COOPERATIVE SENSINGWITH SEQUENTIAL ORDERED TRANSMISSIONS TOSECONDARY FUSION CENTER

Laila Hesham† Ahmed Sultan† Mohammed Nafie† Fadel Digham�

† Wireless Intelligent Networks Center (WINC), Nile University, Egypt� National Telecom Regulatory Authority (NTRA), Egypt

ABSTRACTSuccessful spectrum sharing in a cognitive radio network dependson the correct and quick detection of primary activity. Cooperativespectrum sensing is therefore suggested to enhance the reliabilityof such detection. However, it renders another significant problemof increased detection delay and traffic burden. Moreover, efficientschemes for multi-sensor data fusion should be designed. In this pa-per, we employ sequential detection scheme together with orderedtransmissions from cognitive detectors. We derive two sequentialschemes, one that is capable of achieving the minimum probabil-ity of error, and another that trades-off performance with delay. Forthe latter we derive expressions for the likelihood functions of or-dered observations and compute the thresholds via backward induc-tion. Simulation results demonstrate the relative performance of theapproaches proposed in the paper.

Index Terms— Cognitive Radio, cooperative spectrum sensing,sequential detection, detection delay

1. INTRODUCTION

Cognitive radio technology is one of the key candidate technologiesfor improving the utilization of scarce radio spectrum [1]. Cogni-tive networks consist of licensed primary users and secondary usersknown as cognitive detectors. These cognitive detectors are requiredto efficiently detect spectral holes that dynamically appear when li-censed users are not using their assigned spectrum. Therefore, spec-trum sensing becomes a fundamental and challenging task to reli-ably detect a vacant channel. Shadowing and deep fading on the linkbetween a primary transmitter and a secondary sensor may cause in-dividual sensors to falsely decide that the probed channel is vacant.Secondary transmission on this channel would then cause interfer-ence to existing communications which makes spectrum sensing re-lying on one sensor highly unreliable [2]. Exploiting spatial diversityamong a system of multiple sensors has been therefore proposed, be-cause cooperation of these spatially distributed cognitive detectorstogether dramatically improves the sensing reliability and detectiondelay. However, to get the desired results from such cooperation,efficient schemes should be designed to combine the sensing infor-mation from individual sensors [3]. Central processing of diversesensing results is done at a fusion center which is authorized to makethe final decision regarding the vacancy or occupancy of the sensedchannel. However, reporting these local observations to the fusioncenter over orthogonal channels increases the traffic burden which isa significant drawback.

In contrast to block detection schemes, sequential detectionschemes would reach a quick final decision while maintaining somedetection performance measures (see, e.g., [4, 5]) where the numberof observations samples is determined according to the reliability

of the observations, and it is therefore a random variable. Applyingsequential analysis to distributed detection problems has becomeof considerable interest (see, e.g., [6, 7]), specifically in cognitiveradio context as in [8]. A sequential detection scheme was proposedin [8], where each sensor computes the log-likelihood ratio (LLR)for its observations and reports it to the fusion center over a perfectreporting channel. The LLR’s are accumulated sequentially at thefusion center till their sum is found sufficient to cross either of twopredefined thresholds. In [9], the authors have proposed the ideaof ordering the transmissions from local detectors. By means oftimer backoff mechanism, the local sensor with the most informa-tive observation has the priority to report it to the fusion center.Consequently, transmissions are restricted to highly informative andconfident observations only. This scheme gives the added advan-tage of dramatical reduction of the average number of transmissionsrequired till the final decision is reached. It is to be noted that thesensor selection criterion is based on its instantaneous observationquality and not the average SNR as in [10] for example.

In this paper, we adopt the ordering scheme proposed in [9]. Weextend the work in [9] to the cases where a decision has to be madewhen only K readings out of the M sensors have been received atthe fusion center where K ≤ M . Moreover, since we consider thedetection delay to be one of our main concerns, we propose anothertechnique where a sequential test is performed at the fusion center,and decision thresholds required are obtained via backward induc-tion often utilized in dynamic programming. The proposed schemeimproves the performance without requiring the participation of allsecondary users in the reporting process. In Section 2 we describethe system model of cooperative spectrum sensing in a cognitive ra-dio network. Section 3 describes the previous work utilizing theordered transmissions scheme. We also describe our two schemes.In Section 4 we demonstrate the difference in performance via sim-ulation results.

