/RSS estimation and the RMSE for theSLS algorithm will be derived for a given channel realization.Similarly, we will derive the localization CRB for given channelrealizations at all RXs and the given RX locations. Giventhese assumptions, the RSS at RX k can be expressed as γk =E[|sk(n)|2] = |hk|2σ2

s , and the received signal samples areGaussian distributed according to rk,m(n) ∼ CN (0, ρk,mγk +σ2w) with ρk,m = ζ2k,m. Consequently, we obtain a Gaussian-

distributed sample vector rk ∼ CN (0,Σk) with covariancematrix Σk = E[(rk − E[rk])(rk − E[rk])

H] = diag(ρk,1γk +σ2w, ρk,2γk + σ2

w, . . . , ρk,Mγk + σ2w), which is diagonal due to

our assumption of uncorrelated samples in (7). Overall, oursample model is very similar to the models in related workssuch as [16], where the DoA estimation CRB is derived forDCAAs and a given RSS under the assumption that the receivedsamples are temporarily uncorrelated.

To establish this sample model, we have made two assump-tions that simplify our analysis significantly. In the following,we will motivate these assumptions further and also shortlysketch how they could be relaxed, First, we assumed thatus(τ) = 0 if τ = nTs, n = ±1, 2, . . ., holds for the TX signalACF. As will be seen in the following, this helps us to approxi-mate the sector powers as Gaussian distributed and uncorrelatedat an individual RX. Now, for signals with a nonzero ACFat τ = nTs, it might still be possible to approximate sector-powers values as Gaussian distributed depending of courseon the structure of the ACF (see, e.g., [20] and referencestherein). Similarly, modeling sector-powers at individual RXsas uncorrelated is still a fairly good approximation, e.g., forACFs that strongly decay with τ . Note also that the derivationof the localization CRB does not assume uncorrelated sector-powers. We could thus also obtain the RSS/DoA CRBs forcorrelated sector-powers by simply following the derivation ofthe localization CRB.

The second assumption that we have made is that the energyof the LoS path is dominating. To consider strong non-line-of-sight (NLoS) components explicitly, we would have to sum upover all NLoS paths and the LoS path in (7). The resultingdistribution of the received samples rk,m is, in general, againdependent on the TX signal ACF. To obtain results that areindependent of a specific ACF, we could, however, approx-imate the contributions from different paths as independentand Gaussian distributed. The resulting rk,m(n) is then still

Gaussian distributed but with a mean and variance determinedby all the paths. From there on, the results for scenarios withstrong NLoS paths are obtained in a similar manner as those thatwe derive in this paper. Thus, this approach would significantlysimplify the analysis, whereas at the same time making itpossible to study the influence of NLoS paths arriving fromangles other than the DoA of the LoS path.

However, the given discussed relaxation is beyond the scopeof this paper, and we will, in the following, assume that the twoassumptions hold.

III. SECTOR-POWERS AND THEIR STATISTICS

As discussed earlier, the communication of raw sensor mea-surements is impractical in a network where the communicationbetween the collaborating RXs is wireless. Therefore, we con-sider a localization system, where the received samples of allsectors m = 1, . . . ,M at every RX k are preprocessed intosector-powers according to

εk,m =1N

N−1∑n=0

|rk,m(n)|2 . (8)

The motivation for this particular preprocessing is discussed indetail in Section III-A, whereas the model for the sector-powersis described in Section III-B.

A. Sector-Powers as Compressed Measurements andSufficient Statistics

Here, we will establish the use of sector-powers as com-pressed measurements by means of Neyman–Fisher factoriza-tion [21, pp. 116–117]. Toward that end, we will first identifythe part of the measurements that is relevant for the localization.The received sample vector rk at RX k depends on the TXlocation �P via the following variables: time delay ηk betweenTX and RX, RSS γk, and antenna attenuations ρk,m, m =1, . . . ,M . While the time delay is dependent on the TX–RXdistance via a very simple relationship, it is not very usefulfor the localization due to the assumed lack of synchronizationamong the RXs and due to the assumption that the TX is non-cooperative and its signal ACF unknown. In case of the RSS,the exact dependence on the location is strongly influenced bythe environment and is thus hard to describe with a model thatis both simple and accurate. Therefore, we do not considerlocalization algorithms that extract the TX location from thedelay ηk or the RSS γk. Instead, we focus solely on algorithmsthat extract the TX location from the antenna attenuation ρk,m.Note that it nevertheless makes sense to estimate the RSSat every RX if an intermediate step with DoA estimation istargeted as the RSS is an indicator for the quality of the DoAestimates and can therefore be used, e.g., for weighting the DoAestimates (see Section V-B). With these, we are now ready tostate the following theorem.

Theorem 1: Sector powers are a sufficient statistic for the es-timation of �P and, consequently, an ideal compressed message


for the localization system. Moreover, DoA/RSS estimation atan RX can, equally and without loss of performance, be basedon the received samples or the sector-powers.

Proof: Since the preprocessing of samples into com-pressed message takes place in every individual RX, we willconsider only the samples rk originating from an arbitrary RXk. For a given TX location, rk has the probability distributionfunction (pdf) f(rk|�P) = (πMN |Σk|)−1 exp(−rHk Σ−1

k rk).Given the assumption that the RXs do not oversample, the pdfcan also be written as

f (rk|�P) =1

(π)MN |Σk|

· exp(−

M∑m=1

1ρk,mγk + σ2

w

N−1∑n=0

|rk,m(n)|2). (9)

The Neyman–Fisher factorization theorem states that, if a pdfcan be factorized as f(rk|�P) = u(T(rk), �P)h(rk), whereh(rk) is a factor independent of the parameter �P that we wantto estimate, and u(T(rk), �P) is a factor that depends on themeasured data rk only through a function T(rk), then T(rk)is a sufficient statistic for the estimation of �P [21]. For pdf(9), we can choose the factors h(rk) = 1 and u(T(rk), �P) =f(rk|�P) with T(rk) = εk(rk), which denotes the M × 1 vec-tor composed of the M sector-powers in (8). Thus, we haveshown that the sector powers values are a sufficient statisticfor the estimation of �P. Now, the antenna attenuation dependson the TX location only via the DoA. Hence, �P can bereplaced with ϕk in (9), and in case of sectorized antennas, DoAestimation based on sector-powers is equal to the estimationbased on samples. A similar argument can be made for RSSestimation based on sector-powers. �

A consequence of the given theorem is that the CRB forDoA/RSS estimation based on sector-powers is equal to thesample-based DoA/RSS CRB. Moreover, the localization CRBfor sector-powers also encompasses all localization algorithmsthat employ an intermediate DoA/RSS estimation step such asthe one discussed in Section V.

