This document analyzes the performance of cognitive radio networks using maximal ratio combining over correlated Rayleigh fading channels. It presents a simple analytical method to derive closed-form expressions for the probabilities of detection and false alarm. The key findings are: 1) The detection probability is a monotonically increasing function of the number of antennas, as more antennas provides more diversity gain. 2) Antenna correlation degrades the sensing performance compared to independent antennas. Higher correlation results in lower detection probability. 3) Complementary receiver operating characteristic curves illustrate that both higher signal-to-noise ratio and lower antenna correlation improve detection performance by increasing the detection probability and decreasing the probability of miss at a given false alarm probability.

1. 1
Performance of Cognitive Radio Networks with
Maximal Ratio Combining over Correlated
Rayleigh Fading
Trung Q. Duong‡, Thanh-Tan Le†, and Hans-J¨urgen Zepernick‡
‡Blekinge Institute of Technology, Ronneby, Sweden
E-mail: {quang.trung.duong, hans-j¨urgen.zepernick}@bth.se
†University of Ulsan, Korea
Email: 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. We
present a simple approach to derive the probability of detection
in closed-form expression. The numerical results reveal that the
detection performance is a monotonically increasing function
with respect to the number of antennas. Moreover, we provide
sets of complementary receiver operating characteristic (ROC)
curves to illustrate the effect of antenna correlation on the
sensing performance of cognitive radio networks employing MRC
schemes in some respective scenarios.
I. INTRODUCTION
In cognitive radio technologies, dynamic spectrum access
has gained significant attention in the research community
since it enables much higher spectrum efficiency and reduces
the spectrum scarcity. To be more specific, cognitive radio
network (CRN) systems allow the secondary user (SU) to
operate on “spectrum holes” that are licensed to the primary
users (PUs). One of the challenges in spectrum sensing is how
to detect “spectrum holes” of primary users even in the low
signal-to-noise ratio (SNR) regimes. To cope with this chal-
lenge, multiple antennas applied in CRNs have been recently
considered to improve the sensing performance. However, due
to the correlation between adjacent antennas, the performance
of CRN is significantly degraded.
Related work to energy detection using multiple antennas
with combining technique have primarily been addressed in
[1]–[4]. These works followed the probability density function
(PDF) based approach to derive closed-form expressions for
detection performances taking into account diversity reception
including a series of combining techniques such as maximal
ratio combining (MRC), selection combining (SC), equal
gain 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 generating
function (MGF) based approach and applied an alternative
contour integral representation of Marcum-Q function to
transform the integral to the complex domain. This method
mitigated many difficulties for calculating the integrals to get
the performances of the MRC energy detector over identically
and independently distributed (i.i.d.) Nakagami-m and Rician
fading channels.
One of the severe problems of multiple antenna systems
is the correlation between adjacent antennas. It has been
shown that the correlation of these antennas degrades spatial
diversity gain [5]–[8]. In particular, Digham et al. [1] have
analyzed the effect of the spatial correlation on the detection
performances in case of using SLC over the exponential
correlated Rayleigh fading channels. In addition, the effect
of antenna correlation on the sensing performances has been
examined for cooperative sensing [9]. Moreover, Kim et al.
[10] have analyzed the detection performances in correlated
CRN by using the central limit theorem (CLT).
To circumvent the above problem, we present a simple
analytical method to derive the closed-form expressions for
probabilities of detection PDE and false-alarm PF A over
correlated Rayleigh fading channels by using a PDF based
approach. It will be shown in our paper that the MRC energy
detector achieves significantly high performance in compar-
ison to energy detector using a single antenna. The adverse
effect of spatial correlation on spectrum sensing performance
is also analyzed by comparing the detection probability with
and without correlation. Finally, complementary receiver op-
erating characteristic (ROC) curves are obtained by plotting
probabilities of miss, PM = 1 − PDE, versus probability of
false alarm PF A for different scenarios.
The rest of this paper is organized as follows. Section II
briefly reviews the sensing performance in terms of probability
of detection PDE and probability of a false alarm PF A evalu-
ated over additive white Gaussian noise (AWGN) and Rayleigh
fading channels. An MRC technique applied to a multiple
antenna CRN receiver for both correlated and i.i.d. Rayleigh
fading channels is presented in Section III. In Section IV, we
show the numerical results and analysis. Finally, concluding
remarks are given in Section V.
