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A Novel Collaborative Spectrum Sensing Scheme Based on Covariance Matrix

Kejun Lei?, Xi Yang?,? , Shengliang Peng? and Xiuying Cao?

?

College of Physics Science and Information Engineering. Jishou University Jishou, Hunan Province, P.R.China leikejun-123@163.com

?

National Mobile Communication Research Lab. Southeast University Nanjing, Jiangsu Province, P.R.China ynkej@163.com

Abstract—The statistical covariance matrixes of received signal and noise are usually different, thus the distinguishing property can be used to detect whether the primary user exists or not. In this paper, according to the multivariate statistical theory, a collaborative spectrum sensing method based on covariance matrix is introduced. The proposed scheme exhibits better performance than the conventional cooperative scheme for correlated signal or correlated noise case, especially for correlated noise case. On the other hand, it approaches to the OR-rule fusion scheme when both signal and noise are independent and identically distributed (i.i.d.). In addition, the decision threshold can be easily obtained through theoretical computing whether the received noise samples at cognitive users are correlated or not. Simulations based on various types of received signals are given to verify the method. Keywords—cognitive radio; collaborative spectrum sensing; covariance matrix; generalized likelihood ratio test (GLRT)

The notations conform to the following conventions. Vectors are column vectors and are denoted in lower case bold, e.g., x. Matrices are upper case bold, e.g., A. The superscript T stands for transpose operator. tr(A), det(A) and etr(A) is trace, determinant and exponential trace of A, respectively. Iq is the q × q identity matrix. By A;0 we indicate A is positive definite. Vec{A} means to form a new vector by stacking the columns of A under each other. ? denotes Kronecker product. E{·} and Cov{·} represent the expected value and covariance operator. II. SYSTEM MODEL In CRN with N unlicensed users, the hypothesis test for spectrum sensing at the kth time instant is formulated as H0: xi(k) = ni(k) i = 1, 2, … , N , i = 1, 2, … , N . (1)

I. INTRODUCTION In a cognitive radio network (CRN), spectrum sensing plays a fundamental role. However, detecting the presence of primary users is practically difficult [1]. Thus, collaborative spectrum sensing methods exploiting spatial diversity among cognitive users have been investigated by many researchers [1]-[3]. Classical cooperative schemes include OR-rule and energy fusion (EF, i.e., Square-Law Combining (SLC) [2]), however, both of them adopt energy detector as a building block for its simplicity, which causes they aren’t the optimal scheme for detecting correlated signals. An optimal linear cooperation sensing scheme based on nonlinear optimization problem has been proposed in [1], however it is also with the assumption that noise samples are i.i.d.. To overcome the shortcoming of collaborative sensing schemes mentioned above, we introduce a new collaborative spectrum sensing scheme based on statistical covariance matrix of received signal. Since the covariance matrixes of received signal and noise samples are usually different, the distinguishing property is used in the proposed scheme to detect whether the primary user exists or not. The multivariate statistical theory has been used to analyze the method and obtain the decision threshold. Simulation results indicate that the proposed scheme exhibits much better performance than conventional cooperative scheme for correlated signal or correlated noise, especially for correlated noise case. On the other hand, it approaches to the OR-rule fusion scheme when both signal samples and noise samples are i.i.d..

H1: xi(k) = si(k) + ni(k)

where H0 indicates primary signal does not exist, while H1 indicates primary signal exists, si(k) and ni(k) denote the signal sample transmitted by primary users and noise sample of the ith cognitive user, respectively. In this paper, we assume si(k) and ni(k) are Gaussian random process [4]. Without loss of generality, si(k) and ni(k) are assumed to be independent. In addition, we assume the distance between every two cognitive users is long enough to guarantee si(k)’s of different cognitive users are independent. We consider P consecutive samples at each cognitive user and then define the following vectors: xi(k) = [ xi(k) xi(k-1) " xi(k – P + 1) ]T. ni(k) = [ ni(k) ni(k-1) " ni(k – P + 1) ]T . si(k) = [ si(k) si(k-1) " si(k – P + 1) ]T . (2) (3) (4)

