Cooperative Spectrum Sensing Using Discrete Goodness of Fit Testing for Multi-Antenna Cognitive Radio System Yuxin Li, Yinghui Ye, Guangyue Lu and Cai Xu National Engineering Laboratory for Wireless Security, Xi'an University of Posts and Telecommunications, Xi'an 710121，China Email:
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[email protected] Abstract—In diversity-based multiple antenna cognitive radio system, the performance of the existing cooperative spectrum sensing algorithms based on covariance matrix degrades seriously due to the channel correlation. In this paper, we formulate spectrum sensing as a multinomial distribution test problem, and then apply the Discrete Anderson Darling (DAD) test, one of discrete goodness of fit tests, to examining the above problem. The proposed algorithm dubbed DAD sensing is assessed through Monte Carlo simulations. It is shown that the performance of DAD sensing is robust to the antenna correlation and noise uncertainty. Keywords—cognitive radio, Discrete Anderson Darling test, cooperative spectrum sensing, channel correlation, noise uncertainty
I.
INTRODUCTION
The rapid development of wireless communication technology and the fixed frequency spectrum management policy result in the shortage of spectrum resources. Cognitive Radio (CR), as a newly developing technology, can improve the spectrum utilization dramatically, whose main thought is to detect the presence of primary user (PU) within the desired frequency band and then enable secondary user (SU) to access the vacant channel rapidly without causing interference to PU. Hence, an effective and robust spectrum sensing is a prerequisite and fundamental task for CR system [1-2]. To this end, the classical spectrum sensing algorithms using single antenna mainly include Cyclostationary Feature Detection (CFD), Energy Detection algorithm (ED) and Matched-Filtering (MF) [2]. However, the complex electromagnetic environment results in unreliable spectrum sensing [3]. For example, the performance of ED algorithm is weak as a result of noise uncertainty. To overcome this fact, multi-antenna system has been widely used in cognitive radio system due to the improvement of reliable spectrum sensing. Zeng proposes two kinds of spectrum sensing algorithms by using antenna correlation: spectrum sensing methods based on eigenvalue structure and covariance. Spectrum sensing based on eigenvalue structure exploits the properties of covariance matrix eigenvalues and vectors of received signal, such as Maximum Minimum Eigenvalue (MME), Feature Template Matching and Subspace Projection [46]. Although these methods eliminate the influence of noise uncertainty, they have high complexity due to 978-1-5090-0690-8/16/$31.00 © 2016 IEEE
eigenvalue decomposition. Spectrum sensing based on covariance mainly includes Covariance Absolute Value (CAV), Maximum Correlation Coefficients (MCC) and so on [7-9]. For example, CAV algorithm employs the ratio of the absolute values of all the elements and the absolute value of the diagonal elements as a statistic test to sense available spectrum. According to Ref.[10], antenna diversity gain overcomes channel fading by means of signal space diversity reception in different spatial positions with different fading characteristics, in this way, the distance between the antennas is typically greater than ten wavelengths, which leads to the low correlation among the antennas. The performance of the proposed algorithms [7-9] descends dramatically. Meanwhile, the spectrum sensing problem is considered as a nonparametric hypothesis testing problem and the Friedman test is employed to achieve spectrum sensing. In this paper, we consider the spectrum sensing as a nonparametric hypothesis testing problem, where the received power of each antenna is equal when the spectrum is idle but different when the spectrum is busy, and then exploit another nonparametric hypothesis test (goodness of fit test, GoF test), which has been widely used in spectrum sensing [11-13]. We utilize another GoF test—the Discrete Anderson Darling (DAD) test—to examine whether the received signal comes from the assumed discrete multinomial distribution that is defined in this paper and propose a algorithm called DAD sensing to achieve spectrum sensing. Since we just calculate the antenna power, the proposed algorithm is well suited to situation with low antenna correlation and noise uncertainty. The rest of the paper is organized as follows. In section II, a brief introduction of multi-antenna spectrum sensing system model is given. In section III, the basic principles of spectrum sensing algorithm using DAD test are presented. In section IV, we test and verify the performance of the proposed methods. Finally, we draw the conclusion. II. SYSTEM MODEL A. Signal Model As shown in Fig.1, we consider a SU with multiantenna. Without loss of generality, the spectrum sensing can be expressed as a binary hypothesis test problem: H0 denotes the null hypothesis (PU signal is absent) and H1