II. SYSTEM MODEL AND SINGLE ANTENNA SENSING
PERFORMANCE
Let s(t) be the primary user signal that is transmitted over
the channel with gain h and additive zero-mean and variance
978-1-4244-7057-0/10/$26.00 ©2010 IEEE
65

2. 2
N0 AWGN n(t). Let W be the signal bandwidth, T be the
observation time over which signal samples are collected and
B = TW be the time-bandwidth product. We assumed that B
is an integer. The hypothesis tests for spectrum sensing H0
and H1 related to the fact that the primary user is absent or
present, 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 expressed
as [11]
n(t) =
2B
i=1
nisinc(2Wt − i), 0 < t < T (2)
with ni = n i
2W considered as Gaussian random variable
according to CLT. Under H0, the normalized noise energy
can be modified from [12]
Y = 1/ (2N0W)
2B
i=1
n2
i (3)
Obviously, Y can be viewed as the sum of the squares
of 2B standard Gaussian variates with zero mean and unit
variance. Therefore, Y has a central chi-squared distribution
with 2B degrees of freedom. Under H1, the same approach
is 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, the
hypothesis test (1) can be written as
H0 : Y ∼ χ2
2B
H1 : Y ∼ χ2
2B (2γ)
(4)
Hence, the PDF of Y can be expressed as
fY (y) =
⎧
⎨
⎩
1
2BΓ(B)
yB−1
exp −y
2 , H0
1
2
y
2γ
B−1
2
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 defined
as [12]
PDE = P(Y > λ|H1) (6)
PF A = 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)
PF A =
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 −
x2
+ a2
2
IB−1(ax)dx
(10)
B. Detection and False-Alarm Probabilities over Rayleigh
Channels
In this section, we derive the average detection probability
PDE over a Rayleigh fading channel. Clearly, PF A will remain
the same because it is independent of the SNR. The detection
probability can be given by
PDE =
∞
0
QB 2γ,
√
λ
1
¯γ
exp(−γ/¯γ)dγ (11)
To obtain a closed-form expression of (11), we now introduce
an integral Υ(.) shown in the Appendix A as
Υ(B, a1, a2, p, q) =
∞
0
QB (a1
√
γ, a2) γq−1
exp(−p2
γ/2)dγ
=
B−1
i=0
(a2)
2i
Γ(q) exp(−a2
2/2)
2iΓ(i + 1)(p2 + a2
1)
2q 1F1 q; i + 1; −
a2
1a2
2
4(p2 + a2
1)
+
2q
(q − 1)!
p2q
a2
1
p2 + a2
1
exp −
a2
2p2
2(p2 + a2
1)
×
q−2
n=0
p2
p2 + a2
1
n
Ln −
a2
1a2
2
2(p2 + a2
1)
+ 1 +
p2
a2
1
p2
p2 + a2
1
q−1
Lq−1 −
a2
1a2
2
2(p2 + a2
1)
(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 as
follows:
PDE =
1
¯γ
Υ(B,
√
2,
√
λ,
2
¯γ
, 1) (14)
III. MULTI-ANTENNAS SENSING PERFORMANCE
As mentioned above, spectrum sensing plays an important
role of a CRN system. If an SU does not detect properly the
“spectrum holes”, it unintentionally causes interference to a
PU’s signal. Hence, it is motivated to find an accurate primary
signal detection approach. To obtain a reliable detection,
multiple antennas in a CRN can be used to exploit fully the
amount of diversity offered by the channels.
In this section, we consider a CRN system that includes L
antennas. The channels between the PU transmitter and SU
receiver antennas are i.i.d. Rayleigh fading channels. We now
exploit the spatial diversity of multiple antennas at SU by using
66

3. 3
MRC techniques. However, in CRNs, the long path from the
PU and the SU may cause a small angular spread value at
the SU which creates a correlation between adjacent anten-
nas. Therefore, we also examine the effect of equicorrelated
Rayleigh 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, the
sum of the SNRs on the individual receiver antennas, can be
given 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
variables
with 2B degrees of freedom and non-centrality parameter 2γi,
we observe that YMRC is a non-central distributed variable
with 2LB degrees of freedom and non-centrality parameter
2
L
i=1 γi = 2γMRC. Then, the PDE at the MRC output for
AWGN 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
¯γL
exp(−γ/¯γ) (18)
The average PDE for MRC scheme, PDE,MRC, can be
obtained by averaging (17) over (18) and comparing it with
the integral (12):
PDE,MRC =
1
(L − 1)!
