Cooperative Spectrum Sensing in Cognitive RadioNetworks With Weighted Decision Fusion Scheme

Edward C. Y. Peh†, Ying-Chang Liang‡, Yong Liang Guan† and Yonghong Zeng‡† School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore.

‡ Institute for Infocomm Research, A*STAR, Singapore.Emails: (pehc0004, eylguan)@ntu.edu.sg, (ycliang, yhzeng)@i2r.a-star.edu.sg

Abstract—In cognitive radio networks, both the sensing timeand the fusion schemes used for cooperative spectrum sensingaffect the detection probabilities of the primary users and thethroughput of the secondary users. Therefore, joint optimizationof the sensing time and the cooperative fusion scheme has beenstudied before in terms of sensing-throughput tradeoff design. Inthis paper, different from the previous studies, we consider thecase that each secondary user may have different detection signal-to-noise ratio (SNR), and requires different threshold for energydetection. Weightings are used to weigh the decisions from thesecondary users before combining. A new algorithm is proposedto compute the thresholds for the secondary users and the optimalweightings for the decisions are shown. Computer simulations arepresented to show the performance of the proposed algorithm.

I. INTRODUCTION

Cognitive radio, which senses unoccupied channels for itsown usage, is a promising technology to mitigate the spectrumshortage problem [1], [2]. Spectrum sensing is importantbecause a missed detection of the primary users can causeinterference to the primary users if the secondary users decideto use the occupied channel while a false alarm will cause thesecondary users to lose an opportunity to use the unoccupiedchannel.

Cooperative sensing can be used to improve the spectrumsensing performance [3]–[7]. There are various cooperativeschemes to combine the sensing information from the sec-ondary users. These schemes can be mainly categorized intodecision based fusion [8] and data based fusion [9]. For de-cision based fusion scheme, each secondary user has to makea decision based on its sensing results and sends its decisionto a fusion center which determines the final decision. Fordata based fusion scheme, there is no need for the secondaryusers to make their own decisions. The secondary users senddata, which is usually the test statistic of the sensing resultscollected, to the fusion center that makes the final decision.This paper focuses on the decision based fusion scheme.

Another parameter that affects the sensing performance isthe sensing time. A longer sensing time will improve thedetection performance. However, if the frame duration is fixed,this will reduce the data transmission time of the secondaryusers. Hence, in [10], the sensing-throughput tradeoff prob-lem was formulated to find the optimal sensing time thatmaximizes the secondary users’ throughput while providesadequate protection to the primary user.

Since both the sensing time and the cooperative fusionscheme affect the throughput of the secondary users, a jointoptimization of the sensing time and the parameters of the co-operative fusion scheme is considered in [11] for the sensing-throughput tradeoff problem. In [11], the paper considered thecase where k-out-of-N fusion rule is used and that all usershave the same detection SNR and the same energy detectorthreshold. In this paper, we consider a more general casethat all users may have different detection SNRs and moreimportantly, have different energy detector thresholds. Theoptimal decision fusion rule at the fusion center is derivedin this paper and is shown to be a weighted one based onthe likelihood-ratio test. A new algorithm is proposed in thispaper to obtain the energy detector thresholds and the fusioncenter threshold. The weightings for the decisions, which arefunctions of the energy detector thresholds, will therefore beobtained as well.

The rest of this paper is organized as follows. The systemmodel is introduced in Section II. In Section III, the problemformulation and the proposed algorithm to find the optimalsensing time, individual energy detectors’ thresholds, and theoptimal decision fusion rule are presented. Computer simula-tions are provided in Section IV. Finally, we draw conclusionof our contributions in Section V.

II. SYSTEM MODEL

We consider a cognitive radio network where there are Nsecondary users that act as sensor nodes to cooperatively detectthe presence of the primary user by sending their sensingdecisions to the secondary base station that acts as a fusioncenter. The primary user is assumed to be either idle or activethroughout the sensing period. The hypotheses of the absenceand the presence of the primary user are denoted as H0 andH1, respectively. The sampled signals that are received at theith secondary user during the sensing period are given as

H0 : yi(n) = ui(n), (1)

H1 : yi(n) = his(n) + ui(n) (2)

where s(n) denotes the signal from the primary user and isassumed to be an independent, identically distributed (i.i.d.)random process with zero mean and variance E[|s(n)|2] = σ2

s .The noise ui(n) is assumed to be i.i.d. circularly symmetriccomplex Gaussian with zero mean and variance E[|ui(n)|2] =

