Effect of Quantization and Channel Errors onCollaborative Spectrum Sensing

Sachin Chaudhari, Visa KoivunenSMARAD CoE, Department of Signal Processing and Acoustics

Helsinki Univ. of Technology, FinlandEmail: {sachin, visa}@signal.hut.fi

Abstract—This paper analyzes the effect of quantization andchannel errors on the performance of collaborative spectrumsensing in cognitive radios. Each secondary user (SU) employsa simple and computationally efficient autocorrelation-based de-tector for Orthogonal Frequency Division Multiplexing (OFDM)signals of the primary user (PU). The local decision statistics inthe form of log-likelihood ratio (LLR) from individual detectorsare quantized and sent to the fusion center (FC). The statisticalproperties of the decision statistics in the presence of quantizationare established. The quantized decision statistics are sent througha channel that may cause errors. The effect of channel errors isincorporated in the analysis through Bit Error Probability (BEP).The detection performance at the fusion center is studied usinganalytical tools and simulations.

I. INTRODUCTION

Distributed detection is an important technology in sensornetworks [1]-[5]. It has been applied to cooperative spectrumsensing in cognitive radio [6]-[8]. Cooperative detection im-proves the detector performance, and increase the coveragein a cognitive radio network. In addition, it mitigates theshadowing and the fading effects through diversity [9].

Most of the present and emerging wireless standards arebased on Cyclic Prefix (CP) based Orthogonal FrequencyDivision Multiplexing (OFDM) signals, e.g., IEEE 802.11a/g(WLANs), IEEE 802.16 (WiMAX), IEEE 802.20 (MBWA),Long Term Evolution (LTE). It is, therefore, fair to assumethat most of the primary users (PUs) will be transmittingOFDM waveforms. In [7], an autocorrelation coefficient baseddistributed detection scheme is presented for detecting CP-OFDM signals. In this method, each secondary user (SU)evaluates autocorrelation coefficient based log-likelihood ratio(LLR) and sends it to a fusion center (FC), which then makesthe final decision. Optimum detection performance can beachieved if each user transmits an exact value of LLR to theFC. However, this may consume too much bandwidth. Thereare two practical ways to reduce the amount of data transferbetween a SU and the FC : quantization and censoring. In [8],the censoring based approach is presented for collaborativedetection of CP-OFDM signals in cognitive radio. In this paperwe use quantization to reduce the number of bits used fortransmitting the decision statistics from the SUs to the FC.

The research of S. Chaudhari and V. Koivunen has received funding fromthe European Community’s Seventh Framework Programme (FP7/2007-2013)under grant agreement no. 216076 (SENDORA).

One quantization approach is to send the local decisions,i.e., one bit hard decisions, to the FC. This problem has beenextensively studied in the distributed detection literature [3].Although this approach reduces the communication cost, suchhard decisions result in a substantial loss of performance. See[4] and references therein. This gives motivation for usingmore than 1 bit for quantization. For multi-bit local decisionstatistics, the local decision spaces have to be optimally de-signed. In [5], the author has given a comprehensive overviewof the optimal quantization schemes for decentralized detec-tion theory. However, the computational complexity of suchschemes is high, i.e., exponential in number of levels requiredfor the quantization. We study the effect of quantization on thedetection performance rather than seeking an optimum quan-tizer. So for the convenience and the mathematical tractabilityof the problem, a simple uniform quantizer is used at eachof the SUs. For the cooperative detection at the FC in thepresence of quantization, Neyman-Pearson criterion is used todesign decision rule at the FC.