1
¯γL
Υ LB,
√
2,
√
λ, 2/¯γ, L (19)
B. Equicorrelated Rayleigh Channels
In this case, we consider the slow nonselective correlated
Rayleigh fading channels having equal branch powers and the
same correlation between any pair of branches, i.e., ρij = ρ,
i, j = 1, 2, ..., L, denotes the power correlation coefficient
between the i-th and j-th antennas. For L equicorrelated
Rayleigh 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
10
0
10
−3
10
−2
10
−1
10
0
Probability of a False Alarm PFA
ProbabilityofMissPM
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 correlated
Rayleigh channel at different power correlation coefficient ρ and SNR values
(B = 6, L = 8).
0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
ProbabilityofDetectionPDE
ρ = 0.2
ρ = 0.4
ρ = 0.6
ρ = 0.8
single antenna
IID multiple antennas
PDE increases as
ρ decreases
Fig. 2. Probability of detection versus SNR when MRC applied to
equicorrelated Rayleigh fading channels, B = 6, PF A = 0.01, L = 8,
ρ = 0.2, 0.4, 0.6, 0.8.
The detection probability PDE,MRC,Corr can be obtained by
averaging (17) over (20) and using (12), giving
PDE,MRC,Corr =
a b
b−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 illustrate
the effect of antenna correlation on the sensing performance
of CRNs.
Fig. 1 shows the sensing performance of CRN with MRC
for the time-bandwidth product B = 6 and the number of
antennas L = 8. As can be seen from Fig. 1 where comple-
mentary ROC curves at the given SNR value are presented,
67

4. 4
4 5 6 7 8 9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of antennnas (L)
ProbabilityofDetectionPDE
ρ = 0.2
ρ = 0.4
ρ = 0.6
ρ = 0.8
IID multiple antennas
SNR = 5 dB
SNR=10 dB
Fig. 3. Probability of detection versus number of antennas L when MRC
applied to equicorrelated Rayleigh fading channels, B = 6, PF A = 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
10
0
10
−3
10
−2
10
−1
10
0
Probability of a False Alarm PFA
ProbabilityofMissPM
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 correlated
Rayleigh channel at different L (SNR = 5 dB or 10 dB, B = 6, ρ = 0.2).
antenna correlation between two adjacent antennas makes
detection performance deteriorate. Note that a correlation is
caused not only by a close distance between two adjacent
antennas but also a small angular spread value generated
by the great distance between the primary transmitter and
the sensing node of the CRN. Moreover, spectrum sensing
performance degradation is proportion to the decrease of the
SNR. In particular, the sensing performance at SNR = 7
dB outperforms SNR = 5 dB for all considered correlation
factors.
In order to highlight the influence of number of antennas
and correlation on sensing performance, Fig. 2 shows that
the use of multiple antennas in a CRN system provides
significantly higher gain compared to single antenna system
while an increase in the correlation factor value gives a small
loss. Specifically, in Fig. 2, for the worst case of correlated
channels, i.e., ρ = 0.8, the detection probability in this case
still outperforms single antenna system.
Fig. 3 illustrates the dependence of PDE on the number
of antennas and power correlation coefficient ρ at given SNR
= 5 dB and 10 dB. We easily observe that if we increase
the number of antennas, the CRN achieves higher detection
performance since the MRC is appropriate for the model with
high number of antennas. For example, when ρ = 0.2 and
SNR = 5dB and the number of antennas varies from 4 to
9, the detection performance is approximately improved from
0.45 to 0.9.
Fig. 4 provide the complementary ROC curves at SNR = 5
dB and 10 dB and power correlation coefficient ρ = 0.2.
We can clearly see that the sensing performance is improved
whenever the number of antennas increases despite antenna
correlation. However, reducing the number of antennas makes
the system size suitable in practical applications such as the
mobile terminal, i.e., the trade-off refers to a slight loss of
detection performance by using the appropriate number of
antennas (about less than 8 antennas).
V. CONCLUSION
In this paper, we analyzed sensing performance of an energy
detection approach used in CRNs when multiple antennas are
employed. By exploiting the spatial diversity offered by the
wireless channels, we use the MRC technique to obtain higher
detection performance. To cope with practical applications,
we investigate the effect of equicorrelation between adjacent
antennas on sensing performance. Based on performance anal-
ysis, it is shown that the sensing performance degradation
is proportional to the spatial correlation. However, we can
mitigate 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(−p2
γ/2)dγ
(22)
Let γ = x2
, then (22) can be written as
1
2 Υ(B, a1, a2, p, q) =
∞
0
QB(a1x, a2)x2q−1
exp(−p2
x2
/2)dx
(23)
From (10), we have
QB(a1x, a2) =
1
(a1x)B−1
∞
a2
y exp −y2
+(a1x)2
2 yB−1
IB−1(a1xy)dy
(24)
Now, we use the rule of integration by parts
udv = uv − vdu
with u = yB−1
IB−1(a1xy), dv = y exp(−y2
+(a1x)2
2 ),
and calculate du = a1xyB−1
IB−2(a1xy), v =
68

5. 5
− exp(−y2
+(a1x)2
2 ). Then, recursion method is applied
to (24) to yield
1
2
Υ(B, a1, a2, p, q)
=
B−1
i=0
(a2)
2i
Γ(q) exp(−a2
2/2)
2i+1Γ(i + 1)(p2 + a2
1)
2q 1F1 q; i + 1; −
a2
1a2
2
4(p2 + a2
1)
+
2q−1
(q − 1)!
