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Performance of Cognitive Radio Networks withMaximal Ratio Combining over Correlated

Rayleigh FadingTrung Q. Duong‡, Thanh-Tan Le†, and Hans-Jurgen Zepernick‡

‡Blekinge Institute of Technology, Ronneby, SwedenE-mail: {quang.trung.duong, hans-jurgen.zepernick}@bth.se

†University of Ulsan, KoreaEmail: tanlh@mail.ulsan.ac.kr

Abstract—In this paper, we apply the maximal ratio combining(MRC) technique to achieve higher detection probability in cogni-tive radio networks over correlated Rayleigh fading channels. Wepresent a simple approach to derive the probability of detectionin closed-form expression. The numerical results reveal that thedetection performance is a monotonically increasing functionwith respect to the number of antennas. Moreover, we providesets of complementary receiver operating characteristic (ROC)curves to illustrate the effect of antenna correlation on thesensing performance of cognitive radio networks employing MRCschemes in some respective scenarios.

I. INTRODUCTION

In cognitive radio technologies, dynamic spectrum accesshas gained significant attention in the research communitysince it enables much higher spectrum efficiency and reducesthe spectrum scarcity. To be more specific, cognitive radionetwork (CRN) systems allow the secondary user (SU) tooperate on “spectrum holes” that are licensed to the primaryusers (PUs). One of the challenges in spectrum sensing is howto detect “spectrum holes” of primary users even in the lowsignal-to-noise ratio (SNR) regimes. To cope with this chal-lenge, multiple antennas applied in CRNs have been recentlyconsidered to improve the sensing performance. However, dueto the correlation between adjacent antennas, the performanceof CRN is significantly degraded.

Related work to energy detection using multiple antennaswith combining technique have primarily been addressed in[1]–[4]. These works followed the probability density function(PDF) based approach to derive closed-form expressions fordetection performances taking into account diversity receptionincluding a series of combining techniques such as maximalratio combining (MRC), selection combining (SC), equalgain combining (EGC), switch and stay combining (SSC),square-law selection (SLS), and square-law combining (SLC).Recently, the analysis in [3] has used the moment generatingfunction (MGF) based approach and applied an alternativecontour integral representation of Marcum-Q function totransform the integral to the complex domain. This methodmitigated many difficulties for calculating the integrals to getthe performances of the MRC energy detector over identically

and independently distributed (i.i.d.) Nakagami-m and Ricianfading channels.

One of the severe problems of multiple antenna systemsis the correlation between adjacent antennas. It has beenshown that the correlation of these antennas degrades spatialdiversity gain [5]–[8]. In particular, Digham et al. [1] haveanalyzed the effect of the spatial correlation on the detectionperformances in case of using SLC over the exponentialcorrelated Rayleigh fading channels. In addition, the effectof antenna correlation on the sensing performances has beenexamined for cooperative sensing [9]. Moreover, Kim et al.[10] have analyzed the detection performances in correlatedCRN by using the central limit theorem (CLT).

To circumvent the above problem, we present a simpleanalytical method to derive the closed-form expressions forprobabilities of detection PDE and false-alarm PFA overcorrelated Rayleigh fading channels by using a PDF basedapproach. It will be shown in our paper that the MRC energydetector achieves significantly high performance in compar-ison to energy detector using a single antenna. The adverseeffect of spatial correlation on spectrum sensing performanceis also analyzed by comparing the detection probability withand without correlation. Finally, complementary receiver op-erating characteristic (ROC) curves are obtained by plottingprobabilities of miss, PM = 1 − PDE , versus probability offalse alarm PFA for different scenarios.

The rest of this paper is organized as follows. Section IIbriefly reviews the sensing performance in terms of probabilityof detection PDE and probability of a false alarm PFA evalu-ated over additive white Gaussian noise (AWGN) and Rayleighfading channels. An MRC technique applied to a multipleantenna CRN receiver for both correlated and i.i.d. Rayleighfading channels is presented in Section III. In Section IV, weshow the numerical results and analysis. Finally, concludingremarks are given in Section V.