Commonly in the literature on distributed detection usingthe quantized values, an assumption has been made that theinformation transmitted by the user is received without anyerrors at the FC. This assumption may not be practical and thecontrol channel used for sending information to the FC may besubjected to propagation effects, interference and collisions.These errors may cause a change in the error probabilitiesat the FC and consequently, a significant performance loss[10]. In [11], the authors treat the issue of designing optimallocal sensor decision rules in the presence of a non-idealchannel. However, [10]-[11] consider the case of sending a 1-bit decision to the FC. In this paper, we analyze the detectionperformance at the FC when SUs send the quantized softdecision statistics over an erroneous channel. This paper in-corporates the channel effect in terms of Bit Error Probability(BEP). The distribution of the received LLRs in the presenceof quantization and channel errors is established at the FC.Using these distributions, the performance of the detectionscheme at the FC is analyzed. In essence, a framework isprovided to get the design parameters (1) the number of bitsrequired for the quantization and (2) the BEP, such that theloss in cooperative detection performance in the presence ofquantization and channel errors is negligible as compared tothe ideal case of transmitting the exact LLRs over an error-freechannel.

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This paper is organized as follows. In Section II, the sys-tem model for autocorrelation coefficient based collaborativespectrum sensing of an OFDM signal in cognitive radio ispresented. Section III presents the performance of the detectionscheme in the presence of quantization. After that, we analyzethe performance in the presence of quantization and channelerrors in Section IV. Section V presents the simulation results.Finally, Section VI concludes the paper.

II. SYSTEM MODEL FOR COOPERATIVE DETECTION

We assume that there are Ns number of SUs trying to detecta PU waveform from a CP-OFDM system. Each SU sees anindependent channel between itself and the PU. Let H0 denotethe null hypothesis that the OFDM based PU is inactive, whileH1 denote the alternate hypothesis that the OFDM based PUis transmitting. For a CP-OFDM signal, if we denote Td as thenumber of useful symbols and Tc as the number of symbolscorresponding to the cyclic prefix in an OFDM block, then theautocorrelation coefficient ρ at lag Td is non-zero under H1.Based on this property, the hypothesis test can be formulatedin terms of ρ [7] as:

H0 : ρ = 0

H1 : ρ = ρn, (1)

where ρn = Tc

Td+Tc

SNRn

1+SNRn. Here SNRn is the signal to

noise ratio (power ratio) at the nth user. The autocorrelationcoefficient based LLR for nth SU is given [7] by

Ln = −M log(1− ρ2n) +2Mρn(ρn − ρn)

1− ρ2n, (2)

where ρn is the maximum likelihood estimate of autocorrela-tion coefficient based on M + Td symbols (M � Td) of thereceived signal at the nth SU. Under the two hypotheses, thedistributions of the Ln can be derived from the distribution ofρn [7] and are

H0 : Ln ∼ N (mn0, σ2n0)

H1 : Ln ∼ N (mn1, σ2n1) (3)

where

mn1 = −M log(1− ρ2n),

mn0 = −M log(1− ρ2n)−2Mρ2n1− ρ2n

,

σ2n1 = 2Mρ2n,

and σ2n0 =

2Mρ2n(1− ρ2n)

2.

Now, each SU evaluates Ln and sends it to the FC, whichmakes a final decision. Under the assumption of conditionalindependence, the optimal test statistic at the FC will be Tf =∑Ns

n=1Ln. The distribution of Tf can be obtained from the

distribution of Ln. The Neyman-Pearson Fixed Sample Size(FSS) test can be designed for maximizing the probability ofdetection with a constraint on the false alarm probability. See[7] for details on the detector at the FC and its performance.

III. EFFECT OF QUANTIZATION

In this section, we only consider the effect of quantizationon detector performance at the FC. Consider the followingcollaborative sensing scenario. Each SU quantizes its autocor-relation based LLR Ln and sends it to the FC, which makesthe final decision based on these quantized LLRs. Let d be thenumber of bits available for quantization of the SU statistics,and D = 2d be the number of quantization levels. LetLsu,n denote the quantized value of the LLR (Ln) transmittedby nth SU. Hence, Lsu,n will take one of the D allowedvalues depending on Ln. The quantizers at each of the localsensors are considered independent. For simplicity, an uniformquantization is assumed. The quantization region at each of thelocal sensors is (−mp0,n,mp1,n) where mp0,n = 3σn0+|mn0|and mp1,n = 3σn1 + mn1. The quantization interval isΔn =

mp0,n+mp1,n

D. The quantization region is chosen here

such that 99.9% of the distribution area gets covered for eachhypothesis. The quantization levels for the quantizer at the nth