In our scheme, all of the cognitive users are located in a same cluster [3], so that we can assume that they have same statistical covariance matrix about the primary signal samples or noise samples: ni(k) ~ NP( μ0, Σ0 ) i = 1, 2, … , N , si(k) ~ NP(μ1, Σ1 ) i = 1, 2, … , N , (5) (6)

where E{ni(k)}=μ0, Cov{ni(k)}=Σ0, E{si(k)}=μ1, Cov{si(k)} = Σ1, Σ0;0 and Σ1;0. Here, we have no restriction that the signal samples or the noise samples are i.i.d.. Note that the ni(k) and si(k) are statistical independent, we

978-1-4244-3693-4/09/$25.00 ?2009 IEEE

can then easily obtain the distribution of xi(k) using the property of multivariate normal random distribution, i.e., xi(k) ~ NP(μ0 + μ1, Σ0 + Σ1 ) , (7)

here E{xi(k)} = μ0 + μ1, Cov{xi(k)} = Σ0 + Σ1. Due to Σ0 ; 0 and Σ1;0, thus Cov{xi(k)};0. If the primary signal presents, then Σ1 ≠ 0; conversely, if the primary signal dose not exist, then Σ1 = 0. Hence, the covariance matrix of received signal should not equal to Σ0 when the primary signal is present. In this paper, this distinguishing property is used to detect whether the primary user exists or not. All samples at each cognitive user are transmitted to fusion central through a dedicated control channel [1], and then the fusion central make a decision based on the proposed property. Hence, problem (1) is equivalent to the following representation H0: Rx = Σ0 , H1: Rx ≠Σ0 . (8)

hypothesis H1 and μ are unknown, we can use the maximum likelihood estimation (MLE) to obtain these parameters. The detector based on the MLE is called GLRT detector, which is usually effective for signal detection problems [5]. Then, the test statistic is defined as

L ( X) =

μ∈R P , Σ 0 μ∈R

max p ( X; μ, Σ , H1 )

max p ( X; μ, Σ , H0 ) P

.

(12)

Now the key problem is converted to find those values of μ and Σ to maximize the likelihood function (11). B.J. Muirhead and R. L. Dykstra have found the MLE of μ and Σ [6], which is described in the following proposition. Proposition 1: Assume that signal samples vectors received by each cognitive user, i.e., xi(k)~NP(μ, Σ), i=1,…,k and N > P, then the MLE of μ and Σ are

μ＝ 1 N

∑x

j =1

N

j

(k ) ,

(13) (14)

where Rx denotes the covariance matrix of received signal. We have assumed that the covariance matrix of noise samples is known in (8). In practice, when the primary users are silent, Σ0 can be approximated by the sample correlations matrix, which is the unbiased estimator of the true covariance matrix of received signal [5]. III. COVARIANCE MATRIX-BASED COLLABORATIVE SPECTRUM SENSING A larger probability of detection (Pd) in CRN results in less harmful interference to primary users, and a smaller probability of false alarm (Pf) leads to higher spectrum efficiency. Thus, for a good spectrum sensing method, the common objective is to maximize Pd while satisfying a pre-defined Pf. Usually we have less information about the primary signal, so the threshold is selected based on Pf. In this section, the multivariate statistical theory is adopted to analyze Pf and determine the threshold. The received signal samples of each cognitive user are collected to form N × P received data matrix X = [ x1(k), x2(k), " , xN(k) ]T, (9) the ith row of X is the signal samples of the ith cognitive user. According to the analysis in Section II, we have known that x1(k), x2(k), " , xN(k) are P-dimensional independent normally distributed random vectors, hence, X is normally distributed random matrix , i.e., X ~ NN×P(1NμT, IN ? Σ ),

T

and

Σ＝ 1 Ν

∑ (x

j =1

N

j

(k ) ? μ)(x j ( k ) ? μ )T .