2. SYSTEMMODEL

We consider a slotted primary system where the primary activitydoes not change during the time slot duration, τs, and switches inde-pendently from one slot to the next. There areM cognitive sensorswhich take a number of measurements at the beginning of each timeslot and compute a function of these measurements. A maximum ofK sensors among theM sensors forward the results sequentially toa fusion center at which the final decision regarding primary activityis taken. We consider binary hypothesis testing at the fusion centerwith the following two hypotheses:

H0: Sensed channel is free ;H1: Sensed channel is busy

The prior probabilities of each, denoted by π0 and (1− π0), respec-tively, are assumed to be known. Observations from different sen-

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sors are conditionally independent given either hypothesis but can benon-identical. Let Xi (n) be the received signal at the ith sensor atinstant n, where i = 1, 2, ..., M . At each sensor i, {Xi (n)}N

1 , areindependent given each hypothesis and are identically distributed,such that a total of N samples are taken over a time duration τN .Under the two hypotheses,Xi (n) is given by

H0 : Xi(n) = Wi(n) , n = 1, 2, ..., N

H1 : Xi(n) = Si(n) + Wi(n) , n = 1, 2, ..., N(1)

where Wi is additive white Gaussian noise (AWGN) with the samenoise power, σ2, at all sensors. Without loss of generality, the re-ceived primary signal Si (n) is assumed to be real zero-mean Gaus-sian random variable. The conditional probability distributions ofXi (n) given H0 and H1 are described by fXi(n)(xn | H0) andfXi(n)(xn | H1), respectively, such that

fXi(n)(xn | H0) ∼ N(0, σ2)fXi(n)(xn | H1) ∼ N(0, σ2

si+ σ2)

(2)

where σ2siis defined as the average received primary signal power at

the ith local sensor, and is assumed to be known at the local sensorsand fusion center. Moreover, they are assumed to be fixed over thetime slot duration. The local observations at sensors are defined as

Yi =N∑

n=1

log

[fXi(n)(xn | H1)

fXi(n)(xn | H0)

](3)

such that Yi is the LLR computed by the ith local detector, then re-ported to the fusion center sequentially as explained below. Defining

the local signal-to-noise ratio (SNR) as γi =σ2

si

σ2 , we can compute

the LLR at the ith sensor as

Yi =1

2σ2·

γi

γi + 1

N∑n=1

| Xi(n) |2 − log (1 + γi)N

2(4)

It is clear that the likelihood functions of Yi given H0 or H1

are shifted and scaled chi-square distributions with N degrees offreedom. Note that the information exchange between the cognitivesensors and the fusion center is assumed to be perfect and occurson a low-rate common control channel that is not on the band beingsensed. We assume that the procedure of seizing the control channeland sending the LLR observation obtained by one sensor requires anamount of time which we denote as τ .

Since the transmission of each computed LLR value takes sometime τ , eliciting another measurement from the sensors, though im-proving the reliability of detection, wastes a duration of τ from thepotential secondary transmission opportunity in case the primary isoff. This causes a decrease in secondary throughput. In other words,we have a reliability-throughput trade-off which we can control byallowing only the subset of the cognitive detectors with the most re-liable observations to transmit their LLR’s to the fusion center. Thisis implemented by having two thresholds at the fusion center. TheLLR with maximum magnitude is transmitted first to the fusion cen-ter. A decision can be made, but if the metric is between the twothresholds, the second highest LLR in magnitude is transmitted tothe fusion center and is combined with the first, and then a decisionis attempted again. This continues until a decision in favor ofH0 orH1 is made, or K LLR’s are accumulated at the fusion center. If k

sensors, 1 ≤ k ≤ K, are probed before a decision is reached, thetime to make a decision is τN + kτ . Since the slot duration is τs,

this leaves τs − τN − kτ for secondary transmission. Given the du-rations τN , τ and τs, it is obvious thatK must satisfy the inequalityτs − τN − Kτ ≥ 0.

