64 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 1, JANUARY 2012

Cooperative Spectrum Sensing in Multiple Antenna BasedCognitive Radio Network Using an Improved Energy Detector

Ajay Singh, Student Member, IEEE, Manav R. Bhatnagar, Member, IEEE,and Ranjan K. Mallik, Senior Member, IEEE

Abstract—Performance of cooperative spectrum sensing withmultiple antennas at each cognitive radio (CR) is discussed inthis paper. The CRs utilize selection combining of the decisionstatistics obtained by an improved energy detector for makinga binary decision of the presence or absence of a primary user(PU). The improved energy detector uses an arbitrary positivepower 𝑝 of amplitudes of samples of the PU’s signals. The decisionof each CR is orthogonally forwarded over imperfect reportingchannels to a fusion center, which takes the final decision of aspectrum hole. We derive expressions of the probabilities of falsealarm and missed detection of the proposed cooperative spectrumsensing scheme. By minimizing the total error rate (sum of theprobability of missed detection and the probability of false alarm)we derive a closed-form solution of the optimal number of CRsrequired for cooperation. It is shown by simulations that byusing multiple antennas at the CRs, it is possible to significantlyimprove reliability of spectrum sensing with extremely lowinterference levels to the PU at very low (much less than 0 dB)signal-to-noise ratio of the PU-CR link.

Index Terms—Cooperative spectrum sensing, improved energydetector, multiple antennas, total error rate.

I. INTRODUCTION

COOPERATIVE spectrum sensing with conventional en-ergy detector [1] in single antenna based cognitive radio

networks for improving reliability in detecting a spectrum holehas been studied considerably in recent times [2]–[4]. It isshown in [5], [6] that the performance of a cognitive radionetwork can be improved by utilizing an improved energydetector in the cognitive radios (CRs), where the conventionalenergy detector is modified by replacing squaring operation ofthe received signal amplitude with an arbitrary positive power𝑝. In [7], [8], it is shown that reliability of spectrum sensingcan be improved in the CR by using multiple antennas. In thispaper, we consider optimization of a cooperative spectrumsensing scheme with an improved energy detector, multipleantennas at each CR, and imperfect reporting channels byminimizing the sum of the cooperative probabilities of falsealarm and missed detection referred to as the total error rate inthe paper. The main difference between this paper and [6] is asfollows. In [6], a single antenna based cooperative CR systemwith additive white Gaussian noise (AWGN) channel over thePU-CR links and perfect reporting channels, is considered,whereas, in this paper, we consider a multiple antenna based

Manuscript received September 7, 2011. The associate editor coordinatingthe review of this letter and approving it for publication was F. Jondral.

The authors are with the Department of Electrical Engineering, IndianInstitute of Technology - Delhi, Hauz Khas, New Delhi 110016, India (e-mail: ajaysingh8226@gmail.com, {manav, rkmallik}@ee.iitd.ernet.in).

Digital Object Identifier 10.1109/LCOMM.2011.103111.111884

cooperative CR system with Rayleigh fading primary user(PU)-CR links and imperfect reporting channels.

II. SYSTEM MODEL

We consider a cognitive radio network containing 𝑁 CRs,one PU, and a fusion center (FC). It is assumed that each of theFC and PU contains a single antenna and each CR contains 𝑀antennas. There are two hypotheses 𝐻0 and 𝐻1 correspondingto the signal received in the 𝑖-th antenna at each CR,

𝐻0 : 𝑦𝑖(𝑡) = 𝑣𝑖(𝑡), if PU is absent,𝐻1 : 𝑦𝑖(𝑡) = ℎ𝑖(𝑡)𝑠(𝑡) + 𝑣𝑖(𝑡), if PU is present,

(1)

where 𝑖 is the antenna index (𝑖 = 1, 2, . . . ,𝑀 ) at each CR, 𝑠(𝑡)denotes the signal transmitted by the PU at time instant 𝑡 withenergy 𝐸𝑠, 𝑣𝑖(𝑡) ∼ 𝒞𝒩 (0, 𝜎2

𝑛) is circularly symmetrical com-plex AWGN, and all ℎ𝑖(𝑡) ∼ 𝒞𝒩 (0, 𝜎2

ℎ) are independent andidentically distributed complex normal circularly symmetricalchannel gains implying Rayleigh fading. It is assumed that theCRs do not have any information about the channels of thePU-CR links. Further, it is assumed that each CR contains theimproved energy detector [5]; the statistic at the 𝑖th antennafor deciding the presence or absence of the PU is given by