B. Sector-Power Model

Since consecutively received samples in a single RX areindependent, sector-powers for a given and slowly varyingchannel are chi-squared distributed. It is well known (e.g., in[22]) that such a distribution can be well approximated asGaussian if the number of samples is large enough. Usingthis approximation, we model the sector-powers as εk,m ∼N (gk,m, σ2

k,m) with mean gk,m = E[εk,m] = ρk,mγk + σ2w

and variance σ2k,m = E[(εk,m − gk,m)2]. Overall, this results in

a model at a single RX that is very similar to the commonlyused one, e.g., in the energy detection literature [22]. In thefollowing, we will discuss this model in more detail for the twocases of DoA/RSS estimation and localization.

1) DoA/RSS Estimation: Since DoA/RSS estimation is car-ried out within a single RX, it suffices to model the sector-powers at a single RX alone. At RX k, the M × 1 vector

of all sector-powers values εk = [εk,1, εk,2, . . . , εk,M ]T ∼N (gk,Qk) has a mean vector gk = E[εk] and covarianceQk = E[(εk − gk)(εk − gk)

T] equal to

gk = [gk,1, gk,2, . . . , gk,M ]T (10)

Qk =diag(σ2k,1, σ

2k,2, . . . , σ

2k,M

)(11)

with σ2k,m = (1/N)(ρk,mγk + σ2

w)2. We note that, due to the

assumption of independent samples, we also obtain indepen-dent sector-powers and, as a consequence, a diagonal covari-ance matrix Qk.

2) Localization: In contrast to DoA/RSS estimation, themodel for localization has to be a global one, meaning thatit involves the sector-powers from all RXs. This leads toa KM × 1 vector of sector-powers ε = [εT1 , ε

T2 , . . . , ε

TK ]T ∼

N (g,Q), with a mean vector g = E[ε] = [g1,g2, . . . ,gK ]T ,that is composed of the means gk, from all RXs k = 1, . . . ,Kas in (10). However, in contrast to the sector-power covarianceat an individual RX, the covariance for the sector-powersfrom all RXs Q = E[(ε− g)(ε− g)T] cannot be modeledas a diagonal matrix. A diagonal covariance matrix Qk atRX k implies an incoming signal with ACF usisi(nTs) =E[si(t)s

∗i (t− nTs)] = 0 for n = ±1,±2, . . ., which is true for

the bandlimited Gaussian TX signal without oversampling atthe RX. However, considering the incoming signals si and sj , attwo RXs i and j, the ACF usisj (Δηij) = E[si(t)s

∗j(t−Δηij)]

depends on the difference of delays, i.e., Δηij = ηi − ηj , forwhich practically always holds that Δηij = mTs. In particular,if di ≈ dj , then Δηij � mTs such that the respective ACFusi,sj (Δηij) = 0.

As shown in Appendix A, the sector-powers covariance canthen be written as

Q =ρρT ◦ γγT ◦C+D (12)

D =σ2w

N

(2diag(ρ ◦ γ) + σ2

wI)

(13)

with

ρ =[ρT1 ,ρ

T2 , . . . ,ρ

TK

]T(14)

ρk = [ρk,1, ρk,2, . . . , ρk,M ]T (15)

γ =[γ11

1×M , γ211×M , . . . , γK11×M

]T(16)

and the KM ×KM matrix C composed of the M ×Msubmatrices

Ωij=

⎛⎜⎜⎜⎝cij(0) cij(−1) · · · cij (−(M−1))cij(1) cij(0) · · · cij (−(M−2))

......

. . ....

cij(M−1) cij(M−2) . . . cij(0)

⎞⎟⎟⎟⎠ (17)


with elements

cij(m) =1N

Rs (Δηij +mTa)− 1 +1N2

N−1∑p=1

(N − p)

× [Rs(Δηij +mTa + pTs) +Rs(Δηij +mTa − pTs)](18)

that are dependent on the normalized ACF of the squared en-velope Rs(τ) = (1/σ4

s )E[|s(t)|2|s(t− τ)|2]. For a zero-meancomplex Gaussian signal, Rs(τ) can be expressed in terms ofits ACF as [23, pp. 67–68]

Rs(τ) = 1 +1σ4s

|us(τ)|2. (19)

Hence, for the example case of a bandlimited Gaussian sig-nal that we use in our evaluations in Section VII, we obtainRs(τ) = 1 + sinc2(Bτ) and a matrix C with elements

cij(m) =1N

sinc2 (tij(m)) +1N

N−1∑p=1

(N − p)

·[sinc2 (tij(m) + pBTs) + sinc2 (tij(m)− pBTs)

](20)

where tij(m) = BΔηij +mBTa. However, neither our sector-power model nor the CRB that we will derive in the followingis limited to the bandlimited Gaussian signal assumption.

IV. CRAMER–RAO BOUNDS

The CRB is a lower bound on the covariance matrix ofany unbiased estimator [21]. For the estimation of a P × 1parameter vector q, given an R× 1 vector ε composed ofthe observations, the CRB is obtained as the inverse of theFisher information matrix (FIM) (e.g., [15], [21]). The FIM iselementwise defined as

Fij = −E

(∂2

∂qi∂qjln [f (ε|q)]

)(21)

with i, j = 1, . . . , P . For the estimation problems consideredin this paper, the observation vector ε ∼ N (g,Q) is approx-imately Gaussian distributed, as discussed earlier, and boththe mean g and covariance Q depend on the parametervector, i.e., g = ξ(q), ξ : RP×1 �→ RR×1 and Q = Λ(q),Λ :RP×1 �→ RR×R. As shown in [21, pp. 47], the FIM in (21) canthen be calculated as

Fij = [g]TqiQ−1[g]qj

+12tr{Q−1[Q]qi

Q−1[Q]qj

}. (22)

A. CRBs for DoA/RSS Estimation

As discussed in Section III-A, the CRBs on DoA/RSS esti-mation can be computed using either samples or sector-powers.However, since the algorithms under consideration are basedon sector-powers, we will also derive the respective CRBs,

assuming a preprocessing of samples into sector-powers. Themodel for the sector-powers (10) and (11) involves the DoAas well as the RSS. Hence, theoretically, the estimation ofDoA and RSS cannot be considered independently. Instead therespective parameter vector consists of both DoA and RSS,i.e., q = [ϕk, γk]

T, which results in a 2 × 2 FIM Γ. Given thediagonal matrix (11) and using (22), it follows that

Γ11 =(N + 2)M∑

m=1

ak,mρ2k,m (23)

Γ12 = Γ21 =N + 2γk

M∑m=1

ak,mρk,m (24)

Γ22 =N + 2γ2k

M∑m=1

ak,m (25)

where ak,m = (SNR−1k,m + 1)−2 is a weight for the contribution

from sector m depending on the SNR in sector m at RX k,SNRk,m = ρk,mγk/σ

2w. Furthermore, ρk,m = [ρk,m]ϕ/ρk,m,

which becomes ρk,m = −4M(ϕk − ϑm)/β2 for the Gaussianradiation pattern. Using the FIM elements (23)–(25), theCRBs on the RMSE of DoA estimation RMSEDoA, and therelative RMSE (RRMSE) on RSS estimation RRMSERSS =RMSERSS/γk are finally obtained as follows:

RMSEDoA =√(

Γ−1)11

> RMSEaDoA (26)

RMSEaDoA =

√Γ−111 =

√√√√ 1N + 2

(M∑

m=1

ak,mρ2k,m

)−1

(27)

RRMSERSS =1γk

√(Γ−1

)22

> RRMSEaRSS (28)

RRMSEaRSS=

1γk

√Γ−122 =

√√√√ 1N+2

(M∑

m=1

ak,m

)−1

. (29)

Thereby, RMSEaDoA is the CRB on the RMSE of DoA esti-

mation when the RSS is known, and accordingly, RRMSEaRSS

is the CRB on the RRMSE of RSS estimation when the DoAis known. Thus, these quantities are also lower bounds forthe complete estimation problem where both RSS and DoAare unknown. Moreover, as shown in Section VII, the boundsRMSEa

DoA and RRMSEaRSS are indeed very good approxi-

mations for the CRB of the complete estimation problem,i.e., RMSEDoA ≈ RMSEa

DoA and RRMSERSS ≈ RRMSEaRSS.

This means that Γ212 � Γ11Γ22 and therefore indicates that

the DoA can be estimated independently of the RSS and viceversa. In fact, this observation is also confirmed by the SLSDoA estimator (see Section V-A). In SLS, the measured sector-powers values, which depend on both RSS and DoA, aretransformed into a ratio JSLS ≈ ρk,i/ρk,j that depends only onthe DoA but not on the RSS.

In the following, we will study the asymptotic behavior ofthe CRBs (27)–(29).


1) Asymptotic CRBs for Large SNR: In the derivation ofthe asymptotic bounds for large SNR, we will be using thefollowing lemma:

Lemma 1: The function κp(ϕ) =∑M

m=1 M(ϕ− ϑm)p, p ∈N, is periodic with period Δϑ.

Proof: The periodicity is easily proven by substituting theindex m of the sector orientation with m′ = mod M (m) + 1such that κp(ϕ) =

∑Mm′=1 M(ϕ+Δϑ− ϑm′)p = κp(ϕ+

Δϑ) since ϑm = ϑm′ −Δϑ. �Based on Lemma 1 and following the steps in Appendix B,

we obtain the asymptotic (large SNR) RMSE of DoA estima-tion and the asymptotic RRMSE of RSS estimation as

RMSESNRDoA =

√3π

|ln(as)|√(M5 −M3)(N + 2)

(30)

RRMSESNRRSS =

1√M(N + 2)

√1 +

3 M2 [P(ϕk)]2

π2 (M2 − 1)(31)

where P(ϕ) = mod Δϑ(ϕ+Δϑ/2)−Δϑ/2. The asymp-totic bounds in (30) and (31) were derived for a given DoA.Interestingly, RMSESNR

DoA is independent of the DoA, whereasRRMSESNR

RSS is a function of the DoA. If we assume a uniformlydistributed DoA, i.e., ϕk ∼ U(−π, π), we obtain the followingbounds:

RMSESNRDoA =RMSESNR

DoA (32)

RRMSESNRRSS =

1√(M − 1

M

)(N + 2)

(33)

as shown in detail in Appendix B.2) Asymptotic CRBs for Large N : As evident from

(23)–(29), the FIM for DoA/RSS estimation can be writtenas Γ = (N + 2)Γ′, where Γ′ is the part of the FIM that isindependent of N . Consequently, the CRBs on the RMSE ofDoA and RSS estimation are proportional to

√N + 2

−1since

Γ−1 = (N + 1)−1(Γ′)−1. Therefore, for N → ∞, the boundson DoA and RSS estimation are equal to

RMSENDoA = 0 (34)

RRMSENRSS = 0. (35)

Clearly, (34) and (35) are independent of the DoA, and con-sequently, the bounds for uniformly distributed DoAs become

RMSENDoA = 0 and RRMSE

NRSS = 0 as well. According to the

law of large numbers, N → ∞ means that the measured sector-powers attain their expected values, which are completely de-termined by DoA and RSS. Inversely, this also makes it possibleto estimate both RSS and DoA without error, as reflected by theCRBs in (34) and (35).

B. CRBs for Localization

As discussed in Section III-B2, the sector-power model forlocalization includes the unknown RSSs from all RXs alongwith the TX location. Moreover, the model includes the covari-ance matrix (12) that depends on the TX location via the ACF,

which is unknown to the RXs according to our assumptions.Therefore, the covariance matrix is treated as an unknown thatis part of the overall estimation problem. As a consequence, theFisher information of the RSSs, γ1, γ2, . . . , γK as well as theFisher information of the covariance matrix have to be includedin the localization FIM as the so-called nuisance parameters(see, e.g., [9], [24]), although they are not of direct interest forthe localization. With respect to the covariance matrix, it canbe seen from (12) that it is sufficient to estimate the unique ele-ments of the matrix C. Thereby, we notice that each submatrixΩij given in (17) has a maximum number of 2M − 1 uniqueelements cij(m) as m = −(M − 1),−(M − 2), . . . ,M − 1.Furthermore, Ωij = ΩT

ij since cij(m) = cji(−m). In additioncii(m) = cjj(m), so that there are only M unique elementsfor all the submatrices on the diagonal. The overall numberof unique elements cij(m) is therefore (1/2)K(K − 1)(2M −1) +M . However, the M elements on the main diagonalsubmatrices do not depend on the actual ACF since cii(0) =1/N and cii(m) = 0 for m = 0 due to our assumption thatus(nTs) = 0 for n = ±1,±2, . . .. Therefore, the number ofunknown elements cij(m) that have to be estimated is N =(1/2)K(K − 1)(2M − 1). Now, define c as the N × 1 vectorcomposed of all elements of cij(m). Then, the parameter vectorfor localization may be expressed as q = [xP, yP,γ

T , cT ]T .Using (22) again, we can then calculate the following FIMelements:

Λij = [g]T(�P)iQ−1[g](�P)j

+12

tr{Q−1[Q](�P)i

Q−1[Q](�P)j

}(36)

Λi,2+k = [g]T(�P)iQ−1[g]γk

+12

tr{Q−1[Q](�P)i

Q−1[Q]γk

}(37)

Λi,(2+K+m) =12

tr{Q−1[Q](�P)iQ

−1[Q]cm}

(38)

Λ(2+k),(2+l) = [g]TγkQ−1[g]γl

+12

tr{Q−1[Q]γk

Q−1[Q]γl

}(39)

Λ(2+k),(2+K+m) =12

tr{Q−1[Q]γk

Q−1[Q]cm}

(40)

Λ(2+K+m),(2+K+n) =12

tr{Q−1[Q]cmQ−1[Q]cn

}(41)

where i = 1, 2, j = 1, 2, k = 1, . . . ,K , l = 1, . . . ,K , m =1, . . . ,N , n = 1, . . . ,N and derivatives equal to

