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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 urgen.zepernick}@bth.se E-mail: {quang.trung.duong, hans-j¨ † University of Ulsan, Korea Email:
[email protected]
Abstract—In this paper, we apply the maximal ratio combining (MRC) technique to achieve higher detection probability in cognitive 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. I NTRODUCTION 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 challenge, 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 978-1-4244-7057-0/10/$26.00 ©2010 IEEE
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 comparison 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 operating 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 evaluated 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. S YSTEM M ODEL AND S INGLE A NTENNA S ENSING P ERFORMANCE Let s(t) be the primary user signal that is transmitted over the channel with gain h and additive zero-mean and variance 65
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N0 AWGN n(t). Let W be the signal bandwidth, T be the observation time over which signal samples are collected and B = T W 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:
where Γ(., .) is the upper incomplete gamma function [13, Sec. (8.350)]. QB (a, b) is the generalized Marcum Q-function [14] defined by 2
∞ 1 x + a2 B QB (a, b) = B−1 x exp − IB−1 (ax)dx a 2 b
H0 : Y = n(t) H1 : Y = hs(t) + n(t)
where Y is the received signal and noise n(t) can be expressed as [11] n(t) =
2B
ni sinc(2W t − i), 0 < t < T
(2)
i=1
i considered as Gaussian random variable with ni = n 2W according to CLT. Under H0 , the normalized noise energy can be modified from [12] Y = 1/ (2N0 W )
2B
n2i
(3)
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 ∞
√ 1 exp(−γ/¯ γ )dγ (11) 2γ, λ PDE = QB γ¯ 0
To obtain a closed-form expression of (11), we now introduce an integral Υ(.) shown in the Appendix A as ∞
i=1
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 noncentral distribution χ2 with 2B degrees of freedom and a noncentrality parameter 2γ [12], where γ is the SNR. Then, the hypothesis test (1) can be written as H0 : Y ∼ χ22B H1 : Y ∼ χ22B (2γ)
(4)
Hence, the PDF of Y can be expressed as ⎧ 1 H0 y B−1 exp − y2 , ⎨ 2B Γ(B) B−1 fY (y) = √ 2 ⎩ 1 y exp − 2γ+y IB−1 2γy , H1 2
2γ
(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 Channels The probability of detection and false alarm can be defined as [12] PDE = P (Y > λ|H1 ) PF A = P (Y > λ|H0 )
(6) (7)
where λ is a detection threshold. Using (4) to evaluate (5) and (6) yields [12]
√ 2γ, λ (8) PDE = QB Γ(B, λ/2) Γ(B)
Υ(B, a1 , a2 , p, q) =
√ QB (a1 γ, a2 ) γ q−1 exp(−p2 γ/2)dγ
0 B−1
2i
(a2 ) Γ(q) exp(−a22 /2)
(9)
a21 a22 q; i + 1; − 4(p2 + a21 )
2
2q 1 F1 2i Γ(i + 1)(p2 + a21 ) 2 (q − 1)! a21 a22 p + exp − 2 p2q p2 + a1 2(p2 + a21 )
n
q−2 p2 a21 a22 L − × n p2 + a21 2(p2 + a21 ) n=0
q−1
p2 a21 a22 p2 L − + 1+ 2 (12) q−1 a1 p2 + a21 2(p2 + a21 )
=
i=0 q
where
2
PF A =
(10)
(1)
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 1 F1 (.) is the confluent hypergeometric function [13, Sec. (9.2)]. From (11) and (12), we obtain the closed-form PDE as follows: √ √ 2 1 , 1) (14) PDE = Υ(B, 2, λ, γ¯ γ¯ III. M ULTI - ANTENNAS S ENSING P ERFORMANCE 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
YM RC =
L
Yi =
i=1
L
h∗i ri (t)
(15)
i=1
where L is the number of antennas. The received SNR, the sum of the SNRs on the individual receiver antennas, can be given by γM RC =
L
0
10
SNR = 5 dB
−1
10 Probability of Miss PM
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 antennas. 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
−2
10
SNR = 5, ρ = 0.2 SNR = 5, ρ = 0.4 SNR = 5, ρ = 0.6 SNR = 5, ρ = 0.8 −3
SNR = 7, ρ = 0.2
10
SNR = 7, ρ = 0.4 SNR = 7, ρ = 0.6
SNR = 7 dB
SNR = 7, ρ = 0.8
γi
(16)
−4
10
i=1
where γi is the SNR on the i-th antenna.
