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Theoretical Analysis of the Capture Probability inWireless Systems with Multiple Packet Reception

CapabilitiesAndrea Zanella, Member, IEEE, Michele Zorzi, Fellow, IEEE,

Abstract In this paper, we address the problem of computingthe probability that r out of n interfering wireless signalsare captured, i.e., received with sufciently large Signal toInterference plus Noise Ratio (SINR) to correctly decode thesignals by a receiver with multi-packet reception (MPR) andSuccessive Interference Cancellation (SIC) capabilities. We startby considering the simpler case of a pure MPR system withoutSIC, for which we provide an expression for the distribution of the number of captured packets, whose computational complexityscales with n and r . This analysis makes it possible to investigate

the system throughput as a function of the MPR capabilities of the receiver. We then generalize the analysis to SIC systems. Inaddition to the exact expressions for the capture probability andthe normalized system throughput, we also derive approximateexpressions that are much easier to compute and provide accurateresults in some practical scenarios. Finally, we present selectedresults for some case studies with the purpose of illustrating thepotential of the proposed mathematical framework and validatingthe approximate methods.

Index Terms Capture, wireless, collision, successive interfer-ence cancellation, multi-packet reception

The nal version of this manuscript appeared in IEEE Transactions on Communications, vol.60, no.4, pp.1058-1071, April 2012.DOI: 10.1109/TCOMM.2012.021712.100782

I. INTRODUCTION

O NE of the main problems in wireless systems is themutual interference produced by overlapping radio sig-nals emitted by different transmitters, that might prevent thecorrect decoding of some or all of the signals involved, anevent that is often referred to as collision . This situationmay be observed, for instance, in random access systems,where transmissions from different sources take place withoutcoordination, or in dense wireless sensor networks, wheremultiple sensor nodes may require to transmit their data to thesink node in the same time slot, or yet in ad hoc networks, inparticular in the presence of hidden nodes. When the various

signals are received with signicantly different powers, the so-called capture effect may take place, i.e., the strongest signalsmay survive the collision and be correctly decoded despite theinterference due to the other signals [1].

The capture phenomenon may signicantly impact the sys-tem performance, in particular for systems with Multi-PacketReception (MPR) capabilities, i.e., capable of decoding mul-tiple overlapping packets out of a collision event, provided

The authors are with the Department of Information Engineering, Universityof Padova, Padova, 35131, ITALY e-mail: {zanella,zorzi}@dei.unipd.it.

Preliminary and partial versions of this paper have been presented at IEEEISIT09 and IEEE ICC2011.

that the received signals satisfy the capture condition. 1 MPRhas been recently shown to be a promising solution forhigh-capacity wireless networks [2], [3]. In fact, a betterunderstanding of the ability of the receiver to correctly decodeone or more signals, as a function of the statistical distributionof the received signal powers, may enable a more effectivedesign of the transceiver architecture and the optimization of the transmission protocols.

A. Related literatureThe relevance of the signal capture phenomenon in mobile

radio systems has been recognized since long, as testied bythe rather rich literature on the topic.

In [4] the authors assume that a signal is captured wheneverthe strongest interferer is sufciently far apart from the des-ignated receiver, according to a statistical geometry approach.In [5], capture is assumed to occur if the arrival instants of the rst and second signals are sufciently apart. A capturemodel based on the number of simultaneous transmissions isconsidered in [6], whereas in [7] it is assumed that a packetis captured only if during its overall transmission period noother signal is received with higher power. The stability of the slotted Aloha system with MPR capabilities is studied in[8], where the number r of captured signals is modeled as arandom variable whose probability mass distribution dependsonly on the collision size n , i.e., the overall number of overlap-ping transmissions. The authors show that the MPR capabilitycan stabilize Aloha and the maximum stable throughput whenn goes to innity is equal to the mean value of r . The paper,however, does not explicitly focus on the derivation of thecapture distribution, which is instead obtained for some samplecases by using simple capture models, as those describedabove.

Successively, the analysis of the capture phenomenon wasextended to include basic physical propagation aspects. Inthis case, we nd two different approaches for modelingsignal capture in radio systems, one based on the protocolmodel and the other on the physical model . The protocolmodel gives a geometric interpretation of signal propagation,according to which the capture of a signal only depends onthe distance between the different transmitters and the commonreceiver. In [2], [3], in particular, it is assumed that the receiver

1 In this paper, we use the term capture to indicate the condition underwhich a signal can be decoded by an MPR system. However, if the numberof captured signals exceeds the MPR capability of the receiver, the signalsin excess are actually not decoded, even though they experience the capturecondition.


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can capture multiple signals transmitted within its receptionrange, provided that all other (interfering) transmitters are ata distance from the receiver larger than a given interferencerange. This approach makes it possible to carry out elegantperformance analysis and to derive closed form bounds forthe system capacity in different scenarios, but relies on anidealized and rather unrealistic model.

On the other hand, the physical model, which we adoptin this paper, explicitly includes the physical propagationphenomena and the cumulative character of interference inthe capture model, considering the random distribution of thesignal powers at the receiver and introducing the Signal-to-Interference-plus-Noise-Ratio (SINR) criterion to determinethe capture probability [9], [10]. If P j denotes the power of the j th signal at the receiver, the SINR for that signal is denedas

j =P j

h = j P h + N 0(1)

where N 0 represents the background noise power. A signal jis said to be captured and, hence, it is potentially decodable

despite the interference produced by the other overlappingsignals, if j > b , with b > 0 representing the so-calledcapture threshold of the system.

The capture threshold b is a system parameter, whose valuedepends on the structure of the receiver and, more generally,on the properties of the communication system. For instance,conventional narrowband systems with a single antenna neces-sarily have capture threshold b 1 and, as a consequence, atmost one signal at a time can be captured by the receiver [10],[11]. Conversely, in Code Division Multiple Access (CDMA)systems, the capture threshold can be signicantly less than 1,depending on the length of the spreading codes that are usedto distinguish the signal of each user. These systems, in fact,

trade the spectral efciency of each user with a SINR gainproportional to the length of the spreading code, which mayresult in a capture threshold b < 1. Hence, CDMA systemsare capable of decoding up to 1/boverlapping signals, thusexhibiting multi-packet reception capabilities .

In general, however, the signals at the receiver are affectedby random attenuation factors, so that the number of signalsthat can be actually captured is also random. An analysis of thecapture probability for b < 1 has been proposed in [11], [12].In particular, in [11] the authors derive an expression for theprobability that there is at least one signal above the capturethreshold, which is signicantly more difcult to compute thanin the case b

1.

A different approach to enhance the system capacity in thepresence of interference is based on Successive InterferenceCancellation (SIC), which was rst proposed in [13]. Broadlyspeaking, SIC is an iterative reception scheme where sig-nals are generally decoded one at a time, starting from thestrongest, i.e., the one with the largest SINR [14]. After thesignal is decoded, it is canceled from the aggregate receivedsignal and, then, the next strongest user is decoded in thesubsequent decoding iteration. Therefore, SIC systems areinherently capable of multi-packet reception [15]. A recentanalysis of the network capacity of SIC systems is presentedin [16], where the authors provide bounds on the transmission

capacity of a wireless ad hoc network applying a statisticalgeometry approach. In [17], it is shown that SIC can increasethe stable throughput of Aloha systems up to 0.793 packets perslot by measuring the aggregate signal power at the receiverand using ne-grained power control at the transmitters. How-ever, the effect of non-ideal channel state information or powercontrol on the capture probability with a SIC receiver is notaddressed. B. Novel contributions

In this work, we advance the state of the art in the analysisof the capture phenomena by proposing a novel mathematicalframework that, compared to the existing mathematical ap-proaches, provides more general results and scales better withthe size of the problem, i.e., the number of overlapping signals.This framework makes it possible to readily evaluate theperformance of random access systems with MPR capabilitiesin various scenarios, and provides a useful tool for systemdesign and dimensioning. More specically, in this paper weprovide the following original contributions.