In fact, the imposed condition N > P is used to guarantee that the MLE of Σ , i.e., Σ , is positive definite with probability 1 [6]. Based on Proposition 1, we can easily conclude that: Theorem 1: In collaborative covariance matrix based detection (denoted by CCMD for simplicity) scheme, the numbers of cooperative cognitive users must greater than the samples numbers of each cognitive user. Obviously, the condition in Theorem 1 can be easily satisfied. Now we come to derive the test statistic defined by (12). Under hypothesis H1, we have no knowledge about the primary signal, so that both Σ0 and μ are unknown. From Proposition 1, we can directly obtain

μ∈R P , Σ 0

max p ( X; μ, Σ , H1 ) = p( X; μ, Σ , H1 ) A ?N /2 1 A )） etr(? ( )?1 A ) N 2 N 2π e ? NP / 2 =( ) (det( A)) ? N / 2 N = (2π ) ? NP / 2 (det(

A = ∑ (x j ( k ) ? μ )(x j ( k ) ? μ )T .

j =1 N

(15)

where (16)

Under H0, the primary user does not exist, then Σ = Σ0, therefore, only μ is unknown here, so the problem is converted to maximize p ( X; μ, Σ, H0 ) with parameter μ ∈ \P, hence

max p ( X; μ, Σ , H0 ) = max p( X; μ, Σ 0 , H0 ) P P

μ∈R

(10)

where E{X} = 1NμT, Cov{Vec{X }} = IN ? Σ, μ = μ0 + μ1, Σ = Σ0 + Σ1 0 and 1N = [1, … , 1] ∈ \N. The PDF of X is

N 1 (2π ) ? NP / 2 (det( Σ ))? N / 2 etr(? Σ -1 ∑ (x j (k ) ? μ)(x j (k ) ? μ)T ) . 2 j =1

(11)

1 = max{(2 π ) ? NP / 2 (det( Σ 0 )) ? N / 2 etr(? Σ 0-1B)} (17) μ∈R P 2 1 -1 = (2π ) ? NP / 2 (det( Σ 0 ))? N / 2 max{etr( ? Σ 0 B)} μ∈R P 2

μ∈R

where

B = ∑ (x j (k ) ? μ)(x j (k ) ? μ)T .

j =1 N

A. Test Statistic Because we have known the PDF of received data matrix X, the likelihood ratio test (LRT) based fusion rule can be adopted to acquire the test statistic in our scheme. Since the Σ under

(18)

After some manipulations, we have

B = A + N ( μ ? μ ) ( μ ? μ )T .

(19)

Hence,

1 -1 1 -1 tr( ? Σ 0 B) = tr( ? Σ 0 (A + N ( μ ? μ)( μ ? μ )T )) 2 2 1 -1 N (20) -1 = tr( ? Σ 0 A) ? ( μ ? μ)T Σ 0 ( μ ? μ) 2 2 1 -1 ≤ tr(? Σ 0 A) 2 where in the second equality tr(CB) = tr(BC) is used; the

? T ( X) ? > γ ; H0 ? = 1 ? FFn ,m (γ ) , Pf = Pr ? ? b ?

(28)

inequality follows from the fact that ( μ ? μ)T Σ 0-1 ( μ ? μ) ≥ 0 because of Σ 0?1 0 , the equality is achieved if μ = μ . Hence, 1 ?1 1 ?1 etr(? Σ 0 B) ≤ etr(? Σ 0 A) (21) 2 2 where the inequality follows from the fact that exp(·) is a monotonically increasing function and the equality is achieved if μ = μ . According to (17) and (21), we obtain

max p ( X; μ, Σ , H0 ) = (2π ) ? NP / 2 (det( Σ 0 )) ? N / 2 etr( ? P

μ∈R

where FFn ,m (?) is the cumulative function of the F distribution. Hence, when N > 75 or P > 10 , the threshold of spectrum sensing for the CCMD scheme is (29) γ = FF?1 (1 ? PFA ) .