[Q](�P)i =([ρ](�P)iρ

T + ρ[ρ]T(�P)i

)◦ γγT ◦C

+ 2σ2w

Ndiag

([ρ](�P)i

◦ γ)

(42)

[Q]γk=ρρT ◦

[γγT

]γk

◦C+ 2σ2w

Ndiag

(ρ ◦ [γ]γk

)(43)

[Q]cm =ρρT ◦ γγT ◦ [C]cm (44)

[g](�P)i= [ρ](�P)i ◦ γ, [g]γk

= ρ ◦ [γ]γk, [g]cm = 0 (45)


which in turn depend on the trivial derivative [C]cm and[ρk,m](�P)i =[ρk,m]ϕk

[ϕk](�P)i with [ϕk]x=−(yP − yk)/d2k

and [ϕk]y = (xP − xk)/d2k. Hence, the CRB on the RMSE of

location estimation can be stated as

RMSE =

√(Λ−1)11 + (Λ−1)22 (46)

RMSE > RMSER =√(

Λ−1R

)11

+(Λ−1

R

)22

(47)

RMSE > RMSEL =√(

Λ−1L

)11

+(Λ−1

L

)22

=

√Λ11 +Λ22√

Λ11Λ22 −Λ212

. (48)

The latter RMSEs RMSER and RMSEL are again CRBs that as-sume the knowledge of some of the nuisance parameters. Equa-tion (47) assumes the knowledge of the covariance, i.e., ΛR ∈R(2+K)×(2+K) and (ΛR)ij = (Λ)ij , i, j = 1, . . . , 2 +K , and(48) assumes the knowledge of both the covariance and RSSs,i.e., ΛL ∈ R

2×2 and (ΛL)ij = (Λ)ij , i, j = 1, 2. Note that wecould interrelate RMSE, RMSER and RMSEL also through thework in [25] as alternative expressions to (46)–(48). As willbe seen in Section VII, the approximation RMSE ≈ RMSER

is almost a perfect match and greatly reduces the size of theFIM. For an even more compact FIM (and faster computations),it is also possible to apply the latter approximation RMSE ≈RMSEL that is likewise a lower bound on the performance yetnot quite as tight as the CRB (46). Illustrations will be given inSection VII.

V. ALGORITHMS

In the following, we shortly describe the SLS algorithm, asector-power based DoA/RSS estimator that was proposed in[11], and a modified Stansfield estimator [10], which computesa TX location estimate using DoA/RSS estimates from multipleRXs. We will be referring to SLS DoA/RSS estimation withsuccessive Stansfield location estimation as SLS-ST. SLS andSLS-ST will be used to compare the derived CRBs to practicalDoA/RSS and localization algorithms.

A. SLS DoA/RSS Estimator

Due to the nonlinear dependence of the sector-powerson the DoA and RSS, classical estimation approaches suchas least squares and maximum likelihood (ML) would re-quire iterative solutions. To overcome associated problemssuch as divergence, the SLS DoA/RSS estimator was there-fore proposed in [11]. SLS finds the two neighboring sec-tors i, j ∈ 1, . . . ,M, |i− j| = 1 at RX k, such that εk,i +εk,j > εk,m + εk,n for all other neighboring sectors n,m =1, . . . ,M, n,m = i, j, |n−m| = 1. The DoA ϕk is then, withhigh probability, in between the orientations ϑi and ϑj of thetwo sectors i and j. Given an adequate tuning of the antenna andpractical working conditions [11], the incoming signal at theRX is of meaningful strength only in very few sectors, whereasthe remaining measurements are likely to be very noisy. Hence,SLS discards the sector-powers from all sectors, except εk,i andεk,j , and finds the DoA ϕk that minimizes the norm JSLS =

(pk,i/pk,j − ρk,i/ρk,j)2, where pk,m = εk,m − σ2

w, m = i, j.Using an approximation of the antenna’s main beam as inSection II-A, the DoA at RX k is obtained in closed form as

ϕk = ϑij + κ(ln pk,i − ln pk,j) (49)

where ϑij = (ϑi + ϑj)/2 and κ = −π/(2M ln as) assumingϑi > ϑj for simplicity and without loss of generality. Exploit-ing the direct mapping between the DoA estimate ϕk and itscorresponding RSS estimate γk via the antenna’s radiation pat-tern ρk,m = [ζ(ϕk − ϑm)]2, SLS consequently estimates theRSS at RX k as

γk =12

[(εk,i − σ2

w

)ρk,i

+

(εk,j − σ2

w

)ρk,j

]. (50)

To prevent large estimation errors when the signal is severelyattenuated, we have also added a validation check, as discussedin detail in [11].

B. Stansfield Location Estimator

The Stansfield estimator was first introduced in [3]. It isclosely related to the ML DoA fusion algorithm. The ML DoAfusion scheme is a nonlinear least squares function that has tobe solved by iterative algorithms such as the Gauss–Newtonmethod. The iterative algorithms suffer from divergence, whichmakes the ML DoA fusion scheme not reliable in practical sce-narios [26]. The Stansfield algorithm is a linear approximationof the ML fusion scheme resulting in the following closed-formTX location estimate:

�P = (ATWA)−1ATWb (51)

where

A =

⎡⎢⎣sin(ϕ1) − cos(ϕ1)

......

sin(ϕK) − cos(ϕK)

⎤⎥⎦ (52)

b =

⎡⎢⎣

x1 sin(ϕ1)− y1 cos(ϕ1)...

xK sin(ϕK)− yK cos(ϕK)

⎤⎥⎦ (53)

and the weighting matrix is defined as W = diag[w1, . . . , wK ].In here, we use the modified Stansfield algorithm [10], wherethe contribution of each RX is weighted with the correspondingRSS by setting W = diag[γ1, . . . , γK ], which is equivalent toweighting the DoA estimates with the SNRs, given that thenoise variance σ2

w is the same at every RX. Consequently, eachRX communicates the estimated DoA and the estimated RSS tothe FC. However, at the FC, the RSS is only used to indicate thequality of the DoA estimates, whereas TX location informationis not extracted from the RSS. Therefore, the SLS-ST algorithmis lower bounded by the localization CRB that we derivedearlier in this paper. This will be illustrated in Section VII.