We rst consider the simpler and more classical case of

pure MPR systems without SIC. For these types of systems,we derive an analytical expression of the complete capture probability distribution , i.e., we give the expression of theprobability C n (r ) that exactly r signals out of n are abovethe capture threshold for any 0 r n . The numericalevaluation of this expression is scalable with the values of both n and r , unlike the expression in [11] that involvesn nested integrations, and whose complexity is thereforeexponential in n . From the capture probability, we obtainthe exact expression of the normalized system throughput S n (R), dened as the mean number of successfully decodedpackets out of a collision of size n , when there is a limit R(called MPR capability) to the number of signals that can besimultaneously decoded. 2

Second, we extend the analysis to SIC systems, wherethe decoding process involves successive decoding iterations.At each iteration, the signals with power above the cap-ture threshold are decoded and, then, subtracted from thecompound received signal, leaving a fraction z of residualinterference power. This process is repeated sequentially, untilno further signal is captured or the maximum number K of interference cancellation iterations is reached. We provide theexpression of the probability C

( s )

n (r ; K ) that r signals outof n are decoded by a receiver capable of performing upto K interference cancellation iterations. Once again, fromthe capture probability we can derive the normalized systemthroughput as a function of the maximum permitted numberK of SIC iterations.

Third, we derive simple approximate expressions, based onthe central limit theorem, for a lightweight computation of thecapture probabilities. In particular, we propose a novel, coarseapproximation of the capture probability that can be computedwithout resorting to numerical integration. Furthermore, we

2 As a side result, we generalized the capture probability expression to aheterogeneous scenario, where users may have different capture thresholdscorresponding, for instance, to different modulation schemes or spreadingfactors. For space constraints, however, this generalization has not beenreported here, and can be found in [19].


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propose a simple recursive method to obtain an accurateapproximation of the normalized throughput for SIC systemswith extremely low computational complexity.

Finally, to assess the potential of the proposed method andillustrate the type of analysis it enables, we provide a selectionof results obtained in some reference scenarios.

I I . S YSTEM MODEL

The most natural application scenario for this study is theuplink of a wireless access network, such as a slotted Alohasystem or an IEEE 802.11 cell. In this scenario, n radioterminals simultaneously transmit their signals to a commonreceiver. For the sake of simplicity, we only consider the caseof synchronized transmissions of equal duration, which is aclassical assumption in the related literature [16][18]. Ananalysis of the capture phenomenon in case of asynchronoustransmissions of different length for IEEE 802.11 cells is pre-sented in [20], where however the capture model is based onthe number of simultaneous transmissions, without consideringthe SINR aspect.

The received powers P j , with j = 1 , 2, . . . , n , are assumedto be independent and identically distributed (iid) randomvariables, with Probability Density Function (PDF) f P (x)and Cumulative Distribution Function (CDF) F P (x) that de-pend on the transmit powers, the statistical distribution of the distance between the transmitter and the receiver, andthe stochastic phenomena (fading, shadowing) that affect thesignal propagation. The compound signal at the receiver is thesuperposition of the n overlapping radio signals transmittedby the users, with power equal to

=n

j =1

P j + N 0 , (2)

where N 0 accounts for the noise power. For the sake of simplicity, in the sequel we omit the noise term that is expectedto be negligible with respect to the other terms. 3

A signal with power P j is captured if its SINR j , givenby (1), is larger than the capture threshold b,

j =P j

P j> b . (3)

For pure (no SIC) MPR systems, we assume that thereceiver successfully decodes all the captured signals, up tothe MPR capability R : if the number r of captured signalsexceeds R , the remaining signals are actually not decoded.

For SIC systems, we suppose that at each iteration thereceiver is capable of decoding all the signals that experiencethe capture condition. The decoded signals are then subtractedfrom the aggregate received signal before performing thenext decoding iteration. The signal cancellation requires thereceiver to reconstruct the waveform of the decoded users,an operation that involves the accurate estimation of thechannel impulse response and errorless message decoding.When these operations are imperfect, the signal cancellationleaves some residual power that increases the noise level

3The analysis can be extended to include the noise term, though at the costof a more complex notation with no additional insight.

experienced at the successive decoding iterations. To modelthis idiosyncrasy of the interference cancellation process weassume that the cancellation of a signal received with powerP leaves a residual interference power of zP , with z 1.This proportional model for the residual interference power,though simple, is rather common in the literature [16], [21],and has been justied for some specic modulation schemes[22], [23]. In any case, the derivation presented in this papercan be adapted to more general models as well, at the cost of a more cumbersome notation.

The iterative SIC reception process stops when all signalsare decoded, or no signals satisfy the capture condition in adecoding cycle. Furthermore, we assume that the SIC processcan be performed up to K times, with K = 0 correspondingto the case of a pure MPR system with no SIC.

III . A NALYSIS OF PURE MULTI -RECEIVER SYSTEMS

In this section, we consider pure MPR systems, lacking anySIC capability. The aim is to determine the expression of theprobability

C n (r ) = Pr [ r signals out of n are captured ] . (4)

Computing (4) in MPR systems is difcult because theSINR of the different users are coupled. One possible wayto handle this interdependency is to apply the law of totalprobability, conditioning on each random variable {P j }, asdone in [12]. However, this method generates a number of nested integrals that grows linearly with the number of usersand, therefore, the resulting expression is not practical formore than a few users. With our approach, instead, we obtainfor any n an expression with at most three nested integrals,which can be easily computed with numerical methods.

To begin with, we dene the capture condition in terms of the overall aggregate received power by rewriting (3) in thefollowing form

P j > b (5)where b = b/ (1 + b) is termed modied capture threshold .In this way, we express the capture condition in terms of minimum received signal power, b, which is the same forall the users.

Now, the derivation of the expression of C n (r ) developsalong four basic steps. First, we apply basic combinatorialanalysis to express the unordered joint probability functionC n (r ) in terms of the ordered joint probability cn (r ) of capturing the rst r signals and missing the remaining n r .Second, we condition on = x in such a way that thecapture threshold b on the right-hand side of (5) becomesdeterministically equal to xb so that, due to the independencyof the received signal power from different nodes, we caneasily compute the probability that r nodes are received withpower above xb and the others fall below the threshold. Third,we apply Bayes rule to get the conditional PDF of the totalreceived power being x, given that r received signals havepower in the interval (xb, ), and the other n r have powerin the interval [0, xb]. The conditional received signal powersconstrained in these intervals maintain their independency


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and are represented as auxiliary random variables h (u, v),where u and v denote the extremes of the interval. We canthen express the conditional PDF of the aggregate power evaluated in x, given that r signals have power in the interval(xb, ) and n r in the interval [0, xb], as the n -foldconvolution of the PDFs of the auxiliary random variables h (u, v). This convolution can be efciently performed in thefrequency domain, which represents the last step. We can nowstate the nal result and, then, detail the derivation. 4

Theorem 1: For any positive n , and any values 0 r n ,the probability of capturing r out of n packets can be com-puted as

C n (r ) =nr 0 (1 F P (xb)) r F P (xb)n r (6)