n ,m

For the case of N ≤ 75 and P ≤ 10 , the threshold values had been tabulated in [7]. Note that, in the proposed scheme, we can directly obtain decision threshold through (29) whether the received noise samples at cognitive users are correlated or not, however, in EF or OR-rule scheme, the decision threshold can only be obtained through Monte-Carlo simulations when the noise samples are not i.i.d.. IV. COMPUTER SIMULATION AND DISCUSSION In this section, the CCMD scheme is evaluated numerically and compared with some existing methods. Consider a network of N cognitive users and P samples at each user. Every figure implements 10000 Monte-Carlo simulations. Firstly, we evaluate the relationship between detection performance and N or P. In Fig. 1, we set P=10 and Pf =0.01, it is clear that Pd enhances as the number of cooperative cognitive users increases and the theoretical threshold for CCMD is very accurate when N is far greater than P. In Fig. 2, we set N=100 and Pf =0.01, obviously, Pd improves as sample sizes become larger, however, the practical Pf increases correspondingly. Therefore, in CCMD scheme, we should ensure that N is far greater than P in order to achieve reliably detection result. To simplify simulation, we fix Pf =0.05 and assume the covariance matrix for both noise and signal samples have same form, that is, ρ 1P× P + (1 ? ρ )I P , but with different correlation coefficient ρ for them. In Fig. 3, both noise samples and signal samples are i.i.d., i.e., ρ = 0 , which results in the EF scheme is optimal [6], [12]. Because it is very hard to control threshold for each cognitive user, we use theory curve to replace the simulation result for OR-rule. Clearly, the detection performance of CCMD approaches to OR-rule. In Fig. 4, we assume ρ = 0 for noise samples and ρ = 0.99 for signal samples, the figure shows that the proposed scheme performs a little better than EF scheme at high SNR and approaches to it at low SNR. In Fig. 5 and Fig. 6, we assume ρ = 0.99 for noise samples, ρ = 0.2 and ρ = 0 for signal samples, respectively. The figures show that the proposed scheme performs much better than EF scheme. Compared Fig. 3 with Fig. 4 to Fig. 6, it is clear that the EF scheme is not optimal when noise samples or signal samples are correlated. Obviously, the detection performance improves with more cooperative users from Fig. 4 to Fig. 6. V. CONCLUSIONS A collaborative spectrum sensing scheme based on the noise covariance matrix is introduced in the paper. The multivariate statistical theory is applied to analyze the sensing threshold in

1 -1 Σ 0 A) 2

(22)

Substituting (15) and (22) into (12) gives the test statistic by

e 1 L( X) = ( ) ? NP / 2 (det( Σ 0-1 A)) N / 2 etr ( Σ 0-1 A) , N 2

(23)

and the GLRT statistic for i.i.d. noise samples is

e ? NP / 2 1 (24) ) (det( A)) N / 2 etr ( 2 A) . Nσ 2 2σ Here, assuming the noise power level is σ 2 , i.e., Σ 0 = σ 2 I P , then L ( X) = (

(24) can be obtained by substituting Σ 0 into (23).

B. Threshold and Pf In order to implement the spectrum sensing, the decision threshold must be obtained beforehand. Due to lacking of detailed information about the primary signal, the threshold is selected based on Pf. In this section, the multivariate statistical theory is used to find the distribution of the test statistic and provided theoretical threshold and PFA. To satisfy the unbiased requirement in hypothesis testing theory, after substituting N by N-1 into (23) and (24), then taking natural logarithm of both sides of them and multiplying by 2, we can obtain following equivalent test statistics defined in following theorem. Theorem 2: In CCMD scheme, the GLRT statistic for spectrum sensing defined as follows:

T ( X) = ( N ? 1)[ln ( N ? 1) P det( Σ 0 ) 1 + tr(AΣ 0-1 ) ? P] det( A) N ?1

(25)

the GLRT statistic for i.i.d. noise samples is

T ( X) = ( N ? 1)[ln

σ 2 P ( N ? 1) P

det(A)

+

1 tr(A) ? P] . (26) ( N ? 1)σ 2

B. P. Korin found the distribution for (25) under hypothesis H0 [7]. When the number of cognitive user N > 75 , or number of samples at each cognitive user P > 10 , we have