978-1-4244-2519-8/10/$26.00 ©2010 IEEE

σ2u. The channel gain from the primary user to the ith sec-

ondary user is denoted as hi, which is assumed to undergo flatfading and is constant during the sensing period. Without lossof generality, s(n) and ui(n) are assumed to be statisticallyindependent, and the average SNR at the ith secondary useris given as γi = |hi|2σ2

s

σ2u

.Consider that each of the secondary users employs an

energy detector, from [10], when the primary user’s signalis a complex-valued phase-shift keying signal, the energydetector’s probabilities of detection and false alarm at the ithsensor node are, respectively, approximated as

Pdi (τ, εi) = Q((

εi

σ2u

− γi − 1)√

τfs

2γi + 1

), (3)

Pfi (τ, εi) = Q((

εi

σ2u

− 1)√

τfs

)(4)

where Q(·) denotes the right-tail probability of a normalizedGaussian distribution, τ is the sensing time, and fs is thesampling frequency. The threshold of the energy detector atthe ith secondary user is denoted as εi.

After every secondary user makes its individual decision di,where di = 1 denotes that the primary user is detected anddi = 0 denotes otherwise, all their decisions are transmitted tothe secondary base station. Denote d = [d1 d2 · · · dN ] as thereceived decisions at the secondary base station; there can be2N possible outcomes of d. The secondary base station hasto make a final decision d0, based on d, on the presence ofthe primary user for the secondary network. Denote D+ asthe set of d values that the secondary base station will decidethat the primary user is present, while its complementary setD− contains the set of d values that the secondary base stationwill decide that the primary user is absent. The secondary basestation has to decide which d values should be included intoD+ so that an optimal decision fusion rule can be achieved.The general equations for the probabilities of detection andfalse alarm at the secondary base station using decision fusionrules are, respectively, given as

Pd

(τ, ε,D+

)=∑

d∈D+

Pr (d|H1, τ, ε,γ) , (5)

Pf

(τ, ε,D+

)=∑

d∈D+

Pr (d|H0, τ, ε) (6)

where ε = [ε1 ε2 · · · εN ] and γ = [γ1 γ2 · · · γN ].Assume that the secondary users’ decisions are independentof each other, Pr (d|H1, τ, ε,γ) and Pr (d|H0, τ, ε) can be,respectively, expressed as

Pr (d|H1, τ, ε,γ) =N∏

i=1

(1 − Pdi(τ, εi))(1−di) Pdi(τ, εi)di ,

(7)

Pr (d|H0, τ, ε) =N∏

i=1

(1 − Pfi(τ, εi))(1−di) Pfi(τ, εi)di .

(8)

III. PROBLEM FORMULATION

Consider that the frame structure of a cognitive radionetwork consists of a sensing slot and a transmission slot, andthe total frame time to be fixed at T secs. Denote C0 and C1

as the throughputs of the secondary users if they are allowedto continuously operate in the absence and the presence ofthe primary user, respectively. Hence, the average achievablethroughput of the cognitive radio network is given as [10]

R(τ, ε,D+

)=(1 − τ

T

)C0Pr(H0)

(1 − Pf

(τ, ε,D+

))+(1 − τ

T

)C1Pr(H1)

(1 − Pd

(τ, ε,D+

))(9)

where Pr(H0) and Pr(H1) are the probabilities of the pri-mary user being absent and present in the channel, respec-tively. The goal of this paper is to maximize the throughputof the cognitive radio network by finding the optimal decisionfusion rule D+, sensing time τ , and energy detectors’ thresh-olds at the secondary users ε, subject to the primary user issufficiently protected. This problem is formulated as

Problem P1

maxτ,ε,D+

R(τ, ε,D+

)s.t. Pd

(τ, ε,D+

) ≥ Pd (10)

where Pd is the minimum probability of detection that thesecondary base station needs to achieve to protect the primaryuser. Generally, the sensing-throughput tradeoff problem is anot a convex problem and in order to solve problem P1, wetransform it into subproblems with a lower complexity. First,we try to solve the problem P1 for a specific τ value and theproblem becomes

Subproblem SP1

maxε,D+

C0Pr(H0)(1 − Pf

(τ, ε,D+

))+C1Pr(H1)