SU are given by

li,n = −mp0,n + (i− 0.5)Δn for i = 1, . . . , D. (4)

Let Ln denote the set {li,n, i = 1, . . . , D}. If t0,n, . . . , tD,n

are the thresholds of the quantization region, then

t0,n = −∞, tD,n = ∞,

ti,n =li,n + li+1,n

2for i = 1, . . . , D − 1. (5)

Therefore, the probability mass function (pmf) for the quan-tized LLR Lsu,n under the different hypotheses is given by:

P (Lsu,n = li,n|Hj) =

∫ ti,n

ti−1,n

f(Ln = x|Hj)dx

for i = 1, . . . , D; j = 0, 1. (6)

Here f(Ln|Hj) denotes the distribution of Ln under the Hj

hypothesis.Consider that all the transmitted quantized LLRs from the

SUs are received error-free at the FC. Under the assumption ofindependence of the quantized LLRs, the optimal fusion ruleat the FC is again, their sum [12]. Therefore, the test statisticis Tq =

∑Ns

n=1Lsu,n. For the sum of independent discrete

random variables, the pmf is given by the convolution of thepmf’s of the corresponding individual random variables [13]-[14]. Hence the pmf of test statistic Tq is

p(Tq) = p(Lsu,1) ∗ p(Lsu,2) ∗ . . . ∗ p(Lsu,Ns), (7)

where p(a) denotes the pmf of a discrete random variablea and ‘*’ denotes the convolution of the discrete randomvariables. See [13] for details of the exact convolution ofthe discrete random variables and [14] for the implementa-tion. If Lq denotes the set of values Tq can take, then Lq

contains the sum of possible combinations of the values insets L1, . . . ,LNs

. The maximum number of elements in Lq is

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DNs . Now, the Neyman-Pearson test at the FC is given by

Tq < ηq Decide H0

Tq = ηq Decide H1with probability γ1

Tq > ηq Decide H1, (8)

where the threshold ηq and the randomization parameter γ1depend on the maximum allowed false alarm probability α

and are given by

ηq = minxq∈Lq

{xq : P (Tq > xq|H0) < α},

and γ1 =α− P (Tq > ηq|H0)

P (Tq = ηq|H0). (9)

The probability of detection in the presence of quantization isgiven by

Pd = P (Tq > ηq|H1) + γ1P (Tq = ηq|H1). (10)

IV. EFFECT OF CHANNEL ERRORS

In this section, we study the effect of quantization andchannel errors on the detection performance at the FC. Theobjective is to obtain the design parameters for the quantizationand the channel errors such that the cooperative detectiondoes not suffer significant performance loss as compared tothe ideal case. To achieve this objective, we need to derivethe distribution of the sum test statistic at the FC so that theperformance of the detector can be evaluated. The importantstep in this derivation is to obtain the pmf of the receivedquantized LLR for each SU in terms of the pmf of thetransmitted quantized LLR and Pairwise Error Probability(PEP) for the transmitted and received quantized LLRs. Thepmf of the transmitted quantized LLR can be evaluated from(6). However, PEP specifications complicates the analysis andis inconvenient from the implementation point of view. Theanalysis is simplified by using the BEP model as is shownlater in this section.

Consider the scenario presented in Fig. 1. Here, the nth SUsends a d-bit symbol Ssu,n corresponding to the quantizedLLR Lsu,n. The subscript su in notations Ssu,n and Lsu,n isused to indicate that these values correspond to the transmittedSU statistics. Due to the channel errors, some bits out ofthe transmitted d bits may be received with error at the FC.This received d-bit symbol at the FC from the nth secondaryuser is denoted as Sfc,n. The received quantized LLR valuecorresponding to Sfc,n is denoted by Lfc,n, which takes one ofthe D values in the set Ln. The subscript fc in notations Sfc,n

and Lfc,n is used to indicate that these notations correspondto the received statistics at the FC. We assume that the BEPfor each bit in the d-bit sequence is the same and is denotedby Pb,n for the forward channel between the nth SU and theFC.