p2q
a2
1
p2 + a2
1
exp −
a2
2p2
2(p2 + a2
1)
×
q−2
n=0
p2
p2 + a2
1
n
Ln −
a2
1a2
2
2(p2 + a2
1)
+ 1 +
p2
a2
1
p2
p2 + a2
1
q−1
Lq−1 −
a2
1a2
2
2(p2 + a2
1)
(25)
REFERENCES
[1] F. F. Digham, M.-S. Alouini, and M. K. Simon, “On the energy detection
of unknown signals over fading channels,” IEEE Trans. Commun.,
vol. 55, no. 1, pp. 21–24, Jan. 2007.
[2] A. Pandharipande and J.-P. M. G. Linnartz, “Performance analysis of
primary user detection in a multiple antenna cognitive radio,” in Proc.
IEEE International Commun. Conf., Glasgow, Scotland, Jun. 2007, pp.
6482–6486.
[3] S. P. Herath, N. Rajatheva, and C. Tellambura, “Unified approach for
energy detection of unknown deterministic signal in cognitive radio over
fading channels,” in Proc. IEEE International Commun. Conf., Dresden,
Germany, Jun. 2009, pp. 745–749.
[4] S. P. Herath and N. Rajatheva, “Analysis of equal gain combining in
energy detection for cognitive radio over Nakagami channels,” in Proc.
IEEE Global Commununications Conf., New Orleans, U.S.A., Nov.
2008, pp. 1–5.
[5] G. D. Durgin and T. S. Rappaport, “Effects of multipath angular spread
on the spatial cross-correlation of received voltage envelopes,” in Proc.
IEEE Veh. Technol. Conf., Houston, U.S.A., May 1999, pp. 996–1000.
[6] Z. Xu, S. Sfar, and R. S. Blum, “Analysis of MIMO systems with receive
antenna selection in spatially correlated Rayleigh fading channels,” IEEE
Trans. Veh. Technol., vol. 58, no. 1, pp. 251–262, Jan. 2009.
[7] B. Y. Wang and W. X. Zheng, “BER performance of transmitter antenna
selection/receiver-MRC over arbitrarily correlated fading channels,”
IEEE Trans. Veh. Technol., vol. 58, no. 6, pp. 3088–3092, Jul. 2009.
[8] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading
correlation and its effect on the capacity of multielement antenna
systems,” IEEE Trans. Commun., vol. 48, no. 3, pp. 502–513, Mar. 2000.
[9] A. Ghasemi and E. S. Sousa, “Asymptotic performance of collabora-
tive spectrum sensing under correlated log-normal shadowing,” IEEE
Commun. Lett., vol. 11, no. 1, pp. 34–36, Jan. 2007.
[10] J. T. Y. Ho, “Sensing performance of energy detector with correlated
multiple antennas,” IEEE Signal Process. Lett., vol. 16, no. 8, pp. 671–
674, Aug. 2009.
[11] C. E. Shannon, “Communication in the presence of noise,” Proc. of the
IRE, vol. 37, no. 1, pp. 10–21, Jan. 2009.
[12] H. Urkowitz, “Energy detection of unknown deterministic signals,” Proc.
of the IEEE, vol. 55, no. 4, pp. 523–531, Apr. 1967.
[13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and
Products, 6th ed. San Diego, CA: Academic Press, 2000.
[14] A. H. Nuttall, “Some integrals involving the qm function,” IEEE Trans.
Inf. Theory, vol. 21, no. 1, pp. 95–96, Jan. 1975.
[15] G. L. St¨uber, Principles of mobile communication, 2nd ed. Netherlands:
Kluwer Academic, 2001.
[16] R. K. Mallik and M. Z. Win, “Channel capacity in evenly correlated
Rayleigh fading with different adaptive transmission schemes and max-
imal ratio combining,” in Proc. IEEE Int. Symp. on Inform. Theory,
Sorrento, Italy, Jun. 2000, p. 412.
69