VI. ANALYTICAL ERROR MODELS FOR SIMPLIFIED LEAST

SQUARES DIRECTION-OF-ARRIVAL ESTIMATION

A. Bias and RMSE for a Given DoA

To derive the bias and RMSE of SLS DoA estimation, wewill make the following two simplifying assumptions. First, weassume that SLS always picks the two sectors i and j, suchthat ϑj < ϕk < ϑi. Second, we assume that the variances ofthe power pk,l ∼ N (μk,l, σ

2k,l), μk,l = ρk,lγk, and σ2

k,l = σ2k,l,

l = i, j, are much smaller than their respective means, i.e.,σ2k,l � μk,l. Intuitively, both of these assumptions are valid

if the number of samples N and the SNR are sufficientlylarge. As will be seen in Section VII, these assumptions leadto a very good approximation of the RMSE in most of therelevant cases. In the derivation, the first assumption allowsus to disregard the sector-selection process, which is difficultto model analytically. Due to the second assumption, we canwell approximate the logarithms ln pl, l = i, j, in (49) with afirst-order Taylor series around the mean of pk,l, i.e., ln pk,l ≈ln μk,l + (pk,l − μk,l)/μk,l. This leads to an approximation ofthe DoA estimation (49) that is equal to

ϕk= ϑij+κ

(ln

μk,i

μk,j+pk,i − μk,i

μk,i− pk,j − μk,j

μk,j

). (54)

The expected value of ϕk can be calculated as E[ϕk] = ϑij +κ ln(μk,i/μk,j). We then obtain an approximation for the biasof SLS DoA estimation as (see Appendix C)

bias[ϕk] ≈ bias[ϕk] = E[ϕk]− ϕk = 0. (55)

Therefore, SLS is unbiased for a large SNR, which was alreadyverified by numerical results in [11]. Similarly, we calculatethe variance of ϕk as var[ϕk] = κ2(σ2

k,i/μ2k,i + σ2

k,j/μ2k,j) and

approximate the MSE of SLS DoA estimation for a given DoAusing first-order Taylor series, i.e., MSE[ϕk] ≈ MSE[ϕk] =var[ϕk] + [bias[ϕk]]

2 such that the RMSE becomes

RMSESLS[ϕk] ≈κ√N

√a−1k,i + a−1

k,j . (56)

B. RMSE for Uniformly Distributed DoA

For a uniformly distributed DoA ϕk ∼ U(−π, π), we canapproximate the average RMSE of SLS DoA estimation as

RMSESLS ≈

√√√√√ 1Δθ

Δϑ∫0

MSE[ϕk + ϑj ]dϕ (57)

where we use the assumption of equal sectors and integrateonly over one sector instead of over all DoAs from −π to π.The solution of (57) involves integrals of the form

∫ x

0 et2dt,

which can be expressed using, among others, imaginary errorfunctions [27], defined as

erfi(x) = −i erf(ix) =2√π

x∫0

et2

dt. (58)

TABLE IIEXAMPLE VALUES OF (60) AND (61)

An implementation of the imaginary error function can befound, e.g., in MATLAB’s symbolic toolbox. Using (58) in(57), we then obtain an approximation of the average RMSEof SLS DoA estimation equal to

RMSESLS≈κ

√2√π

N√ln a−1

s

[f(as)

4SNR2 +g(as)√2SNR

]+

2N

(59)

with

f(as) = erfi(

2√ln (a−1

s ))

(60)

g(as) = erfi(√

2 ln (a−1s )

)(61)

where we have highlighted explicitly that f(as) and g(as)depend only on the side-sector suppression as. As will bediscussed in Section VII, as should be set to a fixed value inthe interval [0.2, 0.4] for good performance. Therefore, it is notnecessary to evaluate the imaginary error functions containedin (59). Instead, the functions (60) and (61) can be treated asconstants with a few exemplary values for practically relevantcases of as shown in Table II.

C. Asymptotic RMSE

As shown in (56) and (59), the asymptotic RMSEs for largeSNR are equal to

RMSESNRSLS = RMSE

SNRSLS =

π√2NM | lnas|

(62)

whereas the asymptotic RMSEs for large N are equal to

RMSENSLS = RMSE

NSLS = 0. (63)

We notice that (62) is slightly different from the result in[17], where RMSESNR

SLS,[17] = π√

ln(1 + 1/N)/(√

2M | ln as|)due to a different approximation of the sector-power distribu-tion. However, ln(1 + x) =

∑∞n=1((−1)n+1xn/n) for −1 <

x ≤ 1, such that RMSESNRSLS,[17] ≈ RMSESNR

SLS for sufficientlylarge N (e.g., N > 10). Moreover, when comparing (30) with(62) and (34) with (63), we see that SLS is an asymptoticallyefficient estimator in the number of samples N but not in theSNR. We will discuss this in more detail in the following.

VII. EVALUATION, ILLUSTRATIONS, AND ANALYSIS

A. DoA/RSS Estimation

DoA and RSS estimation performance is studied, assuming auniform distribution of incoming DoAs, i.e., ϕ ∼ U(−π;π) inan individual arbitrary RX. We limit our simulation to 100 steps


Fig. 5. Performance of DoA estimation as a function of the side-sector sup-pression as for a uniform distribution of DoAs. Parameters: M = 6; N = 100;and SNR = 5 dB.

Fig. 6. Performance of RSS estimation as a function of the side-sector sup-pression as for a uniform distribution of DoAs. Parameters: M = 6; N = 100;and SNR = 5 dB.

in the interval ϕk ∈ [−Δϑ/2; Δϑ/2] due to the periodicity ofthe radiation pattern. For each DoA step, we average over 1000realizations to obtain numerical results for the CRBs and RMSEof SLS for uniformly distributed DoA. The CRBs (27) and (29)as well as the RMSE of DoA/RSS estimation with SLS dependon the RSS only via the SNR = γk/σ

2w. Therefore, we set the

RSS to an arbitrary fixed value and adjust the noise power σ2w

to control the SNR.Figs. 5 and 6 show the dependence of DoA/RSS estimation

performance on the side-sector suppression as. For valuesas > 0.2 the CRBs on DoA/RSS estimation can be per-fectly approximated by the other lower bounds in (27)–(29),i.e., RMSEDoA ≈ RMSEa

DoA and RRMSERSS ≈ RRMSEaRSS.

Similarly, the analytical expression for the RMSE (59) ofSLS, i.e., RMSESLS, is also very accurate for as > 0.2. Thiscoincides with the region where SLS is unbiased and thuslower bounded by the CRBs [11]. As a guideline for sectorized

Fig. 7. Performance of DoA estimation as a function of the SNR for a uniformdistribution of DoAs. Parameters: as = 0.4; M = 6; and N = 100.

antenna design, we are interested in finding the value as = as,othat results in the lowest RMSE of DoA and RSS estimation.Unfortunately, as,o is a function of the SNR, whereas as is avalue that is fixed during the design process for most sectorized

antennas. In the case of asymptotically large SNR, RMSESNRDoA

increases monotonically with as, whereas RRMSESNRRSS is inde-

pendent of as. In other words, if the noise can be neglected,the sectors in a sectorized antenna should be as selective aspossible to obtain the most accurate DoA and RSS estimates.However, for a finite SNR, a reduction of as results in a decreasein the effective SNR, i.e., SNRk,m = ρk,mSNRk, for all DoAsϕ = ϑm. This of course affects the performance adversely suchthat the choice of as is a compromise. We have found that aside-sector suppression as in the interval [0.2, 0.4] results ina good tradeoff between low and high SNR values. Therefore,we can conclude that the derived approximation of the CRBs isvalid for adequately tuned antennas.

