ES2544_Fast blind spectrum sensing method based on Determinant of Covariance Matrix

Fast Blind Spectrum Sensing Method Based on Determinant of Covariance

Matrix

Zhiqiang Bao , Qingdong Huang , YongZhi Zhai and Guangyue Lu School of Telecommunication and Information Engineering Xi’an University of Posts and Telecommunications

xi’an City, China

baozhiqiang@https://www.sodocs.net/doc/5815829770.html,,

huangqingdong@https://www.sodocs.net/doc/5815829770.html,,cesltsinghua@https://www.sodocs.net/doc/5815829770.html,, gylu@https://www.sodocs.net/doc/5815829770.html,

Abstract. Spectrum sensing is the key problem for cognitive radio systems. A fast blind sensing method based on Determinant of Covariance Matrix (DCM) of the received signals is proposed to sense the available spectrum for the cognitive users with the help of the multiple antennas at the receiver of the cognitive users. The greatest advantage of the new method is that it requires no information of the noise power and without any eigen-decomposition (or SVD) of sample covariance matrix. Both the simulation and the analytical results demonstrate that the proposed method is effectiveness and robustness.

Keywords: Spectrum sensing, cognitive radio, determinant, blind sensing 1. Introduction

Cognitive radio (CR) [1] has recently emerged as a promising technology to increase the spectrum utilization in wireless communications. In a CR network, secondary users (SUs) continuously sense the spectral environment, reliably detect weak primary signals over a targeted wide frequency band, and adapt transmission parameters (such as the transmitting power, modulation and coding scheme, carrier frequency, etc.) to opportunistically use the available spectrum. The typical sensing methods include the energy detector, the matched filter, the cyclostationary feature detection, and so on.

The typical sensing methods required the knowledge of noise power, LUs’ (License User) waveform or known patterns and signal cyclostationary feature. All the above methods need a subjectively pre-defined threshold, which affects the robustness of the methods.

Recently, some blind sensing algorithms are derived from the eigen-values of the covariance matrix. Among them, the detectors based on the sample covariance matrix, including the MME detector [2], MET detector [3], the information theoretic detector [4-5], and DMM detector [6], have been recently proposed. All of them work well in the case of noise uncertainty, and can even perform better than the ideal ED (with perfect noise power estimate) when the detected signals are highly correlated.

However, these methods suffer from the heavily computational load of the eigen-decomposition, which may be unacceptable in real-time signal processing and large-dimension array system. To deal with this problem, several fast blind sensing methods [7,8] based on covariance Matrix of the received signals is proposed, which requires no information of the noise power and without any eigen-value decomposition (EVD).

In this paper, we propose another fast and blind sensing method, which use the determinant of covariance matrix (DCM). We also derive the threshold of our detector based on Liapunov central limit theorem and Taylor expansion. Simulation and the analytical results demonstrate that the proposed method is effectiveness and robustness. 2. Blind Spectrum Sensing based on Determinant of Covariance Matrix

A. Array model and blind sensing algorithm based on Determinant. Multi-antenna is widely used in wireless communication due to its ability in improving the performance of the system. Here the multi-antenna is also served for sensing the LU signal.

Assume a uniform linear array is employed at the CR receiver side with M antennas. The array output data are

)()()()()()()(k k k k k i p

i i N s a N S A X +=+=∑=1θθ (1)

where )]()()([)(P θθθθa a a A "21= is the steering vector of the array, is the signal-vector, and is the noise-vector. The covariance Matrix of output data is

H 21)]()()([)(k k k k P s s s S "=)]()()([)(k K k k M n n n N "21=NN SS XX k k E R A R A X X R +==H H )()()]()([θθ (2)

where is the covariance of the signals, and H denotes the Hermitian Transpose.

is the noise covariance equal to in Gaussian white noise. Here, we only consider the Gaussian white noise situation.

)]()([k k E SS H S S R =)]()([k k E NN H N N R =I 2n σOnce the covariance of the received signals is obtained, we can derive our detection statistics.