(1 − Pd

(τ, ε,D+

))s.t. Pd

(τ, ε,D+

) ≥ Pd. (11)

The optimal solution occurs when the constraint is at equalityand the proof is similar to that in [11]. Hence, we will provideonly a brief explanation on why it is so here. For a specificτ and D+, the values of Pd (τ, ε,D+) and Pf (τ, ε,D+) areinversely proportional to the individual thresholds εi. From(11), it can be seen that the objective function is maximizedwhen Pd (τ, ε,D+) and Pf (τ, ε,D+) are minimized. Hence,the thresholds εi should always be chosen to meet the mini-mum requirement of Pd (τ, ε,D+) = Pd so that Pf (τ, ε,D+)will also be at minimum for a specific τ and D+.

When Pd (τ, ε,D+) is constrained to be equal to Pd, theright hand side of the objective function in (11) becomes aconstant, and the subproblem SP1 can therefore be simplifiedas:

Subproblem SP2

minε,D+

Pf

(τ, ε,D+

)s.t. Pd

(τ, ε,D+

)= Pd. (12)

Subproblem SP2 has a similar form as the Neyman-Pearsonapproach. However, instead of maximizing Pd (τ, ε,D+) for agiven Pf (τ, ε,D+), this subproblem minimizes Pf (τ, ε,D+)for a given Pd (τ, ε,D+). Therefore, the solution of thissubproblem has to satisfy the likelihood ratio test as follows[12],

LR(d) =Pr (d|H1, τ, ε,γ)Pr (d|H0, τ, ε)

≥ k, (13)

and ∑d:LR(d)≥k

Pr (d|H1, τ, ε,γ) = Pd (14)

where k is the threshold of the fusion rule. From (7) and (8),(13) can be further expressed as

N∏i=1

[(1 − Pdi(τ, εi))(1 − Pfi(τ, εi))

](1−di) [Pdi(τ, εi)Pfi(τ, εi)

]di

≥ k, (15)

and by taking logarithm on both sides,

LLR(d) =N∑

i=1

[(1 − di)w0i + diw1i] ≥ log k (16)

where

w0i(τ, εi) = log1 − Pdi(τ, εi)1 − Pfi(τ, εi)

, (17)

and

w1i(τ, εi) = logPdi(τ, εi)Pfi(τ, εi)

. (18)

The d values that are able to satisfy both (14) and (16)simultaneously should be included into the set D+ to form thefusion rule. Equation (16) corresponds to the optimal decisionfusion rule in [8], where the fusion rule puts a weighting of w1i

onto those secondary users’ decisions that detect the primaryuser while a weighting of w0i onto those decisions that do notdetect the primary user before summing all the decisions up.If the summation of the terms is greater than the thresholdlog k, the secondary base station will then decide that theprimary user is present or else it will decide otherwise. But inhere, there is an additional constraint (14) to be satisfied suchthat the primary user is sufficiently protected which makes theproblem more complex.

There are an infinite number of solutions for k and ε thatcan satisfy both (14) and (16) simultaneously. The goal of thispaper is to determine the fusion rule threshold k and the sec-ondary users’ thresholds ε that satisfy both (14) and (16) whilePf (τ, ε,D+) is being minimized. In a Bayesian problem, e.g.,minimizing the objective function of subproblem SP2 withoutthe constraint, a globally optimum solution for k and ε canbe obtained using person-by-person optimization methodology[13], where each threshold εi is optimized individually whileall the other thresholds are assumed to be fixed. However, thisapproach cannot be applied to a Neyman-Person problem [14],such as subproblem SP2, because optimizing εi while keepingall the other thresholds fixed cannot lead to a new value forεi since the constraint Pd(τ, ε,D+) = Pd will be violated.