T ( X) ? bFn, m

(27)

where Fn,m denotes F distribution with n and m degrees of n n+2 1 freedom; here, b = , n = P( P + 1) , m = , 1- D1 - n/m D2 ? D12 2 2 P + 1 ? 2 /( P + 1) ( P ? 1)( P + 2) D1 = , and D2 = . 6( N ? 1) 6( N ? 1) 2 Therefore, according to (27), the probability of false alarm for the CCMD scheme is

Probability of Detection

the scheme. Simulation results have shown the proposed scheme exhibits better performance than the conventional cooperative scheme for correlated signal or correlated noise case, especially for correlated noise case. On the other hand, it approaches to OR-rule fusion scheme when both signal and noise are i.i.d.. In addition, the decision threshold can be easily obtained through theoretical computing in the CCMD scheme. REFERENCES

[1] [2] [3] [4] [5] [6] [7] Z. Quan, S. Cui, and A. H. Sayed, “An optimal strategy for cooperative spectrum sensing in cognitive radio networks,” in Proc. IEEE GLOBECOM, Washington, DC, Nov. 2007. 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. C. Sun, W. Zhang and K. B. Letaief, “Cluster-Based cooperative spectrum sensing in cognitive radio systems,” in Proc. IEEE Int. Conf. Commun., June 2007. E. G. Larsson and G. Regnoli, “Primary system detection for cognitive radio: does small-scale fading help?,” IEEE Commun. Lett., vol. 11, no.10, pp. 799-801, Oct. 2007. S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, vol. 2. Prentice Hall, 1998. B. J. Muirhead, Aspects of Multivariate statistical theory. John Wiley & Sons, Inc, 1982. B. P. Korin, “On the distribution of a statistic used for testing a covariance matrix,” Biometrika, vol. 55, pp. 171-178, 1968.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -30 CCMD(N=100,P=16) CCMD(N=200,P=40) CCMD(N=300,P=60) EF(N=100,P=16) EF(N=200,P=40) EF(N=300,P=60) OR-the(N=100,P=16) OR-the(N=200,P=40) OR-the(N=300,P=60)

-25

-20

-15 SNR

-10

-5

0

Fig. 3. PD for both noise samples and signal samples are i.i.d.

1 0.9 0.8 Probability of Detection 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -30 CCMD(N=100,P=16) CCMD(N=200,P=40) CCMD(N=300,P=60) EF(N=100,P=16) EF(N=200,P=40) EF(N=300,P=60)

-25

-20

-15 SNR

-10

-5

0

Fig. 4. PD for noise samples are i.i.d. and signal samples are highly correlated

1 0.9 0.8 Probability of Detection 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.1 0.1 0 -30 -25 -20 -15 SNR -10 -5 0 0 -30 Pd(N=50,P=10) Pf (N=50,P=10) Pd(N=150,P=10) Pf (N=150,P=10) Pd(N=350,P=10) Pf (N=350,P=10) 1 0.9 0.8 Probability of Detection 0.7 0.6 0.5 0.4 0.3 CCMD(N=100,P=16) CCMD(N=200,P=40) CCMD(N=300,P=60) EF(N=100,P=16) EF(N=200,P=40) EF(N=300,P=60)

-25

-20

Fig. 1. PD & PFA versus N for different SNR (dB)

-15 SNR

-10

-5

0

Fig. 5. PD for both noise samples and signal samples are correlated

1

1 0.9 0.8 Probability of Detection 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -30 Pd(N=100,P=20) Pf (N=100,P=20) Pd(N=100,P=70) Pf (N=100,P=70) Pd(N=100,P=80) Pf (N=100,P=80)

0.9 0.8 Probability of Detection 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -30

CCMD(N=100,P=16) CCMD(N=200,P=40) CCMD(N=300,P=60) EF(N=100,P=16) EF(N=200,P=40) EF(N=300,P=60)

-25

-20

-15 SNR

-10

-5

0

-25

-20

-15 SNR

-10

-5

0

Fig. 2. PD & PFA versus P at different SNR (dB)

Fig. 6. PD for noise samples are highly correlated and signal samples are i.i.d.

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