1

c det()ln

trace()H XX M

XX H M γ>

=<

??????

R T R (3)

where is determinant of matrix, H det()?0 represents the absence of the LU signal and H 1 represents the presence of

the LU signal

In no signal case, the determinant of the covariance matrix is equal to ()M

2n σand

M

XX )

trace(R is the estimation of noise power . So the detection statistics T c is close to zero, and the threshold of detection statistics can be determined according to the distribution of noise covariance matrix’s determinant.

2

n σIn noisy case, we consider that the received signal at a secondary user is corrupted by the additive white Gaussian noise. According to the definition of determinant, it can seen that,

1det()M

XX i

i λ==∏R (4)

where i λis eigen-value of matrix XX R . In ideal case, the eigen-values of XX R can be expressed as followed

(5)

2n M 22n s 1λλλσσασ===>+="2Equation(3) can be rewritten based on eq.(5),

()()()

2(1)

22

c

2ln

ln

ln(1)ln(1SNR)

where SNR 22M 2s

n n s

n s M

222n

n n

s 2n

ασσσασσασασσσσσ

?++===+=+=

T (6)

If the SNR is big enough, we can always differentiate if there is a signal or not based on T c .

B. Theoretic analysis and the threshold determination. Although ()N R XX converges to as N tends to infinity, for finite N , its properties depart from those of the statistical covariance matrix, then the eigen-values of have the property that . At low SNR, the performance of a sensing algorithm is very sensitive to the threshold. Since we have no information of the signal (actually we even do not know if there is signal or not) and noise, it is difficult to set the threshold based on the P XX R XX R M 2P 1P p 21λλλλλλ>>>>>>>++""d . Hence, usually we choose the threshold based on the P fa . We need to determine the behavior of determinant of covariance matrix under null hypothesis, i.e., 0Η. To derive the threshold, the distribution of test statistics must be derived. Firstly, we start from the following definition.

Definition 1[9]: The random matrix m m ×H XX A =is a (central) real/complex Wishart matrix with n degrees of freedom and covariance matrix Σ, (), if the columns of the ),(~ΣA n W m n m × matrix are zero-mean independent real/complex Gaussian vectors with covariance matrix X Σ. The p.d.f. of a complex Wishart matrix for is

),(~ΣA n W m m n ≥[]m n m

i n

m m tr i n f ??=????=

∏

B B ΣΣ

B A det }{exp )!

(det )(11

2

/)1(π (7)

Lemma1[9]: Assume for and , then ),(~ΣM n W m m n ≥0>Σ∏=m

i i M 1)det(~)det(u Σ, where , which independent for different i and is chi-square distribution with n degrees of freedom.

m i i n i ,,2,1,~21"=+?χu 2n χPractically, the statistical correlation matrix in the detection statistics is estimated through a sample covariance matrix. Introduce N as the number of samples collected by each receiver during the sensing period. The M ×M sample covariance matrix is then de ?ned as

XX R ()N R XX

()∑==

N

k k k N

1

H

)()(1X

X N R XX (8)

In the noise case of H 0, it is not hard to verify that ()N R noise follows a central Wishart distribution

(),

(~2N

I

N W N M noise σN R (9)

According to lemma1, we can derive the p.d.f of , which is the key for determination threshold of our

algorithm.

)det(XX R ()∏∏=?=?=M

i i M

N

M M

i i N noise N I N 12121)det(~)det(u u N R σσ (10)

From eq.(10), the distribution parameters of multiplication of several chi-square distribution, is hard to determine. But with logistic operation and Liapunov central limit theorem, we know that the test statistics approach Gaussian distributions with large M.

[]()()[]21c 221det(())

det(())E E ln

E ln E ln E ln ln trace(())M M M M

N i i XX XX i M

M M i XX

n n N N N M N N M σσσ?==??

?

???????????=≈==??????????????

????????