(xb ,) ( ) r (0 ,xb ) ( ) n r ei2x d dxwhere

(u,v ) ( ) =

v

u

f P (a)F P (v)

F P (u)

ei 2a da (7)

for u v and zero otherwise, and i = 1.Proof: The proof of the theorem develops along the four

basic steps described above.First step: Let U 0 be the set of captured signals and U 1 theset of missed (non-captured) signals. Due to the symmetry of the problem, the r captured signals can be arbitrarily chosen.Hence, without loss of generality, we can write

C n (r ) =nr

cn (r ) (8)

where cn (r ) is the probability that signals U 0 = {1, 2, . . . , r },are captured and signals U 1 ={

r + 1 , . . . , n

}are missed. In

formula:

cn (r ) = Pr[ P 0 > b, P 1 b] (9)where, for brevity, we adopted the compact notation P 0 > vin place of {P j > v, j U 0}and similarly for the oppositeinequalities.Second step: Applying the total law of probability on , weget

cn (r ) = 0 Pr [P 0 > xb , P 1 xb| = x] f (x)dx=

0

Pr [ E

| = x]f (x)dx (10)

where f (x) is the PDF of the aggregate received power ,and we set E = {P 0 > xb , P 1 xb}, for compactness.Third step: Applying the rule of Bayes, we obtain

cn (r ) = 0 f (x|E )Pr[ E ]dx= 0 f (x|E )(1 F P (xb)) r F P (xb)n r dx (11)

4Even though the statement and the proof of Theorem 1 are given herewith reference to continuous distributions of the received powers, the sameresult can be shown to hold true for any probability distributions as well.

where

f (x|E )= limh0Pr nj =1 P j(xh, x ] P 0 > xb , P 1 xb

his the conditional PDF of the aggregate received signal power, given E , i.e., given that the rst r signals have power abovethe threshold b = xb, and the remaining n r have powerbelow such a threshold.Fourth step: We now introduce the parameterized randomvariable (y) dened as

(y) =r

h =1

h (y, ) +n

h = r +1

h (0, y) , (12)

where, for any 0 u v, h (u, v) are iid random variableswith common PDFf (u,v ) (a) =

f P (a)F P (v) F P (u)

, for a(u, v] ; (13)

and zero otherwise. In practice, (13) is the conditional PDF

of P given that P (u, v].5

Due to the statistical independence of the terms in (12), thePDF f (y ) (a) of (y) is equal to the multi-fold convolutionof f (y, ) (a) and f (0 ,y ) (a). In the frequency domain, theFourier Transform (FT) ( y ) ( ) of f ( y) (a) becomes

( y ) ( ) = (y, ) ( )r (0 ,y ) ( )

n r (14)

where (u,v ) ( ) is the FT of f (u,v ) (a), which is given by(7). The function f (y ) (x) can be obtained from (14) throughinverse FT, that is

f ( y ) (x) =

(y, ) ( )r (0 ,y ) ( )

n r ei2x d .(15)

We now notice that, for any x, the function f (y ) (x) withy = xb is equal to f (x|E ). Hence, (11) can be expressed as

cn (r ) = 0 f (xb ) (x) (1 F P (xb)) r F P (xb)n r dx .(16)Replacing (15) into (16) and the result into (8) we nally get(6).

Note that this result is completely general and holds forany spatial distribution of the transmitters and any propagationmodel, provided that the received powers are iid. The actualevaluation of (6) might require numerical methods for the

computation of the two nested integrals and of the FT (7),when it cannot be expressed in closed form. 6In any case, thecomputational complexity of (6) is limited for all the cases of interest and, most importantly, it is essentially independent of r and n , so that our method is very scalable. On the otherhand, the expression provided in [11, Eq. (19)] only gives theprobability of capturing at least one signal (which is equal to1C n (0) ), and requires the explicit computation of n nested

5 If Pr [ P = 0] > 0, the PDF f (0 ,v ) (a ) must be dened for a [0, v]rather than a (0, v].

6 An efcient way to compute the FTs and their inverse is described in [18],[19]


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integrals, whose complexity grows exponentially with n , andtherefore cannot be used except for very small collision sizes. 7

We now turn our attention to the normalized system throughput , dened as the expected number of packets that can besuccessfully decoded in a slot in which n users transmit. Thisperformance gure has been deeply analyzed in the litera-ture, mainly for: (i) systems with single reception capability(R = 1 ), i.e., able to decode only one packet even whenmultiple signals experience SINR > b , or (ii) systems withinnite reception capability ( R = ), i.e., capable of correctlyreceiving all the packets that satisfy the capture condition [11].

In this work, we generalize the analysis to systems thatcan actually decode at most R 1 simultaneous signals(e.g., due to hardware limitations), even when the number of signals above the capture threshold is larger than R . Denotingby S n (R) the normalized throughput of a system with MPRcapability R n , we haveS n (R) =

R 1

r =1r C n (r )+ R

n

r = R

C n (r ) =R 1

r =1r C n (r )+ RQ n (R)

(17)where Qn (R) = nr = R C n (r ) is called full-capacity reception probability , since it denotes the probability that R or moresignals are above the capture threshold and, consequently, themulti-reception capability of the receiver is fully exploited.Using (6) into (17), we can compute the normalized sys-tem throughput for any value of the reception capability R .In particular, the normalized throughput of single receptionsystems is equal to S n (1) = Qn (1) = 1 C n (0) , whereasthe normalized throughput of innite-reception systems isS n () = E [r |n], where E [r |n] denotes the expected value of the number of captured signals (out of n ) and can be computedas in [10].

IV. A NALYSIS OFSUCCESSIVE -I NTERFERENCE -C ANCELLATION SYSTEMS

The derivation of the capture probability to systems withSIC basically follows the same rationale as in Sec. III, butis slightly more complex and requires a more cumbersomenotation due to the iterative nature of the decoding process.

Let k be the iteration at which the reception process ends,i.e., in which the last capture occurs. Furthermore, let U h bethe set of signals that are decoded at the hth iteration, withh = 0 , 1, . . . , k . The signals that remain to be decoded at theend of the reception process, if any, are collected in the setU k+1 . The aggregate power of the signals in set U h will bedenoted by 8

h =jU h

P j , h = 0 , 1, . . . , k + 1 . (18)

Since the decoded signals are cancelled from the overallreceived signal, leaving a fraction z of their power as residual

7The approach proposed in this section can be easily extended to aheterogeneous scenario, where both the capture threshold and the PDF of the received signal powers may differ among users. The extension, whichunfolds exactly as in the homogeneous case but requires a more cumbersomenotation, can be found in [19].

8If the set U k +1 is empty, the aggregate power k +1 is conventionally setto 0.

interference, the overall signal power at decoding iteration hcan be expressed as

h = zh1

j =0j +

k+1

= h

+ N 0 ,

where the rst sum is zero when h = 0 . Once again, in thefollowing the noise term will be omitted. A signal with power

P j will be captured at decoding iteration h if

P j > h = h b (19)where h is called absolute capture threshold for iteration h.