3. PROPOSED RANKED SEQUENTIAL SCHEME

In this section we describe the ordered transmissions scheme. Abackoff timer is set at each of the M sensors according to the mag-nitude of the locally computed LLR. Specifically, the timer is in-versely proportional to the absolute value of the sensor’s LLR [9].Consequently, transmissions are restricted to the more confident ob-servations such that only highly informative measurements are pro-cessed at the fusion center. These combined observations are thencompared to two thresholds and one of three decisions is made ac-cordingly; declareH0, continue taking the next ranked observations,or declareH1. In [9], this idea was adopted such that the processingdone at the fusion center is the accumulation of the LLR observa-tions. Let Y [m] denote the LLR of rank m = 1, 2, ..., K , wherem = 1 is the highest rank. For 1 ≤ k ≤ K, the decision strategy isexpressed as

Ifk∑

m=1

Y[m]

⎧⎪⎨⎪⎩

< t(L)k Declare H0

∈ (t(L)k , t

(H)k ) Continue

> t(H)k Declare H1

(5)

Note that the thresholds are generally time-dependent, i.e., their val-ues depend on the stage index k. When the statistics evidently favorone hypothesis over the other, the decision is made and LLR trans-missions from the local sensors stop. Otherwise, more observationsare taken possibly till the end of the time slot of primary activity.These data-dependent thresholds t

(L)k and t

(H)k are given by

t(L)k = log π0

1 − π0− (M − k) |Y [k]|

t(H)k = log π0

1 − π0+ (M − k) |Y [k]|

(6)

where M − k is the number of sensors that have not transmittedtheir LLR’s yet at the kth stage, and |Y [k]| is the absolute value ofthe LLR received at stage k. However, if this scheme is forced totake a decision in favor of eitherH0 orH1 at stageK, then it shouldwork only for the caseK = M , as can be seen by the fact that onlyt(L)M = t

(H)M . The transmission is ordered in terms of the absolute

LLR value. That is, |Y [m]| < |Y [k]|, ∀m > k. It can be shown thatthis choice of thresholds ensure that this scheme has the same aver-age probability of error as maximum a posteriori (MAP) procedurewhere all the LLR’s are summed and compared to log π0

1 − π0.The

average probability of error when the guessed hypothesis is not thetrue one is given by Pe = (1−π0) Pr (H0|H1)+ π0 Pr (H1|H0).

The advantage of this system is that the average number of trans-missions needed to reach a decision is about half the total number ofthe sensors communicating with the fusion center. Here, we consid-ered extending the scheme in [9] to work forK ≤ M case such thatwe only process theK LLR’s with highest magnitudes out of theM

LLR’s. Assume that the probability density function of LLR valueYi under hypothesis H (either H0 or H1) is given by fYi

(y|H).Define βi,b,H = Pr {[|Yi| > |b|] | H}. The value of βi,b,H can bereadily computed knowing the probability density functions of Yi. Inthis work, we consider only the case of identical sensors, therefore,fYi

(y | H) = fY (y | H) and βi,b,H = βb,H .The optimal probability of error MAP block detector when the

best K out of M LLR values (highest in magnitude) are received

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can be written as

log

[fY1..YK |H1

(y1, y2, ..yK | H1)

fY1..YK |H0(y1, y2, ..yK | H0)

]H1><

H0

logπ0

1 − π0(7)

where, using the independence of LLR’s at various sensors, (7)can be given by

log

[fY (y1|H1) ..fY (yK |H1) (1 − βyK ,H1)

M−K

fY (y1|H0) ..fY (yK |H0) (1 − βyK ,H0)M−K

]H1><

H0

logπ0

1 − π0

(8)such that the given sequence of observations is Y1 = y1, Y2 =y2, ..Yk = yk [11].