∏∑u R R T u R ? (11)

[]()()[21c 221det(())

det(())D D ln

D ln D ln D ln trace(())M M M

M N i i XX XX i M

M M i XX

n n N N N N M σσσ?==?

?

?

???????????=≈=??????????????

????????

∏∑u R R T u R ]= (12)

The parameters of distribution are obtained through 2-order Taylor expansion at []x x E ==μ. Let ()()x ln x h =,

()()()()()()()()23

311h ln h h h O(()26

x x x x )x μμμμμμμμ′′′′′′=+?+?+?+? (13)

So the mean and variance of can be computed as followed.

i u ln ()()()()()22

1E ln x ln h E ln 22D()x x μμμμμ′′≈+?=????? (14) ()()()()()()()()()()()[]()()()()()2

222

2

22

442321D D ln x E ln h h ln 22D()1h E h E 44E ()1h h E h 2()1x x x x x x x D x x μμμμμμμμμμμμμμμμμμ

?′′′≈+?+??+????????

????′′′=?+?+???????????????????′′′′′+?+?? (15) And with some results below we can get the mean and variance of our test statistics.

[][]()

[]()[]()[]

E 1D 21E E 2(1)E 2(2)...E i i k

i i i N i N i k k =?+=?+??=+?+???u u u u u i u (16)

According to the analysis above, we can derive the threshold of our algorithm based on false alarm probability.

()()f P Pr Q c T αγγ=>= (17)

where

(

)2()2Q u t

t e

μσ??+∞

=

∫du (18)

It can be seen that the threshold has nothing to do with the knowledge of noise power and signal information, therefore our method is belong to blind sensing algorithm.

2n σC. Remark. To compute the determinant of matrix, our method and CDC [8] merely requires 33n flops [10]. However the EVD-based algorithms require flops (only for computing eigen-values)[10]. Thus the computation complexity of our method is significantly reduced.

34n 3. Simulation results

To demonstrate the performance of the proposed methods, simulations are provided. A Uniform Linear Array (ULA) is used here with M = 8 sensors and half wavelength inter-element spacing. The QPSK signals are used in the simulations.

In the simulations, the definition of SNR is ()22

1010log s n σσ for conveniently. In the following, all the results are averaged over 100000 Monte Carlo realizations.

Fig.1 shows the statistical distribution of our method under H 0 and H 1. In the left part of the figure, there is distribution only in noise case; we can see that the theoretical distribution coincides with statistical distribution. In the right part of the figure, there is the distribution of noisy signal with SNR=-6dB, which separate well with noise only case.

0500100015002000250030003500

4000Threshold

Fig1. The statistical distribution of our method under H 0 and H 1

SNR(dB)

P d

SNR(dB)

P d

(a)Pfa=0.01, N =1024 (b) Pfa=0.01, N =40

Fig.2 Performance comparison of DCM, CAV, CDC and MME methods

Fig.2 shows the comparison the performance of our DCM detector, CAV detector [7], CDC detector [8] and MME detector [2]. The given Pfa is 10-2 and the number of samples are 1024 and 40 separately. It is clear that, DCM, CDC and MME methods are well done at low SNR for large samples. For example, when SNR=-7dB and samples is 1024, P d of DCM, CAV, and CDC methods are 82%, 74% and 86.5%. It is clear that MME detector is the best for large samples,

and our DCM method slightly better than CDC methods in both large and small samples, CAV method is the worst. And also the computation complexity of our method and CDC are greatly reduced compared with the eigen-based methods.

Samples

D i s t a n c e (d B )

Fig.4 Comparison of distances between theoretical and experimental threshould

To evaluate the effectiveness of theoretical threshold, fig.4 gives the distance of theoretical and experimental thresholds of these four methods in different sample numbers. It is clear that the distances of CAV and MME detector are much larger than our DCM and CDC methods, and greatly degraded with small samples, which because of threshold determination based on the asymptotic assumption that the sample size is infinite. From eq.(9) and (10), it can be seen that our method overcomes the inaccurate threshold resulting from the asymptotic assumption. So our DCM detector and CDC detector work well in small samples situation.