We wish to determine the expression of the probability

C ( s )

n (r ; K ) = Pr[ r signals out of n are capturedwithin at most K SIC iterations ] . (20)

The derivation unfolds along the four phases described inSec. III that, however, becomes slightly more involved.First step: To begin with, we observe that C

( s )

n (r ; 0) = C n (r )and, for any K , C

( s )

n (0; K ) = C n (0) , which can be computedusing (6). In the following, we hence assume K 1 andr > 0. The capture probability can thus be expressed as

C ( s )

n (r ; K ) =min( r 1,K )

k=0

C n (r ; k) , (21)

where C n (r ; k) is the probability of capturing r signals withthe last capture occurring at iteration k. It shall be noted that,according to the denition of k, it must be r k + 1 , sinceat least one signal has to be decoded at each iteration for thereception process to continue. Hence, let r h denote the numberof signals decoded at the hth iteration, with h = 0 , 1, . . . , k ,and let r k+1 = n

r denote the number of undecoded signals

at the end of the reception process. Due to the symmetry of the problem, we can write

C n (r ; k) =r k

r 0 =1

r r 0 k+1

r 1 =1

r r 0 ... r k 2 1

r k 1 =1Ac(r ) (22)

where, for compactness, we used r to denote the vector

{r 0 , r 1 , . . . , r k+1 }, and we set A= n !r 0 !r 1 !...r k +1 ! . The functionc(r ) is the ordered capture probability for the vector r , i.e.,the probability that signals 1 to r 0 are captured at iterationzero, signals r0 + 1 to r 0 + r 1 are captured at iteration one,and so on, and that signals from kh =0 rh + 1 to n remainundecoded at the end of the reception process.

We now need to distinguish the case k < K , when thereception ends because no signal is captured at iteration k +1 ,from the case k = K , when the reception process is terminatedbecause the maximum allowed number of SIC iterations hasbeen reached. In the following, we derive the expression of c(r ) for k < K , and then we explain how to adjust the resultfor the case k = K .

Recalling (19), we can express the ordered capture proba-bility for k < K as

c(r )=Pr[ P 0> 0 P 1 > 1P k > k , P k+1 k+1 ](23)


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where we adopted the same compact notation introduced inSec. III. 9

Second step: Applying the total probability theorem withrespect to the random variables h , we obtain

c(r ) = f (g ) Pr P 0 > 0 P 1 > 1 (24) P k > k , P k+1 k+1 = g dg0 dgk+1

where we used and g in place of {0 , , k+1 }, and{g0 , , gk+1 }, respectively, and we set 10

h = zh1

j =0

gj +k+1

= h

g b , h = 0 , . . . , k + 1 .

The function f (g ) in (24) denotes the joint PDF of evaluated in g .Third step: Now, applying the rule of Bayes we get

c(r ) = f (g |E )Pr[ E ]dg0 dgk+1 (25)

where E = k+1h =0 E h with 11 E h = {P h (h , h 1]} forh = 0 , . . . , k , and E k+1 = {P k+1 [0, k+1 ]}. The functionf (g |E ) is the conditional joint PDF of given E , evaluatedin g , i.e.,f (g |E ) = limh 0

Pr [ (g h , g ]|E ]h0h1 hk+1

.

We observe that the events in E are mutually independent, sothat we can write

Pr [ E ] = F P (k+1 )r k +1k

h =0[F P (h1) F P (h )]r h . (26)

Furthermore, the conditional PDF of given E can also befactorized as

f (g |E ) =k+1

h =0

f h (gh |E h ) . (27)

Fourth step: We now introduce the family of parameterizedauxiliary random variables

h (u, v) =r h

=1

h, (u, v) , h = 0 , . . . , K + 1 ; (28)

where h, (u, v) are iid random variables with PDF as in (13).The PDF f h (u,v ) (x) of h (u, v) is hence given by the rh -

fold convolution of f (u,v ) (x) and can be obtained as

f h (u,v ) (x) = (u,v ) ( ) r h ei 2x d . (29)9If U k +1 is empty, the inequality P k +1 k +1 is trivially veried.10 When h = 0 , the rst sum is zero.11 For preserving the expression symmetry, we introduced the dummy

parameter 1 = .

We now notice that, for any x, the function h (u, v)(x) isequal to f h (x|E h ) when u and v correspond to the limits of the interval in E h . Hence, for k < K we have

c(r ) = 0 F P (k+1 )r k +1k

h =0

(F P (h1) F P (h ))r h

k

h =0

( h , h 1 ) ( )

r he

i2gh

d (30)

(0 , k +1 ) ( ) r k +1 ei 2g k +1 d dg0 dgk+1 .For k = K , the ordered capture probability becomes

c(r )=Pr [ P 0> 0 P 1 > 1P K > K , P K +1 K ](31)which is very similar to (23), except for the fact that the upperbound of P K +1 is now K rather than K +1 . Hence, (30)can be directly extended to the case k = K by dening

K +1 = K = zK 1

j =0

gj +K +1

= K

g b .

Putting all the pieces together, we can nally express (21)as12

C ( s )

n (r ; K ) =min {r 1,K }

k=0

r k

r 0 =1

r r 0 k+1

r 1 =1

r r 0 ... r k 2 1

r k 1 =1A

0F P (k+1 )r k +1

k

h =0

(F P (h1 ) F P (h ))r h

k

h =0 ( h , h 1 ) ( ) r h ei 2g h d (32) (0 , k +1 ) ( ) r k +1 ei2g k +1 d dg0 dgk+1 .

It shall be noted that the number of nested integrals growsproportionally to (and hence the computational complexitygrows exponentially with) K , which is however expected tobe limited in practical systems for complexity, latency andefciency reasons. Conversely, the computational complexityof (32) grows much more slowly as a function of r and n , so

that our method is very scalable with respect to the numberof users in the system, which can take also large values.For SIC systems it makes sense to consider the mean nor-

malized throughput as a function of the maximum permittednumber of SIC iterations K . Hence, in this case the normalizedsystem throughput becomes

S ( s )

n (K ) =n

r =1rC

( s )

n (r ; K ) . (33)

12 Note that, for K = 0 , (32) returns C n (r ) , though with a slightly differentexpression with respect to (6).


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7

V. L OW -COMPLEXITY APPROXIMATIONS OF CAPTUREPROBABILITY AND THROUGHPUT

Although in most cases the numerical evaluation of thecapture probability distributions (6) and (32), and of thenormalized throughput functions (17) and (33) is affordable,sometimes it might be preferable to resort to approximatemethods that provide fairly good results at a much lower

computational cost. In the following we propose some possibleapproximations that trade off results accuracy for numericalcomplexity. We rst propose approximations for the captureprobability distributions and, thereafter, we turn our attentionto the normalized throughput expressions.

A. Capture probability approximations

The main issue in computing (6) and (32) consists in thenumerical evaluation of the PDF of (xb) and h (u, v), re-spectively. However, the computation of these functions can begreatly simplied by resorting to the Central-Limit-Theorem(CLT). In fact, for sufciently large r h , the distribution of terms like r h=1 h, (u, v) that appear in the expressions of

C n (r ) and C ( s )

n (r ; K ) can be approximated by a Gaussiandistribution, with mean r h m (u,v ) and variance r h 2 (u,v ) ,where m (u,v ) and 2 (u,v ) are the mean and variance of therandom variable (u, v), provided that they exist and are nite.

For instance, the PDF of the parameterized random variable(y), dened in (12), can be approximated as

f ( y ) (a)1

2 2r (y)exp

(a m r (y))2

22r (y)

=1

r (y)

a m r (y)r (y)

(34)

where (

) is the standard normal PDF and

m r (y) = rm (y, ) + ( n r ) m (0 ,y ) ;2r (y) = r

2 (y, ) + ( n r )

2 (0 ,y ) . (35)

With a change of variable ( y = xb) and a simple rearrange-ment of the terms, the capture probability C n (r ) for pure MPRsystems can thus be approximated as

C n (r )=nr 0 y bm r (y)br (y) [1F P (y)]

r F P (y)n rbr (y)

dy

(36)

The numerical solution of (36) requires a single integration,which is generally much faster than the numerical solution

of (6), and can therefore be used as a simple approximation.In particular, the approximation is excellent for r = 0 , andC n (0) turns out to be very close to the correct value C n (0)already for n > 4. This result is of particular interest becauseit provides a very simple way to have an accurate estimateof the probability that at least one signal is captured, whichcorresponds to the normalized throughput for single receptionsystems, S n (1) = Qn (1) = 1 C n (0) , and is the performancemetric considered in most of the previous literature on thesubject [9][12].