We propose an algorithm to find the secondary users’thresholds ε that minimize Pf (τ, ε,D+) for a given D+. First,we define a term βi for all the N secondary users as follows:

βi =[∂Pd(τ, ε,D+)

∂εi

] [∂Pf (τ, ε,D+)

∂εi

]−1

∀ i = 1, · · · , N. (19)

The term βi is to measure the impact of the ith secondaryuser’s threshold εi on the overall probabilities of the detectionand the false alarm at the secondary base station. A largervalue of βi means that its corresponding εi has a larger impacton Pd(τ, ε,D+) than on Pf (τ, ε,D+). This means that asmall variation in εi with larger βi value will cause a biggerchange in Pd(τ, ε,D+) than in Pf (τ, ε,D+). For a givenD+, Pd(τ, ε,D+) and Pf (τ, ε,D+) can be expressed intothe following forms:

Pd(τ, ε,D+) = aiPdi(τ, εi) + bi, (20)

Pf (τ, ε,D+) = ciPdi(τ, εi) + ei (21)

where

ai = f1(Pd1, · · · , Pdi−1, Pdi+1, · · · , PdN |D+),bi = f2(Pd1, · · · , Pdi−1, Pdi+1, · · · , PdN |D+),ci = f1(Pf1, · · · , Pfi−1, Pfi+1, · · · , PfN |D+),ei = f2(Pf1, · · · , Pfi−1, Pfi+1, · · · , PfN |D+).

Therefore,

∂Pd(τ, ε,D+)∂εi

= − ai

σ2u

√τfs

2π(2γi + 1)

exp

⎛⎜⎝−

[(εi

σ2u− γi − 1

)√τfs

2γi+1

]22

⎞⎟⎠ ,

and

∂Pf (τ, ε,D+)∂εi

= − ci

σ2u

√τfs

2πexp

⎛⎜⎝−

[(εi

σ2u− 1)√

τfs

]22

⎞⎟⎠ .

(22)The proposed algorithm works as follows. Suppose for a

given D+, there is a set of non-optimal secondary thresholds,ε(0) = [ε(0)

1 , ε(0)2 , · · · , ε

(0)N ] which satisfies the constraint of

subproblem SP2. However, as mentioned earlier, it is not pos-sible to optimize individual secondary user’s threshold εi whilekeeping all the other thresholds fixed due to the constraint.Hence, the proposed algorithm optimizes two secondary users’thresholds at a time while keeping the rest of the thresholdsfixed. When one of the two selected thresholds changes suchthat the value of Pd(τ, ε,D+) increases, the other selectedthreshold should change accordingly to bring the value ofPd(τ, ε,D+) back to Pd so that the constraint (12) can bemaintained while minimizing the value of Pf (τ, ε,D+). Notethat from (5)-(8), it can be seen that the values of Pd(τ, ε,D+)and Pf (τ, ε,D+) change in the same direction as εi changes.

When a threshold with small βi value changes, the magnitudechanged in Pf (τ, ε,D+) is greater than Pd(τ, ε,D+), while athreshold with large βi value changes, the magnitude changedin Pd(τ, ε,D+) is greater than Pf (τ, ε,D+). Hence, choosingthe thresholds with the largest and the smallest βi values tominimize the value of Pf (τ, ε,D+), while maintaining theconstraint, will provide the best tradeoff.

At the start of each iteration of the proposed algorithm, thevalues of βi for all the thresholds are computed. The thresholdεi with the smallest βi value is increased such that the valueof Pf (τ, ε,D+) is decreased while the threshold εi with thelargest βi value is decreased to maintain Pd(τ, ε,D+) = Pd.The pair of threshold values is updated and all the valuesof βi are recomputed at the next iteration. The iterationprocess continues until the difference between the largestand the smallest βi values is smaller than a predeterminedvalue ρ. This stopping criterion is based on the fact thatif two thresholds with the same βi are used to minimizePf (τ, ε,D+) while maintaining the constraint, there can be nogain in the performance. If one of the thresholds is changedto reduce Pf (τ, ε,D+) which also results in Pd(τ, ε,D+)being reduced, the other threshold has to change to increasePd(τ, ε,D+) in order to satisfy the constraint. However, ifthey have the same βi value, Pf (τ, ε,D+) will be increasedto the original value and thereby causes no gain in the tradeoffbetween the two thresholds with the same βi value. Thestopping criterion is triggered when the difference betweenthe largest and smallest βi values is less than a predeterminedvalue, since there is not much further gain in performanceshould the proposed algorithm continues.