Now, because of the channel errors, any possible value ofLsu,n from the set Ln may result in a particular value of Lfc,n,

Figure 1. Secondary users (SUs) cooperate to detect CP-OFDM based primaryuser (PU) transmission. The nth SU evaluates LLR (Ln), and transmits d-bit symbol Ssu,n corresponding to the quantized LLR Lsu,n. Due to thechannel errors, the FC receives symbol Sfc,n corresponding to the quantizedvalue Lfc,n. The FC then combines the received LLRs from the cooperatingsecondary users to make a final decision.

say Lfc,n = li,n. Therefore the pmf of Lfc,n is given by

P (Lfc,n = li,n|Hj)

=D∑

k=1

P (Lfc,n = li,n|Lsu,n = lk,n, Hj)P (Lsu,n = lk,n|Hj)

for i = 1, . . . , D; j = 0, 1. (11)

Therefore the pmf of Lfc,n depends on the pmf of Lsu,n andthe corresponding PEPs P (Lfc,n = li,n|Lsu,n = lk,n, Hj)under both hypotheses. Therefore, in a worst case, to get thepmf of Lfc,n, we need to know D2 number of PEPs foreach SU. In other words, we may need D2 number of designparameters to parameterize the channel errors, which is clearlyinconvenient. The analysis can be simplified by using the BEPmodel for PEP as

P (Lfc,n = li,n|Lsu,n = lk,n, Hj)

= P (Sfc,n = si|Ssu,n = sk, Hj)

= P (dh(si, sk)|Ssu,n = sk, Hj)

= Pdi,k

b,n (1− Pb,n)d−di,k , (12)

where dh(si, sk) is the Hamming distance between d-bitsequences (si, sk) and di,k � dh(si, sk) for convenience. Notethat one-to-one mapping between Lsu,n and Ssu,n is assumed.The same is assumed for Lfc,n and Sfc,n. Substituting (12)in (11), we get the pmf of Lfc,n as

P (Lfc,n = li,n|Hj)

=

D∑k=1

Pdi,k

b,n (1− Pb,n)d−di,kP (Lsu,n = lk,n|Hj).(13)

Thus the pmf of Lfc,n depends on Pb,n, di,k and (6). Laterin the simulation results, it is shown that the mapping andhence the hamming distances does not significantly affect theperformance for large D and low Pb,n. Therefore the channelerrors can be specified in terms of a single parameter, i.e.,BEP Pb,n for the nth SU. Once the target value of BEP

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Pb,n is known, we can decide on the modulation, coding,and interleaving schemes for the particular channel conditions.The analysis makes an assumption that each bit in the d-bitsequence sees the same average BEP. The assumption is validfor using the BPSK and M-ary orthogonal schemes to transmitthe bit sequence in a straight forward manner. It is also validfor other modulation schemes like PSK, PAM and QAM, ifwe assume that interleaving is also applied such that each bitin the d-bit sequence sees the same average BEP. For eachfrequency band scanned, a SU has to transmit statistics likeLLRs, SNR, interference level. Therefore it is reasonable toassume that interleaving can be applied to overcome the bursterrors.

Let the sum test statistic at the FC for independent sensorobservations in the presence of quantization and Pb,n bedenoted as Tb, then

Tb =

Ns∑n=1

Lfc,n. (14)

The set of values Tb can take is Lq . Under the conditionalindependence of the received quantized LLRs Lfc,n, the pmfof the test statistic Tb is

p(Tb) = p(Lfc,1) ∗ p(Lfc,2) ∗ . . . ∗ p(Lfc,Ns). (15)

The pmf’s of Tb can be used to analyze the effect of biterrors on the performance of the detector given by (8) byreplacing Tq by Tb. The probability of detection is given by

Pdb = P (Tb > ηq|H1) + γ1P (Tb = ηq|H1), (16)

while false alarm probability in the presence of channel errorsbecomes

Pfb = P (Tb > ηq|H0) + γ1P (Tb = ηq|H0). (17)