In order to perform sequential detection, we compute the ac-cumulated sum at the fusion center at stage k, which is given by

k∑m=1

log

[fY (ym | H1)

fY (ym | H0)

]. If this accumulated sum is > t̂

(H)k , H1

is declared, and if it is < t̂(L)k , then H0 is declared. Otherwise, if

it does not cross either thresholds, the fusion center continues to ac-cumulate the LLR’s from the next ranked sensors. These thresholds,t̂(L)k and t̂

(H)k , would be given by

t̂(L)k = log π0

1 − π0− (K − k) Zk

max − (M − K) max0≤y≤|yk|

ρ(y)

t̂(H)k = log π0

1 − π0− (K − k) Zk

min − (M − K) min0≤y≤|yk|

ρ(y)

(9)for 1 ≤ k ≤ K − 1, such that ρ(y) = log

1 − βy,H1

1 − βy,H0

is acorrection term that appears at each stage k to account for theprobability that all the other (M − K) sensors would have LLRvalues less than |yk|. Since it is not a monotonic function, we needto span over the range from 0 < y < |yk| and choose the max-imum and the minimum value of ρ(y). However, at k = K,t̂(H)K = t̂

(L)K = log π0

1 − π0− (M − K)ρ(yK). Moreover,

Zkmax = log

fY (|yk| | H1)fY (|yk| | H0)

and Zkmin = log

fY (−|yk| | H1)fY (−|yk| | H0)

due to the monotonicity of logfY (y | H1)fY (y | H0)

. This term representsthe maximum possible contribution of the non-transmitting sensorsto the received LLR at stage k. These thresholds guarantee achievingthe same error probability as the MAP block detector.

In addition to this extension, we consider another method ofcombining ranked observations at the fusion center. We use dynamicprogramming to calculate the thresholds that allows throughput for-mulation to be addressed, which is the focus of current work. Thiswould allow to trade-off probability of error with sensing delay, andhence, one could design an objective function that maximizes thesecondary throughput. Specifically, backward induction technique isemployed to provide the optimal action to be taken in order to mini-mize the overall decision cost [12]. In the analysis below we assumethat the fusion center knows the statistics of all detectors but doesnot use sensors’ identifiers or indices to avoid complicated analysisand computational requirements.

Since we adopt the ordered transmission idea, if an LLR valueis received, and since we assume perfect reporting, all subsequentLLR’s would have values with a lesser magnitude. Recall that Yk

is the LLR from the kth sensor, 1 ≤ k ≤ M , whereas Y [k] is theLLR with the kth highest magnitude that is transmitted to the fusioncenter at the kth stage with k = 1, 2, ..K.

Let πk be the probability of the channel being idle at stage k

given the sequence of observations Y [1] = y1, Y[2] = y2, ..Y

[k] =

yk. Probability πk can be obtained recursively as follows:

πk = Pr (H0|y1, y2, ..yk) =

fY [k]|Y [k−1],H0(yk|yk−1, H0) πk−1∑

v=0,1

fY [k]|Y [k−1],Hv(yk|yk−1, Hv) (v + (1 − 2v) πk−1)

(10)

where in the last step, we use the property induced by orderedtransmissions that Y [k] is independent of Y [1], Y [2], ..Y [k−2] givenY [k−1] and either hypothesis. What is needed for backward induc-tion are the conditional probabilities of Y [m] given Y [m−1] underboth H0 and H1. However, the dependence on the last receivedobservation would make it a computationally extensive task. Toreduce the complexity, we propose here to use the distributions ofY [k] without conditioning to make the problem considerably moremanageable. Distribution fY [m−1] (y|H) can be readily obtained as

fY [m−1] (y|H) =

MfY (y|H)

(M − 1

m − 1

)(βy,H)m−1 (1 − βy,H)M−m

(11)

Assume that the cost of deciding i at stage k when j is the truehypothesis is λk

ij , and the cost to continue taking observations isc ≥ 0. Define J

(K)k as the minimum cost-to-go at stage k of the

finite horizonK.