𝑊𝑖 =∣ 𝑦𝑖 ∣𝑝, 𝑝 > 0, (2)

where we have dropped the time index 𝑡 for simplicity. It canbe seen from (2) that for 𝑝 = 2, 𝑊𝑖 reduces to the statisticcorresponding to the conventional energy detector [1]. Forthe above discussed set-up, cooperative spectrum sensing isperformed as follows:

i) Each CR calculates decision statistic given in (2) for all(𝑖 = 1, 2, . . . ,𝑀) antennas and uses selection combiningfor taking a binary decision of a spectrum hole.

ii) The binary decision of each CR is sent to the FC over animperfect reporting channel.

iii) The FC applies the ‘OR’ rule to the binary decisionsreceived from all CRs and takes a final decision onwhether the PU is present or not.

III. PERFORMANCE ANALYSIS OF MULTIPLE ANTENNA

BASED COGNITIVE RADIO WITH IMPROVED ENERGY

DETECTOR

The cumulative distribution function (c.d.f.) of the improvedenergy detector can be written as

𝑃𝑊𝑖(𝑥) = Pr (∣ 𝑦𝑖 ∣𝑝≤ 𝑥) , (3)

where Pr(⋅) denotes the probability. By using the conditionalprobability density function (p.d.f.) of ∣𝑦𝑖∣2 in (3) and after

1089-7798/12$31.00 c⃝ 2012 IEEE

SINGH et al.: COOPERATIVE SPECTRUM SENSING IN MULTIPLE ANTENNA BASED COGNITIVE RADIO NETWORK USING AN IMPROVED ENERGY . . . 65

some algebra, we get the conditional p.d.f. of 𝑊𝑖 underhypotheses 𝐻0 and 𝐻1, respectively, as

𝑓𝑊𝑖∣𝐻0(𝑦) =

2𝑦2−𝑝𝑝 exp

(− 𝑦

2𝑝

𝜎2𝑛

)𝑝𝜎2

𝑛

, (4)

𝑓𝑊𝑖∣𝐻1(𝑦) =

2𝑦2−𝑝𝑝 exp

(− 𝑦

2𝑝

𝐸𝑠𝜎2ℎ+𝜎2

𝑛

)𝑝(𝐸𝑠𝜎2

ℎ + 𝜎2𝑛)

. (5)

From (4), the probability that the decision statistic 𝑊𝑖 is lessthan 𝑧, under hypothesis 𝐻0 is given by

Pr(𝑊𝑖 ≤ 𝑧∣𝐻0)=

∫ 𝑧

0

𝑓𝑊𝑖∣𝐻0(𝑦)𝑑𝑦=1− exp

(−𝑧

2𝑝

𝜎2𝑛

). (6)

Maximal-ratio combining scheme is not considered since ithas spectrum sensing overhead due to channel estimation.Moreover, a combining scheme based on the sum of thedecision statistics of all antennas in the CR is not analyticallytractable. Therefore, we assume that each CR contains a selec-tion combiner (SC) that outputs the maximum value out of 𝑀decision statistics calculated for different diversity branches as𝑍 = max(𝑊1,𝑊2,𝑊3, . . . ,𝑊𝑀 ). Hence, from (6), the c.d.f.of the SC under hypothesis 𝐻0 is

𝑃𝑍(𝑧∣𝐻0) = Pr[max(𝑊1,𝑊2,𝑊3, ...,𝑊𝑀 ) ≤ 𝑧∣𝐻0]

=

[1− exp

(−𝑧

2𝑝

𝜎2𝑛

)]𝑀. (7)

It can be seen from [9, Fig. 3 and Section VI] that for 𝑝 = 2,the SC and square-law combiner perform almost similarly ifthe channels of the PU-CR links are independent of each other.The conditional p.d.f. 𝑓𝑍∣𝐻0

(𝑧) of the SC can be obtained bydifferentiating (7) w.r.t. 𝑧, resulting in

𝑓𝑍∣𝐻0(𝑧) =

2𝑀𝑧2−𝑝𝑝 exp

(− 𝑧

2𝑝

𝜎2𝑛

)𝑝𝜎2

𝑛

[1−exp

(−𝑧

2𝑝

𝜎2𝑛

)]𝑀−1

. (8)