V. SIMULATION RESULTS

For all simulation cases, discrete-time baseband processingis employed. For CP-OFDM based PU, Td = 32 and Tc = 8is assumed. Therefore, each OFDM block has 40 symbols.The number of OFDM blocks used is 100. Consequently,M = 100(Tc + Td) = 4000. The maximum number of SUsconsidered is 5. The signal-to-noise power ratio at each SU isdefined as SNRn(dB) = 10 log10(SNRn). For the simulationresults in this paper, the same average SNR value is assumedfor all SUs. The LLRs Ln are generated in Matlab based ontheir distribution given by (3). Since Ln are independent andidentically distributed (i.i.d) and as identical uniform quantizeris used at each SU, Lsu,n are also i.i.d ∀n = 1, . . . , Ns. Inthis case, Lq = L1 ∗L2 ∗ . . .∗LNs

and the number of possiblevalues Tq can take reduces from DNs to Ns ∗ (D − 1) + 1.The BPSK is used for transmitting d-bit quantized LLRs onthe forward channels, i.e., the channels from the SUs to theFC. Moreover, the forward channels are considered i.i.d, i.e.,Pb,n = Pb for n = 1, . . . , Ns. The results are averagedover 10000 realizations. The parameters γ1 and ηq for theNeyman-Pearson detector at the FC are evaluated to satisfythe constraint on the probability of false alarm α = 0.05 using(9).

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Theory Unquantized Ns=1Theory Q Ns=1Sim Q Ns=1Theory Unquantized Ns=3Theory Q Ns=3Sim Q Ns=3

Figure 2. Theoretical and simulation results match well for detection per-formance in presence of quantization (D=8, d=3). Cooperative detection(Ns = 3) improves detection performance as compared to local detection(Ns = 1).

A. Effect of Quantization

Fig. 2 shows the theoretical and simulation curves for proba-bility of detection Pd vs average SNR(dB), where quantization(D = 8, d = 3) is employed. In the presence of quantization,the theoretical results are obtained by using (10). Simulatedcurves are quite close to theoretical curves validating theanalysis for the local (Ns = 1) and the cooperative sensingcase (Ns = 3). The curves obtained for the quantization caseare also compared to the case of unquantized version (exactLLRs). The values for the unquantized case are obtained from[7]. There is a minor loss in detection performance due to thequantization for the considered scenario.

Fig. 3 shows the effect of quantization on cooperativedetection performance for various values of D for Ns = 3.There is a considerable loss in detection performance whileusing hard decisions, i.e,, D = 2. However, the loss in theperformance decreases as the number of quantization levels isincreased from D = 2 to D = 16 (i.e., the number of bitsused for quantization is increased from d = 1 to d = 4), theperformance loss decreases considerably. In fact, by using 4bits, the performance is practically the same as that of theunquantized version and only a negligible improvement canbe obtained by using more bits. This shows that there is aclear advantage in using soft decision statistics in the form ofquantized LLRs as compared to using hard decisions.

B. Effect of Channel Errors

As can be seen from (13), the performance of the schemein the presence of bit errors depends on the Hamming dis-tance between the transmitted and the received symbol, i.e.,dh(Ss,n, Sf,n). Hence the performance depends on mappingof the bit sequence to the symbols. We will show this laterby comparing the Binary mapping and the Gray mapping. Forother results, we have used Binary mapping. Although TableI gives labeling only for D = 8 (d = 3), the same can begeneralized for any arbitrary D = 2d.

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Effect of Quantization for Ns=3

Theory Q D=2Theory Q D=4Theory Q D=8Theory Q D=16Theory Unquantized

Figure 3. There is loss in detection performance in presence of quantization.This loss decreases as number of bits d = log

2D increase. Incremental gain

in detection performance (i.e., gain obtained by using 1 more bit) decreaseswith increase in number of bits. A negligible loss is observed by using d ≥ 4bits for quantization as compared to the optimal (unquantized) case.

Table IBINARY AND GRAY MAPPING FOR SYMBOLS Ss,n , Sf,n FOR D = 8

(d = 3).