In general, there are 22N

possible sets of fusion rule D+

when there are N decisions di input to the secondary basestation. However, if the fusion rule is constrained to make adecision based on (16), the number of possible sets of D+

will greatly reduced to 2N − 1 for a given ε. The maximumpossible cardinality of D+ is 2N , that is when all the 2N

possible values of d are elements of D+. However, it isobvious that d = [0, 0, · · · , 0] cannot be an element of D+

since it does not make sense for a fusion rule to decide that theprimary user is present when all the secondary users’ decisionsdecide otherwise. The practical maximum cardinality of D+

is therefore 2N −1. At any particular cardinality of D+, thereis only one possible combination set of d values that satisfies(16) for a given ε. For example, if the cardinality of D+ is one,the only possible element in D+ has to be d = [1, 1, · · · , 1]based on (16) with log k =

∑Ni=1 w1i(τ, εi). This is because

given the fact that Pdi(τ, ε) > Pfi(τ, ε); LLR([1, 1, · · · , 1])is always greater than all the other LLR(d) values. So if anyother LLR(d) value is included as an element in D+, thend = [1, 1, · · · , 1] must be included in D+ too, based on thefusion rule (16). The same happens at every other cardinalityof D+, that there is only one possible combination set of dvalues that satisfies (16). Hence, we conclude that when thefusion rule is constrained to make a decision based on (16),the number of possible sets of fusion rule D+ is reduced from22N

to 2N − 1 for a given ε. The possible number of fusion

rule’s threshold k is therefore also 2N − 1 which correspondto each possible set of fusion rule D+ for a given ε. Theproposed algorithm runs through the 2N − 1 possible setsof D+ and obtains the pair of k and ε that minimizes theobjective function of subproblem SP2 for each D+. The pairof k and ε that has the minimum objective function amongthe different possible D+ is output as the final solution. Thesummary of the proposed algorithm is shown in Algorithm 1.

Algorithm 1 : Find the optimal k and the thresholds ε forsubproblem SP2

Initialization:1) Initialize D+ by having d = [1, 1, · · · , 1] as the onlyelement in D+(1).2) Initialize ε(0) to be any value that satisfy (14) for thegiven D+(1) and k(0) =

∏Ni=1 exp(w1i(τ, ε

(0)i )).

Repeat:3) Compute βi for all εi.4) εS := the εi with the smallest βi.5) εL := the εi with the largest βi.Repeat:6) Increase the εS to decrease Pf (τ, ε,D+) while decreasethe εL to maintain Pd(τ, ε,D+) = Pd until the minimumPf (τ, ε,D+) is obtained between εS and εL.7) Compute βi for all εi.8) εS = the εi with the smallest βi.9) εL = the εi with the largest βi.Until: |εL − εS | ≤ ρOutput: ε(j)

10) Update k(j) with ε(j) and D+(j).11) Using ε(j) compute k(j+1) such that the cardinality ofD+(j+1) is increased by one.

Until: D+(2N−1), ε(2N−1), and k(2N−1) are reached.Output: Every k(j) and ε(j) is a pair of solutions that satisfyboth (14) and (16). Choose the pair of k(j) and ε(j) withthe minimum Pf (τ, ε,D+).

IV. COMPUTER SIMULATION

In the following simulations, we set the number of sec-ondary users to be N = 5, and the frame durationto be T = 20ms. The sampling frequency of the re-ceived signal is assumed to be 6MHz and Pd is set at99%. Assume the secondary users’ SNRs are at γ =[−20,−22,−24,−26,−28]dB. We find the maximum achiev-able throughput of a cognitive radio network with the abovesystem parameters by using the proposed algorithm.

Fig. 1 shows the maximum normalized throughput achievedby using the proposed algorithm to obtain k and ε for theweighted decision fusion rule when sensing time is variedfrom 1ms to 19ms. The normalized throughput is definedas(1 − τ

T

)(1 − Pf (τ, ε,D+)). It can be observed that the

maximum normalized throughput is a quasiconvex function interms of the sensing time τ , when k and ε are obtained fromthe proposed algorithm. Hence, algorithms such as Golden

search method can be used to search for the optimal τ insteadof an exhaustive search. The sensing time τ , k, and ε canbe solved by the proposed algorithm with the Golden searchmethod at the outer loop. The results of OR fusion rule andAND fusion rule with all the energy thresholds constrained tobe the same are shown for comparison.

Fig. 2 shows the average weightings allocated to eachsecondary user when the proposed algorithm is used. Averageweighing is defined as (|w0i(τ, εi)| + |w1i(τ, εi)|) /2, whichshows the average weight each secondary user has on thefusion rule (16). From fig. 2, it is not surprising to observe thatthe algorithm gives higher average weightings to secondaryusers with higher SNR values than to secondary users withlower SNR values, since the decisions from secondary userswith higher SNR values should be more accurate and reliable.