The output of the SC is applied to a one-bit hard detectorwhich takes decision of a spectrum hole as

𝑍1

≷0𝜆, (9)

where 𝜆 is the decision threshold in each CR and binary bits 1and 0 correspond to the decision about presence and absence,respectively, of the PU. From (8), (9), [10, Eq. (41), Chap-ter 2], and after many algebraic manipulations, the probabilityof false alarm 𝑃𝑓 in each CR can be obtained as

𝑃𝑓 =1

𝑀− 1

𝑀

[1− exp

(−𝜆

2𝑝

𝜎2𝑛

)]𝑀. (10)

Similarly, the conditional p.d.f. of the output of the SC under𝐻1 is

𝑓𝑍∣𝐻1(𝑧)=

2𝑀𝑧2−𝑝𝑝 exp

(− 𝑧

2𝑝

𝐸𝑠𝜎2ℎ+𝜎2

𝑛

)𝑝(𝐸𝑠𝜎2

ℎ + 𝜎2𝑛)

[1−exp

(−𝑧

2𝑝

𝐸𝑠𝜎2ℎ+𝜎2

𝑛

)]𝑀−1

.

(11)

From (9), (11), [10, Eq. (41), Chapter 2], the probability ofmiss 𝑃𝑚 in each CR is

𝑃𝑚 =1

𝑀

[1− exp

(− 𝜆

2𝑝

(1 + 𝛾)𝜎2𝑛

)]𝑀, (12)

where 𝛾 = 𝐸𝑠𝜎2ℎ/𝜎

2𝑛 is the average signal-to-noise ratio

(SNR) of the PU-CR link.

IV. OPTIMIZATION OF COOPERATIVE SPECTRUM SENSING

SCHEME OVER IMPERFECT REPORTING CHANNELS

It is assumed that the imperfect reporting channel betweeneach CR and the FC is a binary symmetric channel with anerror probability of 𝑞. The probability of false alarm 𝑄𝑓 andthe probability of missed detection 𝑄𝑚 in the FC is givenby [4, Eq. (3)]

𝑄𝑓 = 1− [(1− 𝑃𝑓 )(1 − 𝑞) + 𝑞𝑃𝑓 ]𝑁 ,

𝑄𝑚 = [𝑃𝑚(1− 𝑞) + 𝑞(1 − 𝑃𝑚)]𝑁

. (13)

Define a function 𝑍(𝑝, 𝜆,𝑁) obtained by adding 𝑄𝑓 and𝑄𝑚 with equal weights (assuming equiprobable hypotheses),which denotes the total error rate of this scheme and is twicethe probability of bit error from an on-off keying point ofview. Thus, the total error rate is given by

𝑍(𝑝, 𝜆,𝑁) ≜ 𝑄𝑓 +𝑄𝑚. (14)

The optimized value of 𝑝 for given 𝜆 and 𝑁 can be obtained bytaking the first order partial derivative of (14) with respect to(w.r.t.) 𝑝, setting the result to zero, and then using a fixed-pointiteration method. Similarly, for given 𝑝 and 𝑁 , the optimizedvalue of 𝜆 can be found. The optimized number 𝑁opt of CRsfor a given value of 𝜆 and 𝑝 is obtained from

𝛥𝑍(𝑝, 𝜆,𝑁) = 𝑍(𝑝, 𝜆,𝑁 + 1)− 𝑍(𝑝, 𝜆,𝑁) = 0. (15)

From (10), (12), (13), (14), and (15), we have

𝑁opt ≈⌈

ln𝑓2(𝑞, 𝑃𝑓 , 𝑃𝑚)

ln𝑓1(𝑞, 𝑃𝑓 , 𝑃𝑚)

⌉, (16)

where

𝑓1(𝑞, 𝑃𝑓 , 𝑃𝑚) =𝑃𝑚(1− 𝑞) + 𝑞(1− 𝑃𝑚)

(1− 𝑃𝑓 )(1 − 𝑞) + 𝑞𝑃𝑓,

𝑓2(𝑞, 𝑃𝑓 , 𝑃𝑚) =2𝑞𝑃𝑓 − 𝑞 − 𝑃𝑓

𝑃𝑚 − 2𝑞𝑃𝑚 + 𝑞 − 1, (17)

and ⌈⋅⌉ denotes the ceiling function. The optimized values of𝑝, 𝜆, and 𝑁 can be obtained jointly by using first order partialderivatives of (14) w.r.t. 𝑝 and 𝜆, (15), and by the numericalmethod given in [11].