II. SYSTEM MODEL AND SINGLE ANTENNA SENSING

PERFORMANCE

Let s(t) be the primary user signal that is transmitted overthe channel with gain h and additive zero-mean and variance

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N0 AWGN n(t). Let W be the signal bandwidth, T be theobservation time over which signal samples are collected andB = TW be the time-bandwidth product. We assumed that Bis an integer. The hypothesis tests for spectrum sensing H0

and H1 related to the fact that the primary user is absent orpresent, respectively, are formulated as follows:

H0 : Y = n(t)H1 : Y = hs(t) + n(t) (1)

where Y is the received signal and noise n(t) can be expressedas [11]

n(t) =2B∑i=1

nisinc(2Wt− i), 0 < t < T (2)

with ni = n(

i2W

)considered as Gaussian random variable

according to CLT. Under H0, the normalized noise energycan be modified from [12]

Y = 1/ (2N0W )2B∑i=1

n2i (3)

Obviously, Y can be viewed as the sum of the squaresof 2B standard Gaussian variates with zero mean and unitvariance. Therefore, Y has a central chi-squared distributionwith 2B degrees of freedom. Under H1, the same approachis applied and the received decision statistic Y follows a non-central distribution χ2 with 2B degrees of freedom and a non-centrality parameter 2γ [12], where γ is the SNR. Then, thehypothesis test (1) can be written as

H0 : Y ∼ χ22B

H1 : Y ∼ χ22B (2γ) (4)

Hence, the PDF of Y can be expressed as

fY (y) =

⎧⎨⎩

12BΓ(B)

yB−1 exp(−y

2

), H0

12

(y2γ

)B−12

exp(− 2γ+y

2

)IB−1

(√2γy

), H1

(5)where Γ (·) is the gamma function [13, Sec. (13.10)] and In(.)is the nth-order modified Bessel function of the first kind [13,Sec. (8.43)].

A. Detection and False Alarm Probabilities over AWGN Chan-nels

The probability of detection and false alarm can be definedas [12]

PDE = P (Y > λ|H1) (6)

PFA = P (Y > λ|H0) (7)

where λ is a detection threshold. Using (4) to evaluate (5) and(6) yields [12]

PDE = QB

(√2γ,√λ)

(8)

PFA =Γ(B, λ/2)

Γ(B)(9)

where Γ(., .) is the upper incomplete gamma function [13, Sec.(8.350)]. QB(a, b) is the generalized Marcum Q-function [14]defined by

QB(a, b) =1

aB−1

∞∫b

xB exp(−x

2 + a2

2

)IB−1(ax)dx

(10)

B. Detection and False-Alarm Probabilities over RayleighChannels

In this section, we derive the average detection probabilityPDE over a Rayleigh fading channel. Clearly, PFA will remainthe same because it is independent of the SNR. The detectionprobability can be given by

PDE =

∞∫0

QB

(√2γ,√λ) 1γ

exp(−γ/γ)dγ (11)

To obtain a closed-form expression of (11), we now introducean integral Υ(.) shown in the Appendix A as

Υ(B, a1, a2, p, q) =

∞∫0

QB (a1√γ, a2) γq−1 exp(−p2γ/2)dγ

=B−1∑i=0

(a2)2iΓ(q) exp(−a2

2/2)

2iΓ(i+ 1)(p2 + a21)

2q 1F1

(q; i+ 1;− a2

1a22

4(p2 + a21)

)

+2q(q − 1)!

p2q

a21

p2 + a21

exp(− a2

2p2

2(p2 + a21)

)

×[q−2∑

n=0

(p2

p2 + a21

)n

Ln

(− a2

1a22

2(p2 + a21)

)

+(

1 +p2

a21

)(p2

p2 + a21

)q−1

Lq−1

(− a2

1a22

2(p2 + a21)

)](12)

where

Ln(x) =n∑

i=0

(−1)i

(n

n− i)xi

i!(13)

is the Laguerre polynomial of degree n [13, Sec. (8.970)]and 1F1(.) is the confluent hypergeometric function [13, Sec.(9.2)]. From (11) and (12), we obtain the closed-form PDE asfollows:

PDE =1γ

Υ(B,√

2,√λ,

√2γ, 1) (14)

III. MULTI-ANTENNAS SENSING PERFORMANCE

As mentioned above, spectrum sensing plays an importantrole of a CRN system. If an SU does not detect properly the“spectrum holes”, it unintentionally causes interference to aPU’s signal. Hence, it is motivated to find an accurate primarysignal detection approach. To obtain a reliable detection,multiple antennas in a CRN can be used to exploit fully theamount of diversity offered by the channels.

In this section, we consider a CRN system that includes Lantennas. The channels between the PU transmitter and SUreceiver antennas are i.i.d. Rayleigh fading channels. We nowexploit the spatial diversity of multiple antennas at SU by using

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MRC techniques. However, in CRNs, the long path from thePU and the SU may cause a small angular spread value atthe SU which creates a correlation between adjacent anten-nas. Therefore, we also examine the effect of equicorrelatedRayleigh fading channels on sensing performance.