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Sensing time τ (ms)

Nor

mal

ized

thro

ughp

ut

Weighted fusionOR fusionAND fusion

Fig. 1. Comparisons of the maximum normalized throughputs obtained bythe weighted decision fusion rule, the OR fusion rule and the AND fusionrule.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

Sensing time τ (ms)

Ave

rage

wei

ghtin

gs

User 1 = −20dBUser 2 = −22dBUser 3 = −24dBUser 4 = −26dBUser 5 = −28dB

Fig. 2. Average weighing of each secondary user obtained with the proposedalgorithm at different sensing times.

V. CONCLUSION

In this paper, we have shown that in a cognitive radionetwork with secondary users having different SNR levels,the optimal decision fusion scheme is based on likelihoodratio test where weightings will be given to secondary users’decisions based on their probabilities of detection and falsealarm. An algorithm is proposed to design the secondaryusers’ threshold levels such that the fusion rule based on thelikelihood ratio test is maintained and the detection probabilityat the secondary base station meets the minimum targetedvalue while Pf (τ, ε,D+) is minimized. It is also shown thatsecondary users with higher SNRs will have higher averageweightings in the fusion rule when the proposed algorithm isapplied.

REFERENCES

[1] J. Mitola, III and G. Q. Maguire, Jr., “Cognitive radio: making softwareradios more personal,” IEEE Pers. Commun., vol. 6, no. 4, pp. 13–18,Aug. 1999.

[2] S. Haykin, “Cognitive radio: brain-empowered wireless communica-tions,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb.2005.

[3] S. M. Mishra, A. Sahai, and R. W. Brodersen, “Cooperative sensingamong cognitive radios,” in Proc. IEEE Int. Conf. Commun. (ICC),Istanbul, Turkey, Jun. 2006, pp. 1658–1663.

[4] G. Ganesan and Y. Li, “Cooperative spectrum sensing in cognitive radionetworks,” in Proc. IEEE First Int. Symp. New Frontiers Dynamic Spectr.Access Netw. (DySPAN), Baltimore, MD, Nov. 2005, pp. 137–143.

[5] A. Ghasemi and E. S. Sousa, “Collaborative spectrum sensing foropportunistic access in fading environments,” in Proc. IEEE First Int.Symp. New Frontiers Dynamic Spectr. Access Netw. (DySPAN), Nagoya,Japan, Jan. 1997, pp. 290–294.

[6] G. Ganesan and Y. (G.) Li, “Cooperative spectrum sensing in cognitiveradio, part i: Two user networks,” IEEE Trans. Wireless Commun., vol. 6,no. 6, pp. 2204–2213, Jun. 2007.

[7] E. C. Y. Peh and Y.-C. Liang, “Optimization for cooperative sensing incognitive radio networks,” in Proc. IEEE Wireless Commun. Netw. Conf.(WCNC), Hong Kong, Mar. 2007, pp. 27–32.

[8] Z. Chair and P. K. Varshney, “Optimal data fusion in multiple sensordetection systems,” IEEE Trans. Aerosp. Electron. Syst., vol. 22, no. 1,pp. 98–101, Jan. 1986.

[9] Z. Quan, S. Cui, and A. H. Sayed, “Optimal linear cooperation forspectrum sensing in cognitive radio networks,” IEEE J. Sel. Topics SignalProcess., vol. 2, no. 1, pp. 28–40, Feb. 2008.

[10] Y.-C. Liang, Y. Zeng, E. C. Y. Peh, and A. T. Hoang, “Sensing-throughput tradeoff for cognitive radio networks,” IEEE Trans. WirelessCommun., vol. 7, pp. 1326–1337, Apr. 2008.

[11] E. C. Y. Peh, Y.-C. Liang, Y. L. Guan, and Y. Zeng, “Optimization ofcooperative sensing in cognitive radio networks: A sensing-throughputtradeoff view,” IEEE Trans. Veh. Technol., to be published.

[12] E. L. Lehmann and J. P. Romano, Testing statistical hypotheses, 3rd ed.Springer, 2005.

[13] P. K. Varshney, Distributed Detection and Data Fusion. New York:Springer-Verlag, 1997.

[14] J. N. Tsitsiklis, “Decentralized detection,” Adv. Statistical Signal Pro-cess., vol. 2, pp. 297–344, 1993.