As shown in (23)–(29), the dependence of the DoA andRSS estimation CRBs on the SNR are similar. Therefore, weonly show the performance of DoA estimation as a function ofthe SNR in Fig. 7. Again, the approximations of the CRBs in(27)–(29) are highly accurate. Generally, SLS performs closeto the CRB, with the smallest gap to the CRBs in the interval0 dB < SNR < 15 dB. In contrast to the earlier asymptoticexpression for the RMSE [17] (not explicitly shown in Fig. 7 forclarity), RMSESLS describes the performance of SLS very wellalready for low SNR > 5 dB instead of only for the saturatedregion SNR > 15 dB.

In the saturated region, SLS starts to diverge slightlyfrom the CRB. This is because SLS discards the sector-powers from all sectors, except for the two neighbor-ing sectors i and j (see Section V-A). Thus, for SLS,the CRB approximation in (27) becomes RMSESLS,b

DoA =√(N + 2)−1(ak,iρk,i + a2k,j ρk,j)

−1. For high SNR, ak,i and

ak,j attain their maximum with ak,i ≈ ak,j ≈ 1 such that

RMSESLS,bDoA =

√(N + 2)−1(ρ2k,i + ρ2k,j)

−1, which is indepen-

dent of the SNR and ultimately limits the performance of


SLS. Naturally, we can make a similar argument for theCRB in general and conclude that the RMSE saturates at

RMSESNRDoA ≈

√(N + 2)−1(

∑Mm=1 ρk,m)−1 for a very large

SNR. However, for an increasingly high SNR, those sectorsthat were discarded by SLS are responsible for the performanceincrease in the CRB as the respective weighting ak,m, m =1, . . . ,M,m = i, j, will attain its maximum significantly laterthan ak,i and ak,j . This also manifests itself in the ratio of theasymptotic RMSEs in (62) and (32). For N + 2 ≈ N , we can

write RMSESNRSLS /RMSE

SNRDoA ≈

√(M3 −M)/6, which is an

increasing function with M because the number of sectors thatSLS discards also grows with M . For the case of M = 6, SLSsaturates at an almost sixfold larger RMSE compared with theCRB.

For comparison, we have also added a curve for the CRBof DoA estimation with DCAAs and parallel processing of allantenna branches in Fig. 7. As an example DCAA, we havechosen a uniform circular array (UCA) with Na = 8 anten-nas. Assuming next that the signal bandwidth is significantlysmaller than the carrier frequency, we can then approximatethe Na × 1 steering vector b(ϕ) of the UCA as bi(ϕ) =1/Na exp(−j2πr/λ cos(ϕ− ϕi)), i = 1, . . . , Na where theantennas are arranged on a circle with radius r and angles ϕi,and λ is the wavelength. Now, we can obtain the respective DoACRB for parallel processing CRBPL, using [16, Eq. (5)]. Inaccordance with our signal model for sectorized antennas (seeSection II-B), we moreover set the array output covariance [16,Eq. (1)] to R = γkbb

H + σ2wI. Through straightforward calcu-

lations, we then notice that the CRB is inversely proportional tothe SNR, i.e., CRBPL ∼ SNR−1. Therefore, asymptotically forlarge SNRs, we have CRBSNR

PL = 0. This is different comparedwith the CRB for sectorized antennas, which is nonzero evenin the asymptotic case, as discussed earlier. This behavior isalso visible in the respective RMSEPL curve in Fig. 7. Fur-thermore, one can observe that, under the assumptions made,sequential processing with sectorized antennas actually outper-forms parallel processing with the UCA for low SNRs. Thisresult is, however, strongly dependent on the assumed antennacharacteristics, namely, the number of antenna units Na andsteering vector characteristics in the UCA and the exact radia-tion pattern in the sectorized antenna deployed in the sequentialprocessing. At low SNRs, in general, it is evident that the levelsof the sidelobes in the spatial processing characteristics playan important role since the sector power values are stronglyinfluenced by noise. Thus, drawing universal and generallyapplicable conclusions between the relative performances ofparallel and sequential processing approaches is difficult andis always subject to made assumptions related to the radiationpatterns.

A detailed illustration of the influence of M on the perfor-mance of DoA estimation is shown in Fig. 8. As before, theapproximation for the CRB and the RMSE of SLS are veryaccurate for all considered values of M . From (27), (29), and(59), we observe that RMSEDoA, RMSERSS, and RMSESLS

behave approximately proportional to 1/√N (forN � 2). Due

to space limitations, we have not included a separate plot for thedependence of the estimation performance on N .

Fig. 8. Performance of DoA estimation as a function of the number of sectorsM for a uniform distribution of DoAs. Parameters: as = 0.4; N = 100; andSNR = 5 dB.

B. Localization

Localization performance is analyzed by numerically averag-ing the derived CRBs in (46)–(48) over 1000 randomly drawnTX and RX locations. For each set of RX locations, we alsodraw a separate channel realization with channel coefficientsthat we model according to |hk|2 = c0d

−αk 10−Fk/10, where

c0 is the (constant) average multiplicative gain at referencedistance, α is the path-loss exponent, and 10−Fk/10 is thei.i.d. lognormal shadowing with Fk ∼ N (0, σ2

f ). Similarly, weobtain the RMSE of SLS-ST location estimation by numericallyaveraging over 500 randomly drawn RX locations, each with100 random sector-power realizations. Thereby, we assume thefollowing basic parameter settings. The RXs are uniformlyplaced inside a circle of radius R = 150 m with a protectiveregion R0 = 5 m, which is centered around the TX with loca-tion �P = [0, 0]T for simplicity and without loss of generality.The TX transmit power σ2

s is +20 dBm (100 mW), and thesignal occupies a bandwidth of B = 20 MHz. On the RXside, the measurement noise power σ2

w is −70 dBm (100 pW),the sampling frequency is assumed fs = (1/B), the RX an-tennas have M = 6 sectors, and a side sector suppression ofas = 0.4. The path-loss exponent is α = 4, and for now, theshadowing standard deviation is σf = 0 dB. Later, we will alsoanalyze the impact of shadowing on the localization perfor-mance. Based on these parameters, the average SNR can easilybe shown to be SNR = 10 log10[(2σ

2s (R

2−α0 −R2−α))/((α −

2)σ2w(R

2 −R20))] = 33 dB.