Assume that the output signal of MRC can be obtained by

YMRC =L∑

i=1

Yi =L∑

i=1

h∗i ri(t) (15)

where L is the number of antennas. The received SNR, thesum of the SNRs on the individual receiver antennas, can begiven by

γMRC =L∑

i=1

γi (16)

where γi is the SNR on the i-th antenna.

A. I.I.D. Rayleigh Channels

Since Yi is the sum of L i.i.d. non-central χ2 variableswith 2B degrees of freedom and non-centrality parameter 2γi,we observe that YMRC is a non-central distributed variablewith 2LB degrees of freedom and non-centrality parameter2∑L

i=1 γi = 2γMRC . Then, the PDE at the MRC output forAWGN channels can be evaluated from (8) as

PDE,MRC = QLB

(√2γMRC ,

√λ)

(17)

It is well known that the PDF of γMRC is given by [15, Eq.(6.23)]

fMRC(γ) =1

(L− 1)!γL−1

γLexp(−γ/γ) (18)

The average PDE for MRC scheme, PDE,MRC , can beobtained by averaging (17) over (18) and comparing it withthe integral (12):

PDE,MRC =1

(L− 1)!1γL

Υ(LB,√

2,√λ,√

2/γ, L)

(19)

B. Equicorrelated Rayleigh Channels

In this case, we consider the slow nonselective correlatedRayleigh fading channels having equal branch powers and thesame correlation between any pair of branches, i.e., ρij = ρ,i, j = 1, 2, ..., L, denotes the power correlation coefficientbetween the i-th and j-th antennas. For L equicorrelatedRayleigh channels, the PDF of γMRC is given by [16]

fMRC(γ) = abL−1[

exp(−aγ)

(b−a)L−1

− exp(−bγ)L−1∑k=1

γk−1

(b−a)L−k(k−1)!

], γ ≥ 0

(20)

where

a =1

γ(1 + (L− 1)

√ρ)

b =1

γ(1−√ρ)

10−4

10−3

10−2

10−1

100

10−3

10−2

10−1

100

Probability of a False Alarm PFA

Pro

babi

lity

of M

iss

PM

SNR = 5, ρ = 0.2

SNR = 5, ρ = 0.4

SNR = 5, ρ = 0.6

SNR = 5, ρ = 0.8

SNR = 7, ρ = 0.2

SNR = 7, ρ = 0.4

SNR = 7, ρ = 0.6

SNR = 7, ρ = 0.8

SNR = 5 dB

SNR = 7 dB

Fig. 1. Complementary ROC curves for MRC scheme over correlatedRayleigh channel at different power correlation coefficient ρ and SNR values(B = 6, L = 8).

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pro

babi

lity

of D

etec

tion

PD

E

ρ = 0.2

ρ = 0.4

ρ = 0.6

ρ = 0.8single antennaIID multiple antennas

PDE increases asρ decreases

Fig. 2. Probability of detection versus SNR when MRC applied toequicorrelated Rayleigh fading channels, B = 6, PFA = 0.01, L = 8,ρ = 0.2, 0.4, 0.6, 0.8.

The detection probability PDE,MRC,Corr can be obtained byaveraging (17) over (20) and using (12), giving

PDE,MRC,Corr =

a(

bb−a

)L−1

Υ(LB,√

2,√λ,√

2a, 1)− abL−1

×L−1∑k=1

1(b−a)L−k(k−1)!

Υ(LB,√

2,√λ,√

2b, k)

(21)

IV. NUMERICAL RESULTS AND DISCUSSIONS

In this section, we provide the numerical results to illustratethe effect of antenna correlation on the sensing performanceof CRNs.

Fig. 1 shows the sensing performance of CRN with MRCfor the time-bandwidth product B = 6 and the number ofantennas L = 8. As can be seen from Fig. 1 where comple-mentary ROC curves at the given SNR value are presented,

67

4

4 5 6 7 8 90.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of antennnas (L)

Pro

babi

lity

of D

etec

tion

PD

E

ρ = 0.2

ρ = 0.4

ρ = 0.6

ρ = 0.8IID multiple antennas

SNR = 5 dB

SNR=10 dB

Fig. 3. Probability of detection versus number of antennas L when MRCapplied to equicorrelated Rayleigh fading channels, B = 6, PFA = 0.01, SNR= 5 dB or 10 dB, ρ = 0.2, 0.4, 0.6, 0.8