The same approach can be used to avoid the computationof the innermost numerical integrals in (32). In particular, for

sufciently large values of r h , the CLT approximation for theexpression (29) yields

f h (u,v ) (x)

exp (xr h m ( u,v ) ) 2

2 r h 2 ( u,v )

2 r h 2 (u,v ). (37)

The expression (36) can be further simplied, at the cost of

some additional loss of accuracy. For sufciently small valuesof the coefcient of variation r (y)/m r (y), the Gaussiandistribution function is very narrow around the mean bm r (y),which depends on y as shown by (35). Accordingly, the terms yb

m r (y )b r (y ) in (36) are generally very small when y lies far

away from the mean bm r (y), whereas for ybm r (y) theGaussian term can be approximated by an impulse with areabr (y). Using this approximation in (36), we obtain this newapproximate expression of the capture probability distribution:

C n (r )nr

yY

[1F P (y)]r F P (y)n r , (38)

where Y

is the set of points such thaty = bm r (y) . (39)

The approximate expression (38) does not involve any nu-merical integration, though computing Y may still requirenumerical methods in some cases. Even though this approxi-mation is rather coarse in general, it gives an idea of the shapeof the probability distribution and provides a good estimate of some useful metrics. In particular, the zero-capture probabilityC n (0) and the full capture probability C n (n) are quite wellapproximated by (38), that thus enables the asymptotic anal-ysis of these two important performance measures, as will bediscussed later on.

B. Throughput approximationsWe here propose a rst-order approximation of the capture

probabilities that yields a simple, recursive method to estimatethe normalized throughput of both pure and SIC-capable MPRsystems.

1) Pure multi-receiver case: Here we focus on the normal-ized throughput S n () of pure MPR systems with innitereception capability. The number of decoded signal can beexpressed as r = nj =1 j where j = 1 if the j th signal iscaptured and zero otherwise. Hence, we can write 13

S n () = E [r |n] = n E [ j ] = n Pr [P 1 > I 0] (40)where I 0 = ( N 0 +

nj =2 P j )b . The right-most term of (40)is the probability that a certain signal, say the rst one, is

captured. This probability can be computed in different ways.A rst possibility is to determine the PDF of I 0 as the (n 1)-fold convolution of f P () (neglecting, as usual, the noise term).Otherwise, as in the previous section, we can resort to thecentral limit theorem and approximate I 0 with a Gaussianrandom variable, with mean (n 1)m (0 ,) b and variance(n 1)2 (0 ,) b2 . However, we here prefer to follow yet an-other strategy that yields an even simpler solution. In practice,

13 Note that here we consider the capture threshold b rather than the modiedcapture threshold b .


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8

we approximate I 0 with its mean I 0 = ( n 1)m (0 ,) b .Hence, from (40), we get the following normalized throughputapproximation

S n () = n 1 F P (n 1)m (0 ,) b , (41)which can be directly computed from the CDF of P in anextremely simple manner. As we will see in Sec. VII, despite

its simplicity, (41) provides an accurate approximation of S n () in many cases, in particular when the number of nodesis large and the capture threshold b is small. More importantly,this trivial normalized throughput approximation for pure MPRsystems paves the way for the derivation of the throughputapproximation in SIC systems, which is presented next.

2) Successive Interference Cancellation case: The basicidea consists in computing an estimate of the mean number rhof signals decoded at each iteration h, taking into account theeffect of interference cancellation. For the generic iterationh , we dene the following parameters: n h is the numberof signals that are not yet decoded at the beginning of theiteration, I h is the approximated value of the absolute capture

threshold I h for these signals and h is the additional residualinterference that will result from the cancellation of the rhsignals at the end of decoding iteration h. For h = 0 , we haven 0 = n and

r 0 = n0 Pr [P j > I 0 ] = n 0[1F P (I 0)] , (42)as given by (41). The residual interference left by the cancel-lation of the r0 signals, in turn, will be approximated as

0 = zr 0 E [P |P > I 0] = zr 0m (I 0 ,) .Notice that, when computing the residual interference, themean power of a captured signal is constrained to be larger

than the (approximated) absolute capture threshold I 0 . Analo-gously, the power of the signals considered in the subsequentiteration will be constrained to be lower than I 0 , and thesame applies to the following iterations. Hence, for the genericiteration h > 0, we dene

nh = n h 1

j =0r j ; I h =

h1

j =0j + ( nh 1)m (0 ,I h 1 ) b ;

F (0 ,I h 1 ) (I h ) = Pr [ P I h |P I h 1] ; (43)r h = nh 1 F (0 ,I h 1 ) (I h ) ; h = zr h m (I h ,I h 1 ) .Finally, the approximate normalized throughput for K SIC

iterations is given by

S ( s )

n (K ) =K

j =0

r j . (44)

For b 1 the approximation can be further improved asdescribed in [19].

VI . D EFINITION AND CHARACTERIZATION OF REFERENCESCENARIOS

To exemplify possible applications of the developed analy-sis, we consider three reference scenarios, namely Path Loss

(PL), Rayleigh Fading (RF), and Lognormal (LN), which aredetailed in the following. 14

A. Path Loss model

For the sake of comparison with the previous literature, therst scenario included in our analysis is the so-called Path Loss(PL) model and refers to the case proposed in Section II.E of [11] and in other recent works [24]. The scenario consists of n users uniformly distributed in a circle of radius Dcenteredat the common receiver (e.g., a Base Station of a cellularnetwork, or an Access Point in a WLAN), so that the PDFof the distance r j from the j th user to the common receiveris given by

f r j (a) =2a

D2, for 0 a D,

and zero otherwise. The radio propagation is governed bya simple deterministic path-loss law, with neither fading norshadowing. As in [11], we assume that the received power at adistance r from the transmitter is equal to P (r ) = (1 + r ) ,

where is the path loss coefcient whereas the constant unitterm is added to avoid a physically absurd behavior for r 0.Note that, for the sake of notation simplicity, we omit anymultiplicative constant in the path-loss equation that, in anycase, would be irrelevant in our analysis. Hence, the powerP received from a generic node can be modeled as a randomvariable that takes values in the interval [(D+ 1) , 1], withPDF and CDF given by 15

f P (x) =2

D2 x 2 1 x

1 1 , F P (x) = 1

1 x1

2

D2(45)for (

D+ 1)

x

1. From (45), it is then easy to derive

the PDF of the auxiliary random variable (u, v) that turnsout to be equal to

f (u,v ) (a) =2 a 2 1 a

1 1

D2(F P (v) F P (u)), (46)

for max(( D+ 1) , u) < a min( v, 1), and zero otherwise.The FT of the PDF of (u, v) depends on . For space

constraints, we report here only the result for = 2 that, for(D+ 1) 2 u v 1, turns out to be given as in (47)(see next page) where [g(x)]vu = g(v) g(u), and Ei(m, z ) =

1exp( za )a m da is the exponential integral function.