J(K)k (πk) = min { λ

k00 πk + λ

k01 (1 − πk) ,

λk10 πk + λ

k11 (1 − πk) ,

c + EYk+1

[J

(K)k+1(πk+1)

]} (12)

where πk and πk+1 are related through the expression

πk+1 =πkfY [k+1] (y | H0)

πkfY [k+1](y | H0) + (1 − πk)fY [k+1](y | H1)(13)

The third term in (12) is interpreted as the expected cost when the fu-sion center decides that it should continue taking more observationsfrom local sensors. Thus, we have

EYk+1

[J

(K)k+1(πk+1)

]=

∫J

(K)k+1(πk+1)fY [k+1] (y) dy (14)

and

fY [k+1] (y) = πkfY [k+1](y | H0) + (1 − πk)fY [k+1](y | H1)(15)

The parameter c represents the trade-off between the time takentill a decision is made and the average probability of error. As c

increases, the fusion center becomes more likely to favor one of thetwo hypotheses in a shorter time using only a few LLR’s from thecognitive detectors. However, this cost has no effect at the last stagewhen k = K, where the fusion center has two choices only; todeclare H0 or H1 allowing backward induction from the last stagewhere J

(K)K (πK) = min { λK

00 πK + λK01 (1 − πK) , λK

10 πK +λK

11 (1 − πK) }.

4. SIMULATION RESULTS

In this section, we investigate via numerical simulations the perfor-mance of our proposed schemes. Assume σ2 = 1 and τ = 0.1. Thetime taken at the beginning of a time slot with duration τs = 0.5 tocollect N observations is τN = τ which leaves K = 4 stages in

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4 6 8 10 12 14 16 18 20

10−2

10−1

Number of sensors in CR network

Ave

rage

Pro

babi

lity

of e

rror

DP SchemeModified B&S Scheme

Fig. 1. Average probability of error vs. M ,N = 3, σ2si

= 2,K = 4,and c = 0.01.

the slot. We consider equal local signal to noise ratio over the chan-nels between the local sensors and the primary user, σ2

si= 2, i =

1, 2, ..., M and π0 = 0.5. Each sensor takes N = 3 samples inthe duration τN . The cost, c, to continue without deciding on oneof the two hypotheses is set to 0.01. We have performed computersimulations to plot the average error probability and average sens-ing time versus the number of local sensors in the network. Thedecision costs are assumed to be as follows: λk

ij = 1, i �= j andλk

ij = 0, i = j, where λkii = 0 means that no cost is incurred if the

guessed hypothesis is true.In Figure 1, it is clear that the modified scheme of [9], which we

hereafter refer to as B&S, achieves a lower probability of error sinceit achieves the same probability of error as the MAP block detector,while the dynamic programming scheme, denoted as DP scheme,achieves a higher error probability. However, as elaborated in Figure2, the number of sensors involved in the sensing process in the dy-namic programming scheme is lower than that in the B&S Modifiedscheme. Therefore, the dynamic programming scheme representsa trade-off of the sensing time with the error probability, via the c

parameter. This can be a potentially significant enhancement if thesecondary throughput is to be considered, which is the focus of ourcurrent work.

5. CONCLUSION

In this paper, we have considered ordering the transmissions fromcognitive users according to the information they carry about theprimary user. We devised two sequential schemes, one achieves theoptimum probability of error of a MAP block detector for the bestK out of M LLR’s, and the other would trade-off the error proba-bility for less average sensing time. Simulation results have demon-strated that the dynamic programming scheme achieves a significantdecrease in the delay of a decision at the expense of an increasedprobability of error.

6. REFERENCES

[1] S. Haykin, “Cognitive Radio: Brain-Empowered WirelessCommunications,” IEEE J. Select. Areas in Commun., vol. 23,

4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

4

Number of sensors in CR network

Ave

rage

Num

ber o

f sen

sors

invo

lved

in s

ensi

ng

DP SchemeModified B&S Scheme

Fig. 2. Average number of probed sensors vs. M , N = 3, σ2si

= 2,K = 4, and c = 0.01.

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[8] Q. Zou, S. Zheng, and A. H. Sayed, “Cooperative spectrumsensing via sequential detection for cognitive radio networks,”in IEEE 10th Workshop on Signal Processing Advances inWireless Communications, (SPAWC’09), Perugia, Italy, June21–24, 2009, pp. 121–125.

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