−2
−1
0
10
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).
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 YM RC is a non-central distributed variable with L2LB degrees of freedom and non-centrality parameter 2 i=1 γi = 2γM RC . Then, the PDE at the MRC output for AWGN channels can be evaluated from (8) as
√ 2γM RC , λ (17) PDE,M RC = QLB It is well known that the PDF of γM RC is given by [15, Eq. (6.23)] (18)
The average PDE for MRC scheme, PDE,M RC , can be obtained by averaging (17) over (18) and comparing it with the integral (12): √ √
1 1 Υ LB, 2, λ, 2/¯ γ , L (19) PDE,M RC = L (L − 1)! γ¯
1 0.9 0.8 Probability of Detection PDE
A. I.I.D. Rayleigh Channels
γ L−1 1 exp(−γ/¯ γ) fM RC (γ) = (L − 1)! γ¯ L
−3
10 10 10 Probability of a False Alarm PFA
0.7 0.6
PDE increases as ρ decreases
0.5 ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8 single antenna IID multiple antennas
0.4 0.3 0.2 0.1 0
0
5
10
15
SNR (dB)
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.
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 γM RC is given by [16] exp(−aγ) fM RC (γ) = abL−1 (b−a) L−1 L−1 (20) γ k−1 − exp(−bγ) , γ≥0 (b−a)L−k (k−1)! k=1
where 1 √ γ¯ 1 + (L − 1) ρ 1 b= √ γ¯ 1 − ρ
a=
The detection probability PDE,M RC,Corr can be obtained by averaging (17) over (20) and using (12), giving PDE,M RC,Corr = L−1 √ √ √ b a b−a Υ(LB, 2, λ, 2a, 1) − abL−1 L−1 √ √ √ 1 × Υ(LB, 2, λ, 2b, k) (b−a)L−k (k−1)!
(21)
k=1
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 complementary ROC curves at the given SNR value are presented, 67
4
1
Probability of Detection PDE
0.9
0.8
0.7
SNR=10 dB 0.6
0.5
ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8 IID multiple antennas
SNR = 5 dB 0.4
0.3
4
5
6 7 Number of antennnas (L)
8
9
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
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 SN R = 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
0
10
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 analysis, 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.
SNR = 5dB
−1
Probability of Miss PM
10
−2
10
L = 4, SNR = 10dB L = 5, SNR = 10dB L = 6, SNR = 10dB L = 7, SNR = 10dB L = 8, SNR = 10dB L = 4, SNR = 5dB
−3
10
L = 5, SNR = 5dB L = 6, SNR = 5dB L = 7, SNR = 5dB
SNR = 10 dB
A PPENDIX
L = 8, SNR = 5dB −4
10
−3
−2
−1
10 10 10 Probability of a False Alarm PFA
0
10
A. Evaluation of Υ(B, a1 , a2 , p, q) in (12) We consider the following integral
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 SN R = 7 dB outperforms SN R = 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.
Υ(B, a1 , a2 , p, q) = √ ∞ QB a1 γ, a2 γ q−1 exp(−p2 γ/2)dγ
(22)
0
2
Let γ = x , then (22) can be written as 1 2 Υ(B, a1 , a2 , p, q) ∞ 0
=
QB (a1 x, a2 )x2q−1 exp(−p2 x2 /2)dx
(23)
From (10), we have QB (a1 x, a2 ) = 2 ∞ y +(a1 x)2 1 y B−1 IB−1 (a1 xy)dy y exp − B−1 2 (a x) 1
(24)
a2
Now, we use the rule of integration by parts udv = uv − vdu 2
2
1 x) ), with u = y B−1 IB−1 (a1 xy), dv = y exp(− y +(a 2 B−1 IB−2 (a1 xy), v = and calculate du = a1 xy
68
5 2
2
1 x) − exp(− y +(a ). Then, recursion method is applied 2 to (24) to yield
1 Υ(B, a1 , a2 , p, q) 2
B−1 (a2 )2i Γ(q) exp(−a2 /2) a21 a22 2 = F q; i + 1; − 1 1 i+1 Γ(i + 1)(p2 + a2 )2q 4(p2 + a21 ) 1 i=0 2
2q−1 (q − 1)! a21 a22 p2 + exp − p2q p2 + a21 2(p2 + a21 )
q−2 n p2 a21 a22 Ln − × p2 + a21 2(p2 + a21 ) n=0
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