The statistical mean and power of (u, v), still for = 2 ,are given respectively by

m (u,v ) =log(v) 2 v log(u) + 2 u

D2(F P (v) F P (u)), (48)

M (u,v ) =3(v u) 2( v u)

3D2(F P (v) F P (u)).

14 In [18] we considered an additional scenario, where we combine theclassical path loss model with the Rayleigh fading model.

15 We observe that, for D 10, the PDF obtained with this model isbasically equivalent to that returned by the classical path loss model r when user locations are constrained to be at least one meter apart from thetransmitter, i.e., r [1, D ].


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9

(u,v ) =2 e

i 2 xf

x +i2 3 / 2 f 2erf i2f x

if e i 2 xf

x + i 2f Ei(1 , i 2xf )v

u

D2(F P (v) F P (u))(47)

B. Rayleigh Fading model

In the Rayleigh Fading (RF) scenario, we assume that thepath loss attenuation is the same for all users, as if all transmit-ters were at the same distance from the receiver or they used apower control mechanism that is capable of compensating thelong term path loss attenuation. Therefore, the received signalshave equal mean power, that we normalize to one. However,signals are affected by multi-path fading, so that the receivedpower for the j th user will be an exponentially distributedrandom variable, with PDF and CDF given by

f P (a) = ea , F P (a) = 1 ea , a 0. (49)In this case, the FT of the PDF of (u, v) is given by

(u,v ) ( ) = eu ( i 2 +1)

ev( i 2 +1)

(1 + i 2 )(eu ev ) ,whereas the statistical mean and power are

m (u,v ) =(u + 1) eu (v + 1) ev

eu ev, (50)

M (u,v ) =(2 + 2 u + u2)eu (2 + 2 v + v2 )ev

eu ev.

We observe that, due to the memoryless property of theexponential distribution, the PDF of the auxiliary randomvariable h (u, ), dened in (28), admits the following closedform expression:

f h (u, ) (a) =(a

ur

h)r h

1

(r h 1)! e(a ur h ) ; a ur h .

It is thus possible to avoid the numerical computation of theinverse FT of ( 0 ,) ( )

r h in (32). In particular, it ispossible to get a closed form expression of the full-captureprobability, which is C n (n) = (1 nb)n 1 for nb < 1 andzero otherwise.C. Lognormal power distribution model

Another interesting statistical distribution of the receivedsignal power P is the lognormal model (LN), according towhich the received signals have iid powers with PDF and CDF

f P (x) =1

x 2 e [ln( x ) ]2

2 2

, (51)

F P (x) =12

erfc ln(x)

2 = ln(x)

,

where ln() is the natural logarithm, erfc() is the comple-mentary error function, and () is the standard normal CDF.The parameters and are the mean and standard deviation,respectively, of the associated normal distribution.

The LN model may be considered, for instance, when theterminals are at the same distance from the receiver, but thereceived signals are affected by independent shadowing fadingterms. In this case, the parameters of the PDF can be expressed

as = dB /A and = dB /A where A = 10 / ln(10) ,dB is the mean received power, in dB, whereas dB is thestandard deviation of the shadowing fading terms that rangesfrom 4 dB to 13 dB in outdoor environments [25, Ch. 2, p.50]. The LN model can also be applied to CDMA systemsthat adopt power-control schemes to harmonize the power of the different received signals. Considering for simplicity thehomogeneous-rate case, an ideal power control scheme shallmake all the signals be received with equal target power P ,so that the capture condition (3) is satised by all signals.In practice, however, the actual power received from node j = 1 , 2, . . . , n , in dB, will be equal to P j [dB ] = P [dB ] + j ,where j are iid zero-mean gaussian random variables, withstandard deviation . Perfect power control corresponds to

= 0 dB, whereas imperfect power control usually yields from 1 dB to 4 dB [26]. Thus, P j has lognormal distributionwith = P [dB]/A = ln( P ), = /A .

Unfortunately, the FT of the PDF of (u, v) cannot beexpressed in closed form in this case and, hence, needs tobe computed numerically, e.g., using the recursive methodproposed by Leipnik in [27] and based on the double Taylorexpansion of e(ln x) 2 / (2 2 ) . Conversely, the mean andstatistical power of the auxiliary random variable can beexpressed in closed form as m (u,v ) = I (u, v;1) ; M (u,v ) =I (u, v; 2) , where I (u, v; h) = E (u, v)h =

eh + h2 2 / 2

2( F P (v)F P (u )) erfc+ h 2 ln v 2 2 erfc + h

2 ln u 2 2 .

VII. P ERFORMANCE ANALYSIS

Here we present only a selection of the results obtained inthe three scenarios above, with the purpose of illustrating howthe method proposed in this paper can be used. We rst discussthe case of pure MPR systems and, later, we address the SICcase. Unless otherwise specied, all the gures presented inthis section have been obtained by setting D= 10 , = 2 , = 3 /A , = 0 , and b = 0 .02 or b = 0 .1. Changing the valuesof these parameters will clearly yield different results that,however, preserve the fundamental behavior and propertiesillustrated in the following.

In Figs. 1, 2 and 3 we report the capture probability C n (r )for the PL, RF, and LN scenarios, respectively, as a functionof r . We plot a set of curves obtained by varying n , withb = 0 .02. We observe that in the PL scenario, when n =15, which is well below the upper bound 1/b= 50 on themaximum number of signals that can be potentially captured,the capture probability presents a spike in r = n because themost likely event is that all the n signals are captured (fullcapture). The full capture probability in the RF and LN casesis, instead, lower. The reason is that the full capture requires allthe signals to be received with similar powers, an event that,with the parameters considered in this analysis, is unlikely


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0 5 10 15 20 25 3010

3

10

2

101

100

Number of captured signals (r)

C n

( r )

Path Loss (b=0.02 =2)

n= 15n= 20n= 30n= 100n= 250n= 500n=1000

Fig. 1. Capture probability distributions C n (r ) vs. r in the PL scenariowhen varying the collision size n (b = 0 .02, = 2 ).

0 5 10 15 20 25 3010

3

102

101

100


C n

( r )

Rayleigh Fading (b=0.02)

n= 15n= 20n= 30n= 100n= 250n= 500n=1000

Fig. 2. Capture probability distributions C n (r ) vs. r in the RF scenariowhen varying the collision size n (b = 0 .02).

to occur in the RF and LN scenarios, since the range of thereceived signal power is larger than in the PL case. 16

When n increases, the full capture probability decreases andthe distribution roughly assumes a bell-shaped form in all threescenarios, with mean and variance that progressively decrease.Finally, for very large values of n , C n (0) tends to increase,and the system can capture fewer and fewer signals.

Fig. 4 reports the normalized throughput for the PL andRF scenarios, when varying the number of simultaneoustransmissions n and with b = 0 .1. (To reduce clutter, wedo not report the results for the LN scenario that, with theparameters considered in this analysis, fall in-between the PLand RF results.) It is interesting to note that increasing the

reception capability beyond a certain point yields diminishingreturns. For example, in the case shown, R = 6 alreadyprovides a normalized throughput very close to the maximumpossible, though the number of signals that can be potentiallydecoded with capture threshold b = 0 .1 is 1/b= 10 . Thisresult suggests that it is possible to design radio systems withpartial reception capability that practically attain the sameperformance as systems with innite-reception capability.

Fig. 5 compares the normalized throughput S n (1) = 1 16 This conclusion depends on the relationship between the coverage range

D in the PL scenario and the variance in LN, and different choices can beexpected to yield different results.