Fig. 9 shows the localization performance as a functionof the number of RXs K . Increasing K leads to a betterperformance; however, the biggest gain is achieved in the rangeK < 20. In general, the CRB on localization is practicallyidentical with the approximation (47), i.e., RMSE ≈ RMSER.This means that the knowledge of the sector-power covariance(12) is essentially not significant for the localization. In con-trast to DoA estimation, the knowledge of the RSSs, however,is actually beneficial for localization, i.e., RMSEL < RMSE.Nevertheless, the difference between the bounds (46) and (48)is not very significant such that, in addition to RMSER, RMSEL


Fig. 9. Performance of localization as a function of the number of RXs fora uniform distribution of RXs. Parameters: as = 0.4; M = 6; N = 100; andσf = 0 dB.

also serves as an approximation of the localization CRB withreasonable accuracy. In Fig. 9, we furthermore observe that theperformance of SLS-ST seems to saturate at a higher level thanthe CRBs. This can in part be explained by the fact that SLSdoes not efficiently utilize large SNRs (see Section VII-A). As aconsequence, SLS-ST also does not fully benefit from RXs withhigh SNR that become increasingly likely with increasing K .Additionally, the gap between the CRBs and SLS-ST indicatesthat the intermediate step of turning the sector-powers into DoAestimates and communicating those to the FC limits the perfor-mance of localization. This is supported by the fact that, for agiven set of DoAs and RSSs, Stansfield is known to achieve therespective CRB asymptotically in K [4]. Furthermore, as wehave seen earlier, the performance of SLS is quite close to theDoA/RSS CRB. Hence, the combination of SLS and Stansfieldshould yield close-to-optimal results in a localization systemwhere estimated DoAs and RSSs are communicated insteadof sector-powers. In Section III, we have shown, by means ofNeyman–Fisher factorization, that sector-powers at individualRXs are a sufficient statistic for localization. However, to thebest of our knowledge, such a factorization is not possible forDoA/RSS estimates. Thus, the observed gap in performancebetween SLS-ST and the CRB can also be understood as a conse-quence of sector-powers being a sufficient statistic for localiza-tion (at individual RXs), whereas DoA/RSS estimates are not.

Fig. 10 shows the relationship between the number of sam-ples and the localization performance. Up to N = 200 samples,the performance of the localization system increases signifi-cantly with increasing N , for both the CRB and SLS-ST. How-ever, for N > 200, the increase in performance flattens out. Wehence conclude that a localization system with N < 200 pro-vides a good performance/complexity tradeoff. Fig. 11 showsthe performance of localization as a function of the numberof sectors M . As expected, the performance improves withincreasing M . Similarly, to what we have observed for N , theimprovement in localization performance flattens out for largerM such that, e.g., antennas with M = 10 sectors seem like agood performance/complexity tradeoff.

Fig. 10. Performance of localization as a function of the number of samplesN for a uniform distribution of RXs. Parameters: as = 0.4; K = 10; M = 6;and σf = 0 dB.

Fig. 11. Performance of localization as a function of the number of sectorsM for a uniform distribution of RXs. Parameters: as = 0.4; K = 10; andσf = 0 dB.

Fig. 12 shows the influence of shadowing on the localizationperformance. Since shadowing results in additional randomnessin the simulations, we have increased the number of randomlydrawn TX locations, RX locations, and channel realizationsfrom 1000 to 5000. Note that, for a shorter run time of thesimulations, we have not calculated the exact CRB but onlythe two approximations RMSER and RMSEL. As discussedearlier, RMSE ≈ RMSER, which we confirmed also for σf = 0by simulations with less realization than those shown in Fig. 12.A channel with shadowing standard deviation σf = 0 resultsin a spread of the SNRs such that it becomes more probablefor the RXs to observe larger and smaller SNRs. As shownin Fig. 12, this SNR spread does not affect the localizationCRB significantly. While this might seem surprising at first,it is well in line with [4], where the localization CRB forDCAAs was observed to even decrease with increasing σf .For the SLS-ST estimator, on the contrary, an increase inσf results in a decreased performance. This is explained bythe nonlinear relationship between SNR and DoA estimation


Fig. 12. Performance of localization as a function of the shadowing standarddeviation σf for a uniform distribution of RXs. Parameters: as = 0.4; K = 10;M = 6; and N = 100.

performance, which we have observed in Fig. 7. As a con-sequence of this nonlinearity, the spread of the SNR resultsin worse DoA estimates due to lower SNRs that, on average,cannot be compensated by the better DoAs due to higher SNRs.

VIII. CONCLUSION

In this paper, we have thoroughly investigated the perfor-mance of DoA/RSS estimation and TX localization using RXdevices that are equipped with sectorized antennas. Towardthat end, we first derived the CRBs on DoA/RSS estimationwith sectorized antennas and studied their asymptotic behavior.We then developed an accurate expression for the MSE of apractical DoA estimation algorithm, i.e., the SLS algorithm,compared its performance to the bound and determined thealgorithm’s asymptotic properties. Finally, we derived the lo-calization CRB in a general form that encompasses scenarioswhere the location is directly estimated from the so-calledsector-powers and two-step scenarios where each RX deviceindividually estimates the DoA/RSS, which are then centrallyfused into a location estimate. A comparison of a two-steplocalization algorithm to the CRB reveals that skipping theintermediate DoA/RSS estimation step may improve the overalllocalization performance. Moreover, we provided simple andaccurate approximations for all CRBs. Overall, we believe thatthe insights gained in this paper and the derived performancebounds will help to understand, design, and improve the perfor-mance of sectorized antenna-based localization algorithms andsystems.

APPENDIX ADERIVATION OF SECTOR-POWER COVARIANCE

The covariance of sector-powers originating from sector j atRX i and sector l at RX k is equal to

E [(εi,j−gi,j)(εk,l−gk,l)] =E [εi,jεk,l]− gi,jgk,l

= 2Rijkl + 2Mijkl − gi,jgk,l (64)

with the terms

Rijkl =1N2

N−1∑m=0

N−1∑n=0

E[�{ri,j(nTs)}2 �{rk,l(mTs)}2

]

=1N2

N−1∑m=0

N−1∑n=0


](65)

Mijkl =1N2

N−1∑m=0

N−1∑n=0


]

=1N2

N−1∑m=0

N−1∑n=0


](66)

where the interchangeability of real and imaginary signal partsis due to the assumption of a circularly symmetric signal. Wethen obtain for i = k ∨ j = l

Rijkl =ρi,jρk,lN2

N−1∑m=0

N−1∑n=0

E[�{si,j(nTs)}2 �{sk,l(mTs)}2

]

+14(ρi,jγi + ρk,lγk)σ

2w +

14σ4w (67)

Mijkl =14ρi,jγiρk,lγk +

14(ρi,jγi + ρk,lγk)σ

2w +

14σ4w

(68)

and for i = k ∧ j = l

Rijij =ρ2i,jγ

2k

N2

N−1∑m=0

N−1∑n=0

E[�{s2i,j(nTs)

}�{s2i,j(mTs)

}]

+ ρi,jγiN + 2

2Nσ2w +

N + 24N

σ4w (69)

Mijij =14ρ2i,jγ

2i +

N + 22N

ρi,jγiσ2w +

N + 24N

σ4w. (70)

Given our stationary signals and noting the easily verifiedidentity

N−1∑m=0

N−1∑n=0

f(n−m) = Nf(0) +N−1∑p=1

(N − p) [f(p) + f(−p)]