10−4

10−3

10−2

10−1

100

10−3

10−2

10−1

100

Probability of a False Alarm PFA

Pro

babi

lity

of M

iss

PM

L = 4, SNR = 10dB

L = 5, SNR = 10dB

L = 6, SNR = 10dB

L = 7, SNR = 10dB

L = 8, SNR = 10dB

L = 4, SNR = 5dB

L = 5, SNR = 5dB

L = 6, SNR = 5dB

L = 7, SNR = 5dB

L = 8, SNR = 5dB

SNR = 5dB

SNR = 10 dB

Fig. 4. Complementary ROC curves for MRC scheme over the correlatedRayleigh channel at different L (SNR = 5 dB or 10 dB, B = 6, ρ = 0.2).

antenna correlation between two adjacent antennas makesdetection performance deteriorate. Note that a correlation iscaused not only by a close distance between two adjacentantennas but also a small angular spread value generatedby the great distance between the primary transmitter andthe sensing node of the CRN. Moreover, spectrum sensingperformance degradation is proportion to the decrease of theSNR. In particular, the sensing performance at SNR = 7dB outperforms SNR = 5 dB for all considered correlationfactors.

In order to highlight the influence of number of antennasand correlation on sensing performance, Fig. 2 shows thatthe use of multiple antennas in a CRN system providessignificantly higher gain compared to single antenna systemwhile an increase in the correlation factor value gives a smallloss. Specifically, in Fig. 2, for the worst case of correlatedchannels, i.e., ρ = 0.8, the detection probability in this casestill outperforms single antenna system.

Fig. 3 illustrates the dependence of PDE on the numberof antennas and power correlation coefficient ρ at given SNR= 5 dB and 10 dB. We easily observe that if we increasethe number of antennas, the CRN achieves higher detectionperformance since the MRC is appropriate for the model withhigh number of antennas. For example, when ρ = 0.2 andSNR = 5dB and the number of antennas varies from 4 to9, the detection performance is approximately improved from0.45 to 0.9.

Fig. 4 provide the complementary ROC curves at SNR = 5dB and 10 dB and power correlation coefficient ρ = 0.2.We can clearly see that the sensing performance is improvedwhenever the number of antennas increases despite antennacorrelation. However, reducing the number of antennas makesthe system size suitable in practical applications such as themobile terminal, i.e., the trade-off refers to a slight loss ofdetection performance by using the appropriate number ofantennas (about less than 8 antennas).

V. CONCLUSION

In this paper, we analyzed sensing performance of an energydetection approach used in CRNs when multiple antennas areemployed. By exploiting the spatial diversity offered by thewireless channels, we use the MRC technique to obtain higherdetection performance. To cope with practical applications,we investigate the effect of equicorrelation between adjacentantennas on sensing performance. Based on performance anal-ysis, it is shown that the sensing performance degradationis proportional to the spatial correlation. However, we canmitigate this problem by increasing the number of antennas.

APPENDIX

A. Evaluation of Υ(B, a1, a2, p, q) in (12)

We consider the following integral

Υ(B, a1, a2, p, q) =∞∫0

QB

(a1√γ, a2

)γq−1 exp(−p2γ/2)dγ (22)

Let γ = x2, then (22) can be written as

12Υ(B, a1, a2, p, q) =

∞∫0

QB(a1x, a2)x2q−1 exp(−p2x2/2)dx (23)

From (10), we have

QB(a1x, a2) =1

(a1x)B−1

∞∫a2

y exp[−y2+(a1x)2

2

]yB−1IB−1(a1xy)dy

(24)

Now, we use the rule of integration by parts∫udv = uv −

∫vdu

with u = yB−1IB−1(a1xy), dv = y exp(−y2+(a1x)2

2 ),and calculate du = a1xy

B−1IB−2(a1xy), v =

68

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− exp(−y2+(a1x)2

2 ). Then, recursion method is appliedto (24) to yield

12Υ(B, a1, a2, p, q)

=B−1∑i=0

(a2)2iΓ(q) exp(−a2

2/2)

2i+1Γ(i+ 1)(p2 + a21)

2q 1F1

(q; i+ 1;− a2

1a22

4(p2 + a21)

)

+2q−1(q − 1)!

p2q

a21

p2 + a21

exp(− a2

2p2

2(p2 + a21)

)

×[q−2∑

n=0

(p2

p2 + a21

)n

Ln

(− a2

1a22

2(p2 + a21)

)

+(

1 +p2

a21

)(p2

p2 + a21

)q−1

Lq−1

(− a2

1a22

2(p2 + a21)

)](25)

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