0 5 10 15 20 25 3010

3

10

2

101

100


C n

( r )

LogNormal (b=0.02 /A=3)

n= 15n= 20n= 30n= 100n= 250n= 500n=1000

Fig. 3. Capture probability distributions C n (r ) vs. r in the LN scenariowhen varying the collision size n (b = 0 .02, /A = 3 ).

100

101

102

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Number of overlapping signals (n)

T h r o u g

h p u

t : S

n ( R )

R=1R=2R=3R=4R=5R=6

Fig. 4. Throughput S n (R ) for different capture capabilities R vs. n (b =0.1) in the PL (solid) and RF (dashed) scenarios. Note that the normalizedthroughput curve for R = (innite-reception system) is superimposed tothat obtained with R = 6 .

C n (0) of systems with single reception capability R = 1 (themetric considered in most of the previous literature) for differ-ent values of the capture threshold b. Solid curves refer to thePL case, whereas dashed curves are used for the RF scenario(once again, the LN case is omitted here). The exact results(lines) are compared with the approximate values (markers)obtained using C n (0) given by (36) in place of C n (0) in (17).As can be noted, the accuracy of the approximation is verygood in all cases considered. Although not shown in the gure,we have also veried that using the approximate value C n (0)given by (38) still yields excellent results for b

0.1, whereas

for larger values of b the approximation becomes less accurate.For SIC systems we only show the results obtained in the

RF scenario, due to space constraints. In Fig. 6 we reportthe capture probability C

( s )

n (r ; K ) as a function of r , forn = 20 , which is twice the maximum number of signals

1/b that could be captured by a pure MPR system. Thedifferent curves have been obtained by varying K , as indicatedin the legend of the gure, and setting z = 0 .1. The solidlines with white markers refer to the exact solution given by(32), whereas the black markers correspond to the approximatecapture probability values obtained as described in Sec. V-A.

First of all, we can appreciate that the Gaussian approxima-


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100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


T h r o u g

h p u

t :

S n

( 1 )

b=0.1b=0.5

b=2

b=1

b=0.02

Fig. 5. Throughput S n (1) vs. n for single reception systems in the PL(solid lines) and RF (dashed lines) cases, for different values of the capturethreshold b. Markers correspond to the approximate values obtained by using(36) in place of (6): empty squares for PL, and lled circles for RF.

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Number of captured signal (r)

C ( s )

n

( r ; K

)

K=0K=1K=2K=3K=4K=5

Fig. 6. Capture probability distributions C ( s )n (r ; K ) vs. r for n = 20 in

the RF scenario, when varying K (b = 0 .1). Solid line with white markers:exact values. Black markers: approximate values.

tion (37) already yields very good results when the numberof captured signals is just a few. In fact, for K 2 theapproximate results are basically overlapping with the exactones.

Second, we can clearly see that, in this case, SIC is veryeffective, since the maximum number of signals that can becaptured with non negligible probability increases with K .However, the gain becomes progressively less signicant asK grows, because of the imperfect interference cancellation.

Although we do not report here the results for spaceconstraints, we observed that SIC is less effective in highly

congested scenarios, i.e., when n 1/b. In this case, the

probability of early ending of the decoding process is muchlarger, so that it is likely that only a few SIC iterations can beperformed before the reception process ends.

Fig. 7 shows the normalized throughput results in theRF scenario. Solid lines with markers have been obtainedusing the exact equation (17), whereas dashed lines report theapproximate normalized throughput given by (44). The plotabove in the gure refers to the case of perfect SIC, i.e., z = 0 ,whereas the plots below report the normalized throughput forz = 0 .1 and z = 0 .5, respectively. First of all, the ratherclose match between exact and approximate results conrms

100

101

102

0

5

10

15

20

25


T h r o u g

h p u

t

Rayleigh Fading

K=0K=1K=2K=3K=4K=5

(a) Perfect SIC: z = 0

100

101

102

0

5

10

15

20

25


T h r o u g

h p u t

Rayleigh Fading

K=0K=1K=2K=3K=4K=5

(b) Imperfect SIC: z = 0 .1

100

101

102

0

5

10

15

20

25


T h r o u g

h p u

t

Rayleigh Fading

K=0K=1K=2

K=3K=4K=5

(c) Imperfect SIC: z = 0 .5

Fig. 7. Normalized throughput S n (K ) vs. n , when varying K , with b = 0 .1:exact (solid) vs approximate (dashed) results.

the accuracy of the proposed approximation method, at leastfor reasonable values of K . Comparing the different plots,furthermore, we note that, for a certain K > 0, decreasingz yields two benets: rst, the normalized throughput S n (K )increases for any n ; second, the system becomes more robustto multi-signal collision events, as indicated by the growth of the value of n beyond which the normalized throughput startsdecreasing. Nonetheless, we observe that, irrespective of the


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102

101

100

101

1

1.5

2

2.5

3

3.5

4

Capture threshold (b)

S I C

e f f i c i e n c y

(

( K ) )

Rayleigh Fading

K=0K=1K=2K=3K=4K=5

Fig. 8. SIC efciency (K ) vs. b, when varying K and with z = 0 .1.

0 1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

7

8

Number of SIC iterations (K)

S I C

e f f i c i e n c y

( K ) )

Rayleigh Fading

b=0.01b=0.02b=0.04b=0.06b=0.08b= 0.1

z=0

z=0.1

z=0.5

Fig. 9. SIC efciency (K ) vs. K , when varying b and z .

value of z , the normalized throughput gain tends to reduceafter a few SIC iterations. In the case of ideal SIC ( z = 0 ),for example, increasing the SIC capability from K = 4 toK = 5 yields less than 10% of peak normalized throughputgain.

As known from the literature, SIC is effective in widebandsystems, i.e., when b 1 [16]. To better appreciate theimpact of the capture threshold b on the SIC performance,we introduce the SIC efciency metric, dened as

(K ) =max n {S

( s )n (K )}

max n {S ( s )n (0)}

.

In practice, (K ) is the ratio between the peak throughputof a SIC system capable of performing up to K interference-cancellation iterations, over the peak throughput of the samesystem without SIC, and thus provides the maximum relativethroughput gain that can be obtained by performing SIC.Fig. 8 shows the SIC efciency (K ) as a function of thecapture threshold b, for different values of K . White markersconnected by solid lines refer to exact results, whereas black markers have been obtained with the approximate methoddescribed in Sec. V-B. The gure clearly shows that the benetof SIC decreases rapidly as b grows beyond 0.1 and, fornarrowband systems, i.e., when b > 1, the SIC benet isalmost negligible.

An intuitive explanation of this behavior is the following.

First of all, we observe that the larger b, the higher theprobability C n (0) of capturing no signals at the rst decodingcycle and, in turn, the probability that SIC is not performedat all. If (at least) one signal is decoded at the rst iteration,then SIC might be effective to capture some other signals afterinterference cancellation. However, if the captured signal haspower P , then according to (3), the aggregate power of theother signals must be lower than P/b . The larger b, the smallerthe range of power values that the remaining received signalscan take and, in turn, the less effective the SIC mechanism,which instead requires very disparate signal strengths [15].

In Fig. 9 we analyze the impact of the capture thresholdb and the residual interference factor z in SIC systems byreporting (K ) vs K when varying b and for three values of z , as indicated in the gure. The curves have been actuallyobtained using the approximate expression (44), which allowsfor a much faster, though still accurate, computation of theresults. We see that, for a certain z , the curves obtained withdifferent values of b are bundled. This means that the peak normalized throughput achievable with and without SIC scales

almost linearly with1/b

. Moreover, (K ) grows with K butthe gain becomes negligible beyond a certain number of SICiterations, which depends on z .