(71)

that holds for any function f : N → R, we can then obtain(12)–(18) by using (67)–(70) in (64). �


APPENDIX BDERIVATION OF ASYMPTOTIC BOUNDS

First, we derive the DoA estimation CRB for asymptoticSNR. With limSNRk→∞ ak,m = 1, we get

limSNRk→∞

(Γ−1)11=β4

16(N+2)[κ2(ϕk)− 1

M [κ1(ϕk)]2] . (72)

Due to the periodicity of κp(ϕk), we can equally writeκp(ϕk) =

∑Mm=1[P(ϕk)− ϑm]p, which is easier to handle

mathematically than the earlier expression in Lemma 1. Wethen obtain

limSNRk→∞

(Γ−1)11 =β4

16(N + 2)∑M

m=1 ϑ2m

(73)

since∑M

m=1 ϑm = 0. For both even and odd M , we ob-tain

∑Mm=1 ϑ

2m = (Δ2ϑM/12)(M2 − 1), where we have

used the well-known identities∑M

m=1 m = (1/2)M(M + 1)and

∑Mm=1 m

2 = (1/6)M(M + 1)(2M + 1). Finally, we cansolve (72) and are then able to obtain the RMSE (31). In asimilar fashion, we can derive the RSS estimation CRB forasymptotic SNR as

limSNRk→∞

(Γ−1)22=γ2k

N+ 2

[1∑M

m=1 ϑ2m

[P(ϕk)]2+

1M

]. (74)

In contrast to (72), (74) depends on the DoA. The average RSSCRB for asymptotic SNR is calculated through integration overthe period of κp(ϕk), i.e.,

1Δϑ

Δϑ/2∫−Δϑ/2

limSNRk→∞

(Γ−1)22dϕk (75)

which finally yields the RRMSE (33). �

APPENDIX CDERIVATION OF SIMPLIFIED LEAST SQUARES

DIRECTION-OF-ARRIVAL ESTIMATION BIAS

It is easy to show that E[ϕk] can be written as

E[ϕk] = ϑij +2κβ2

[M(ϕ− ϑj)

2 −M(ϕ− ϑi)2]. (76)

Next, we choose ϑj = 0 and ϑi = Δϑ. Since our antenna wasassumed to have equal sectors and since the mapping of anglesis arbitrary, this choice does not result in a loss of generality butsimplifies the handling of M(ϕ− ϑm), m = i, j. Recallingour earlier assumption that ϑj < ϕk < ϑi, we can now writeM(ϕ− ϑm)2 = (ϕ− ϑm)2. Therefore, we obtain

E[ϕk] =Δϑ

22κβ2

[2Δϑϕ−Δ2ϑ

](77)

from where we arrive at (55) through straightforwardcalculations. �

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Janis Werner was born in Berlin, Germany, in 1986.He received the Dipl. Ing. degree in electrical en-gineering from Dresden University of Technology(TUD), Dresden, Germany, in 2011. He is currentlyworking toward the Ph.D. degree with Tampere Uni-versity of Technology, Tampere, Finland.

He is currently a Researcher with the Depart-ment of Electronics and Communications Engineer-ing, Tampere University of Technology. His mainresearch interests include localization with an em-phasis on directional antenna-based systems and the

utilization and further processing of location information in future-generationmobile networks.

Jun Wang received the B.Eng. degree in electronicinformation engineering from Beijing University ofTechnology and the M.Phil. degree in electricalengineering from City University of Hong Kong,Hong Kong, in 2006 and 2009, respectively, andthe Ph.D. degree in electrical engineering from theUniversity of California, Los Angeles, CA, USA, in2014, under the supervision of Prof. D. Cabric.

Since 2014, he has been with Broadcom Corpo-ration as a Research Scientist in wireless communi-cation systems design. His research interests include

the acquisition and utilization of primary-user location information and variousissues related to spectrum sensing in cognitive radio networks.

Aki Hakkarainen received the M.Sc. degree (withhonors) in communication electronics from TampereUniversity of Technology (TUT), Tampere, Finland,in 2007. He is currently working toward the Ph.D.degree with TUT.

From 2007 to 2009, he was a Radio-Frequency(RF) Design Engineer with Nokia, Salo, Finland.From 2009 to 2011, he was a Radio System Special-ist with Elisa, Tampere. He is currently a Researcherwith the Department of Electronics and Commu-nications Engineering, TUT. His research interests

include digital signal processing methods for RF impairment mitigation anddigital signal processing for localization.

Danijela Cabric (SM’14) received the Dipl. Ing.degree from the University of Belgrade, Belgrade,Serbia, in 1998; the M.Sc. degree in electri-cal engineering from the University of California,Los Angeles, CA, USA, in 2001; and the Ph.D. de-gree in electrical engineering from the University ofCalifornia, Berkeley, CA, in 2007.

She was also a member of the Berkeley WirelessResearch Center, University of California, Berkeley.Since 2008, she has been with the Departmentof Electrical Engineering, University of California,

Los Angeles, where she is currently an Associate Professor. Her researchinterests include novel radio architecture, signal processing, and networkingtechniques to implement spectrum sensing functionality in cognitive radios.

Dr. Cabric served as the Technical Program Committee Cochair for theEighth International Conference on Cognitive Radio Oriented Wireless Net-works in 2013. She currently serves as an Associate Editor for the IEEEJOURNAL ON SELECTED AREAS IN COMMUNICATIONS (cognitive Radioseries) and the IEEE COMMUNICATIONS LETTERS. She received the SamueliFellowship in 2008, the Okawa Foundation Research Grant in 2009, and theHellman Fellowship and the National Science Foundation Faculty Early CareerDevelopment (CAREER) Award in 2012.

Mikko Valkama (M’01) was born in Pirkkala,Finland, on November 27, 1975. He received theM.Sc. and Ph.D. degrees (both with honors) in elec-trical engineering from Tampere University of Tech-nology (TUT), Tampere, Finland, in 2000 and 2001,respectively.

In 2003, he was a Visiting Researcher with theCommunications Systems and Signal Processing In-stitute, San Diego State University, San Diego, CA,USA. He is currently a Full Professor and the Depart-ment Vice Head with the Department of Electronics

and Communications Engineering, TUT. His main research interests includecommunications signal processing, estimation and detection techniques, signalprocessing algorithms for software-defined flexible radios, cognitive radio, full-duplex radio, radio localization, fifth-generation mobile cellular radio, digitaltransmission techniques such as different variants of multicarrier modulationmethods and orthogonal frequency-division multiplexing, and radio resourcemanagement for ad-hoc and mobile networks.

Dr. Valkama has been involved in organizing conferences. He served asthe Publications Chair for the 2007 IEEE Workshop on Signal ProcessingAdvances in Wireless Communications, held in Helsinki, Finland. He receivedthe Best Ph.D. Thesis Award from the Finnish Academy of Science and Lettersfor his dissertation entitled “Advanced I/Q signal processing for widebandreceivers: Models and algorithms” in 2002.

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