V. NUMERICAL RESULTS

It is assumed that the average SNR of all PU-CR links isthe same and is labeled as ‘SNR’ in the plots. Fig. 1 showsthe total error rate versus 𝑝 for 𝑀 = 2, normalized threshold𝜆𝑛 = 𝜆/𝜎2

𝑛 = 30, SNR=10 dB, 𝑞 = 0.001, and varyingnumber of cooperative CRs 𝑁 = 1, 2, . . . , 8. It can be seenfrom Fig. 1 that there exists a unique value of 𝑝 ∕= 2 andnumber of cooperative CRs for which the total error rate isminimum. The optimized value of 𝑝 is numerically found to

66 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 1, JANUARY 2012

1 2 3 4 5 6 7 8 9 10

10−2

10−1

100

p

Tot

al e

rror

rat

e

N=1N=2N=3N=4N=5N=6N=7N=8

Fig. 1. Total error rate of the proposed cooperative spectrum sensing versus𝑝 for varying number of cooperative CRs, 𝜆𝑛 = 30, 𝑀 = 2, 𝑞 = 0.001,and SNR=10 dB.

be 3.0490 as discussed in Section IV and optimized numberof cooperative CRs is obtained from (16) as 4. The total errorrate versus SNR plots of the proposed scheme with jointlyoptimized and sub-optimal values of 𝑝, 𝜆, and 𝑁 are shownin Fig. 2. Fig. 2 shows that the cognitive system with multipleantennas achieves less error rate specially at low SNR values(-20 - 0 dB) as compared to the single antenna based CRsystem. By using jointly optimized values of 𝑁 , 𝜆, and 𝑝,the total error rate of the CR system can be further reducedto very low values at all SNRs, as considered in Fig. 2. Itcan be seen from Fig. 3 that by using the total error rateminimization criterion and multiple antennas at each CR it ispossible to achieve arbitrarily low values of the probabilitiesof false alarm and missed detection at very low SNR. Forexample, for 𝑞 = 0.001, 𝑀 = 2, and −20 dB≤SNR≤ 0 dB,values of the missed detection probability and false alarmprobability in the FC are approximately 3 × 10−3 and 10−2,respectively. It can be noticed that the value of the misseddetection probability obtained by using multiple antennas andimproved energy detector is much less than the specified valueof 0.1 by the IEEE 802.22 cognitive wireless regional areanetwork (WRAN) standard [12]. Therefore, by using the totalerror rate criterion, it is possible to jointly optimize the valuesof 𝑝, 𝜆, and 𝑁 such that the opportunity of using a spectrumhole significantly improves and interference to the PU stayswithin the standard specified limits.

VI. CONCLUSION

Optimization of a cooperative spectrum sensing schemewith an improved energy detector and multiple antennas basedCRs over imperfect reporting channels is discussed. It isshown that by using the total error rate minimization criterionit is possible to achieve significant improvement in utilizationof the spectrum hole and reduction in interference level forthe PU at very low SNR range.

−20 −15 −10 −5 0 5 10 15 2010

−3

10−2

10−1

100

SNR ( dB )

Tota

l err

or ra

te

M=1, sub−optimal M=1, optimizedM=2, sub−optimal M=3, sub−optimal M=2, optimizedM=3, optimized

sub−optimal

optimized

Fig. 2. Total error rate versus SNR plots of the proposed scheme with jointoptimization and without optimization. (For the sub-optimal scheme, 𝜆𝑛 = 5,𝑁 = 5, 𝑝 = 2, and 𝑞 = 0.001 are used.)

−20 −15 −10 −5 0 5 10

10−4

10−3

10−2

10−1

100

SNR ( dB )

Pro

babi

lity

of fa

lse

alar

m a

nd m

isse

d de

tect

ion

M=1, Probability of false alarmM=1, Probability of missed detectionM=2, Probability of false alarmM=3, Probability of false alarmM=2, Probability of missed detectionM=3, Probability of missed detection

Fig. 3. Probability of false alarm and probability of missed detection versusSNR of the proposed scheme for 𝑞 = 0.001.

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