To exemplify a possible utilization of the proposed analysis,we consider the problem of dimensioning an MPR-enabledaccess point in a hot-spot scenario. We assume that usersare uniformly distributed around the access point, within acell of radius D, and that they all use the same modulationscheme, which yields a capture threshold b. The number of users that simultaneously transmit in a certain slot is modeledas a spatial Poisson process, with density , so that the numbern of overlapping signals is a Poisson random variable withaverage D= D2 . Note that the population of users servedby that access point and, in turn, the mean collision size growquadratically with the cell radius D. The aim is to dimensionthe coverage range D, MPR capability R , and SIC capabilityK of the access point in order to provide good performance,while avoiding costly and useless over-provisioning of thereceiver capabilities. To begin with, we dene the mean uplink throughput as

S D(K ) =

n =1S

( s )

n (K )nDn!

exp(D) .We then use (40) (or (41)) to determine the mean throughputfor a pure (no SIC) MPR system with innite receptioncapability R =

, when varying the cell radius

D. The result

is represented by the curve with white markers in Fig. 10,in three cases, namely i) all nodes transmit with equal andxed power and the received radio signals are affected by pathloss attenuation only (PL), ii) nodes apply long-term powercontrol to compensate path loss attenuation, but signals arestill affected by Rayleigh fading (RF), iii) nodes apply short-term power control that perfectly compensates path loss andfading, except for a residual zero-mean Gaussian error withstandard deviation = 1 .5 dB (LN). Curves refer to densityof = 1 / 200 transmissions per square meter per slot.

From the gure, we can see that there exists an optimalvalue of the cell radius, which depends on the power control


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capabilities of the users. For smaller values of the coveragerange, the MPR capability of the receiver is not fully exploitedbecause the number of parallel transmissions is low. Con-versely, when the coverage range is greater than the optimalvalue, the mutual interference will decrease the throughput.

We also observe that power control yields higher peak throughput, but less robustness to variations of the transmis-sion density. For example, the optimal performance in the LNscenario with = 1 / 200 is obtained with a cell radius of approximately 50 meters. However, doubling the cell radiusthat, in the LN case, is equivalent to fourfold increase onthe user density, the throughput drops almost to zero. Whenusers do not apply power control and signals are affected bypath loss attenuation only, the peak throughout is signicantlylower, but less sensitive to the variations of the transmissiondensity. Therefore, this conguration is suitable when thefocus is on the minimization of the infrastructure complexity(and cost) rather than on throughput maximization.

In the following, we consider the latter scenario and xthe coverage range to D= 50 m. Therefore, the expectednumber of simultaneous transmissions per slot is D = 40with = 1 / 200. From (17), we can then determine theminimum MPR capability Rm of the receiver beyond whichthe performance gain becomes negligible. For instance, wecan set Rm = argmin R {S n (R)/S n () 1 }where is the maximum acceptable performance loss. In our example,setting Rm = 15 yields less than 10% of throughput loss fortransmission density larger than = 1 / 200. Note that the useof SIC largely compensates the throughput loss incurred bylimiting R , since signals exceeding the MPR capability of thereceiver at a certain decoding cycle will be decoded at thesuccessive SIC iteration. The mean throughput S D(K ) whenvarying the number K of admissible SIC iterations can be

estimated using (33), or the approximation (44). The lines withlled markers in Fig. 17 report the throughout S D(1) after asingle SIC iteration, whereas the results for larger values of K are omitted to reduce clutter. As already noted, the rst SICiteration brings along a signicant performance gain, reachingabout 50% of the peak throughput that is attained with K 5iterations.

VIII. D ISCUSSION AND FINAL REMARKS

In this paper, we proposed a novel approach for the compu-tation of the probability that r out of n overlapping signals arereceived with SINR above the capture threshold and, hence,

can be correctly decoded. Different from previous approachespresented in the literature, our method deals with the interde-pendency among the SINR values experienced by the differenttransmitters in a simple and scalable manner. Furthermore, itis rather exible and can be used, with marginal adjustments,in a variety of scenarios.

We rst applied the proposed mathematical framework to amulti-receiver system in a homogeneous scenario, where allusers have the same capture threshold and received signalpower distribution, whereas an extension to heterogeneoussystems is presented in [19]. We then generalized the modelto systems with SIC capabilities. Besides the exact analysis,

20 40 60 80 100 120 1400

5

10

15

20

25

30

35

40

45

50

Cell radius (D)

T h r o u g

h p u

t

PL (K=1)RF (K=1)LN (K=1)PL (K=0)RF (K=0)LN (K=0)

Fig. 10. Mean throughput S D (K ) vs. cell radius D in PL ( ), RF ( ), andLN ( ) scenarios, for an access point with pure (no SIC) MPR capabilities(empty markers) and single SIC (lled markers). The other parameters are:spatial density of transmitting nodes = 1 / 200 users per square meter,capture threshold b = 0 .02, residual interference factor is z = 0 .1.

we also derived approximate expressions for the captureprobability and the system normalized throughput that proveto be in very good agreement with the exact results, whilebeing much easier to compute.

The results provided in this paper can be used to betterassess the performance of a large variety of MAC and ARQtechniques under more realistic channel models, and to investi-gate the effect of a number of parameters, such as the capturethreshold b, the multi-packet reception capability R and theSIC capability K , thus offering a valuable tool for the design,dimensioning and optimization of multi-receiver systems. Forexample, in our case-study analysis, we observed that, withoutSIC, the achievable normalized throughput is generally wellbelow the maximum theoretical value 1/b allowed by a

capture threshold b, so that increasing R beyond a certain levelyields marginal benets. Furthermore, the SIC mechanismbrings signicant performance gains, in particular for wideband systems (with b1), though the normalized throughputincrement rapidly diminishes at each SIC iterations.

Furthermore, the proposed approach makes it possible toanalyze the asymptotic behavior of the zero-capture probabilityC n (0) and, in turn, of the single-receiver throughput S n (1) =1C n (0) , for large values of n . In fact, using (38), the zero-capture probability C n (0) can be approximated as C n (0)

yY F P (y)n F P (y)n where yis the largest valuein Y . From (39) and (35), it is now easy to realize that,

for large n , we have y

nbm (0 ,) , so that we can writeC n (0)F

P (nbm (0 ,) )n . Applying the one-side variant of the Chebishev inequality, we get

C n (0) 1 2 (0 ,)

2 (0 ,) + m2 (0 ,) (nb 1)2

n

(52)

1 n 2 (0 ,)

2 (0 ,) + m2 (0 ,) (nb 1)2

1 W 2

nb2,

where the second inequality follows from (1 x)n 1nx ,whereas in the last step we introduced W = (0 , )m (0 , ) , whichis the coefcient of variation of the received signal powerdistribution, and simplied the negligible terms under the


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assumption nb1. Therefore, in congested scenarios (i.e.,when nb is large), we have that the normalized throughputfor single-receiver systems is upper bounded by S n (1) =1C n (0) W

2

nb 2 which is larger in the case of high-variancesignal power distributions. This observation is in line with theresults reported in Fig. 10 for the hot-spot scenario, accordingto which the throughput for large values of the cell radius Dis better in the PL case than in the RF or LN cases. In fact,increasing D, the mean collision size n grows as quickly asD2 , whereas the coefcient of variation W of the receivedsignal power increases as quickly as D/ log(D) in the PLcase, or remains constant in the RF and LN cases. Therefore,the probability of capturing at least one signal decreases aslog(D)2 in the PL case, and as D2 in the other cases.

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