Quantization Levels Binary mapping Gray mappingl1 000 000l2 001 001l3 010 011l4 011 010l5 100 110l6 101 111l7 110 101l8 111 100

Fig. 4 shows the theoretical and the simulated curves forprobability of detection in the presence of quantization andchannel errors as function a of average SNR(dB). Theoret-ical results for the case of quantization and channel errorsare obtained by using (16). For different values of Ns andPb, the theoretical and the simulated curves are sufficientlyclose to validate the theoretical analysis. Also, as expected, aperformance loss is observed for the case of sending quantizedLLRs over an erroneous channel as compared to the case ofsending exact LLRs over an error free channel (denoted bytheory).

Fig. 5 presents the detection performance for the differentvalues of Pb. As seen earlier, there is a performance loss in thepresence of quantization as compared to using the exact LLRs.The presence of channel errors further reduces the detectionprobability. In addition, the false alarm probability increaseswith an increase in the channel errors. However, this changein the detection and the false alarm probabilities is negligiblefor low values of Pb (≤ 10−2).

Fig. 6 presents the effect of number of secondary users Ns

on the detection performance. It is interesting to note that theeffect of channel errors on the detection performance reducesas the number of SUs cooperating increases. In other words,robustness and diversity gain are obtained.

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Figure 4. Simulation and theoretical curves for Pd vs SNR(dB) in the presenceof quantization and channel errors are close validating the theoretical analysis.

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Theory Q Pb=0

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Figure 5. In the presence of channel errors, probability of detection decreasesand false alarm probability increases. The performance loss is considerablefor Pb = 0.1 and makes it difficult to satisfy the false alarm constraint. Theloss is negligible for low values of Pb (≤ 10−2).

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Theory Pb=0 Ns=1

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Theory Pb=10−1 Ns=3Theory Pb=0 Ns=5

Theory Pb=10−1 Ns=5

Figure 6. As the number of secondary users (SUs) cooperating increases, theperformance loss due to the channel errors reduces. Thus cooperation mitigatesthe effect of errors due to the forward channel.

Next, we compare the performance of Binary and Graymapping to study the effect of mapping on the detectorperformance at the FC in the presence of Pb. Table I gives the

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Figure 7. Gray mapping gives significant gain over Binary mapping when thenumber of bits d for quantization is small and BEP Pb is high.

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Figure 8. For large D and low Pb, the performance gain of Gray mappingover Binary mapping becomes negligible.

Gray mapping for the case of D = 8 (d = 3). Fig. 7 showsthat the Gray mapping gives better performance compared tothe Binary mapping for Pb = 10−1 and D = 4 (d = 2).This result is not surprising as the optimality of Gray mappinghas been proved for the signal constellations in PSK, PAM,and QAM communications systems [15]. However, the gain inusing Gray mapping is negligible for Pb = 10−2 and D = 8as observed from Fig. 8. Therefore the effect of mapping issignificant only for the case of high values of Pb and smallvalues of D.

VI. CONCLUSION

In this paper, the effects of quantization and channel errorsare analyzed for autocorrelation-based distributed detection ofa CP-OFDM primary signal in a cognitive radio. A frameworkis presented for obtaining the number of bits for the quanti-zation and the target BEP for channel errors in the practicalcooperative detection scenario.

The distribution of the quantized LLR at the SUs is estab-lished. The distribution is also derived for the test statistic atthe FC in the presence of only quantization. It is observed that,with an increase in the number of bits used for quantization

d, the detection performance improves. However, the perfor-mance loss, while using 4 or more bits (d ≥ 4), is negligiblecompared to the unquantized case.

Next the effect of channel errors is incorporated usingthe BEP model. The distribution of the test statistic at theFC is established in the presence of channel errors andquantization. Due to the channel errors, the probability ofdetection decreases and the false alarm probability increases.The effect of channel errors on the detection performance isnegligible for a low BEP, Pb ≤ 10−2. For a high probabilityof error (of the order Pb > 10−2), however, a considerabledegradation is observed. This necessitates the use of channelerror coding for high BEP values. It is also observed thatuser cooperation provides diversity gain and thus mitigatesthe effects of channel errors on the detection performance atthe FC. The effect of bit mapping to symbols is also studied.The mapping of bits to symbol is important only when a fewbits are used and bit errors are greater.

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