stands for the alternative hypothesis (PU signal is present). The received signal can be described as
H0 wm ( n), xm (n) , n 1, 2, , N (1) hm s (n) wm (n), H1 where s(n) represents the PU signal. wm(n) is additive white Gaussian noise(AWGN), with mean zero and variance σ². N is the number of samples. hm denotes the channel coefficient of m-th antenna which is assumed to be constant during the sensing duration and obeys slow fading Rayleigh. Then, the signal from the m-th antenna can be denoted as
x1 x1 (1) x x (1) X 2 2 x M xM (1) Similarly,
x1 ( N ) x2 ( N ) (2) xM (2) xM ( N ) x1 (2) x2 (2)
W w1 , w2 , , w M
T
Channel gains of M antennas can be expressed as
h h1 , h2 , , hM
T
Instantaneous power matrix of received signal is
p1 (1) p (1) P XX 2 pM (1)
p1 (2) p2 (2)
pM (2)
p1 ( N ) p2 ( N ) (3) pM ( N )
e 23
( d / c )2
(6) where Λ is the angle expansion, λc is the wavelength, d is the distance between two adjacent antennas. Thus, the antenna correlation matrix H is a Toeplitz matrix. Since detection methods based on covariance take advantage of correlation between the signals to sense spectrum, the correlation descends and the sensing performance drops dramatically when the distance between antennas increases. C. Energy Detection Multi-antenna assisted by ED algorithm is a classical spectrum sensing algorithm. The total energy of received signal is regarded as the test statistic, namely N
M
TED pm (n) n 1 m 1
H1
ED
(7)
H0
It is not hard to find that the total energy of the received signal increases when the PU signal is present, therefore scholars compare statistic test TED with threshold γED to verify whether the spectrum is idle or not. Note that the threshold γED is relative to noise variance, which results that ED algorithm is affected by noise uncertainty. D. CAV algorithm In order to overcome the impact of noise uncertainty in ED algorithm, the difference of statistics covariance matrix between H0 and H1 is employed to detect whether PU exists or not. And the test statistic is
where * denotes Hadamard product of two matrices. Generally, we assume that S is independent of W, then the statistics covariance matrix of the received signal can be denoted as
G XX H
2
TCAV
(4)
1 M M Gmk M m 1 k 1 1 M Gmm M m 1
(8)
where Gmk represents the elements of m-th row, k-th column in G. CAV algorithm is popular due to the fact that CAV does not need of any priori knowledge and be free of noise uncertainty. However, the performance of CAV algorithm descends markedly or even fails when antenna correlation is low, which hinders its application in the real CR system. Fig.1 Multi-antenna system model of cognitive radio
B. Antenna Correlation Model The exponential correlation model is widely used in multi-antenna system to describe the correlation among multiple antennas. For the M antennas system, the antenna correlation matrix R is an M×M matrix with its components described as
i j , i j , i, j 1, , M , 0 1 (5) Rij * R ji , i j where ρ is a correlation coefficient between two adjacent antennas. The definition of ρ in Ref.[14] is defined as
III. DISCRETE ANDERSON DARLING SENSING ED algorithm is sensitive to noise uncertainty and the performance of CAV algorithm drops on account of low antenna correlation. In this section, we propose DAD sensing to circle the above problems. After obtaining frequency vector by using the rank of instantaneous power matrix, we transform spectrum sensing problem into whether ranks in P obey the assuming multinomial distribution or not, and exploit DAD test to test it. A. Spectrum sensing as a discrete GoF test When H0,
pm (n) wm2 (n) . Since there is only noise,
the average power of each antenna level remains the same
theoretically. When H1,
pm (n) (hm sm (n) wm2 (n)) 2 .
The channel gain hm is different over a period of time, which leads to the differences in the average power level for different antennas. Therefore, spectrum sensing can be achieved via comparison of the antenna instantaneous power. To compare instantaneous power of each antenna, we sort each column of matrix P in order to get the rank matrix, that is,
r1 (1) r1 (2) r (1) r (2) 2 R 2 rM (1) rM (2)
r1 ( N ) r2 ( N ) f (P ) rM ( N )
(9)
where the elements of R range from 1 to M. For example, we suppose that there are n columns in P, p3(n)