4. Conclusions

This paper proposes a fast and blind spectrum sensing method based on the determinant of the received signal covariance matrix. The employed test statistic requires no information of the noise power and without any eign-decomposition (or SVD). And our method also has the robust theoretical threshold, which unsensitive with sample size. The simulation results demonstrate its effectiveness and robustness.

5. Acknowledgment

This work was supported in part by the Natural Science Research Program of Shaanxi’s Provincial Education Department (11JK0925), the Nature Science Foundation of China (60602053), Program for New Century Excellent Talents in University (NCET-08-0891), the Natural Science Foundation of Shaanxi Province (2010JQ80241),the Natural Science Foundation of Hubei Province (2009CDB308) and the Fund from Education Department of Shaanxi Government.

References

[1]

Haykin S. Cognitive radio: brain-empowered wireless communications. IEEE J. Select. Areas Commun., 2005, 23(2): 201-220.

[2] Zeng Y H and Liang Y C. Eigenvalue based spectrum sensing algorithms for cognitive radio. IEEE Trans.

Commun., 2009, 57(6): 1784-1793.

[3] Zeng Y H, Liang Y C and Zhang R. Blindly combined energy detection for spectrum sensing in cognitive radio.

IEEE Signal Process. Lett., 2008, 15: 649-652.

[4] Rui W and Meixia T. Blind Spectrum Sensing by Information Theoretic Criteria for Cognitive Radios. IEEE

transactions on vehicular technology, 2010, 59(8): 3806-3817.

[5] Zayen B, Hayar A and Nussbaum D. Blind spectrum sensing for cognitive radio based on model selection.

CrownCom 2008, 3rd International Conference on Cognitive Radio Oriented Wireless Networks and Communications, Singapore, May, 2008: 15-17.

[6] Yingxi W, Guangyue L. DMM Based Spectrum Sensing Method for Cognitive Radio Systems. Journal of

Electronics and Information Technology, 2010, 32 (11): 2571-2575.

[7] Zeng Y H and Liang Y C. Spectrum-sensing algorithms for cognitive radio based on statistical covariances. IEEE

Trans. Veh. Commun., 2009, 58(4): 1804-1815.

[8] Yang X, Lei K J, Peng S L and Cao X Y. Blind Detection for Primary User Based on the Sample Covariance

Matrix in Cognitive Radio. IEEE communications letters, 2011, 15(1): 40-42.

[9] Anderson TW. An introduction to multivariate statistical analysis. 2nd

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CA（美化学文摘）网络版数据库使用说明 一、数据库简介 CA（Chemical Abstracts）由美国化学文摘社(CAS--Chemical Abstracts Service)编辑出版，是涉及学科领域最广、收集文献类型最全、提供检索途径最多、部卷也最为庞大的一部著名的世界性检索工具。CA报道了世界上150多个国家、56种文字出版的9500多种科技期刊、科技报告、会议论文、学位论文、资料汇编、技术报告、新书及视听资料，摘录了世界范围约98%的化学化工文献，所报道的内容几乎涉及化学家感兴趣的所有领域。 SciFinder Scholar数据库为CA（化学文摘）的网络版数据库，收录内容比CA更广泛，功能更强大。利用现代机检技术，进一步提高了化学化工文献的可检性和速检性,更整合了Medline医学数据库、欧洲和美国等50几家专利机构的全文专利资料、以及化学文摘1907年至今的所有内容。它涵盖的学科包括应用化学、化学工程、普通化学、物理、生物学、生命科学、医学、聚合体学、材料学、地质学、食品科学和农学等诸多领域。它可以透过网络直接查看“化学文摘”1907年以来的所有期刊文献和专利摘要；以及八千多万的化学物质记录和CAS注册号。 SciFinder Scholar可检索数据库包括： CAplusSM:包含来自150 多个国家、9000 多种期刊的文献，覆盖1907 年到现在的所有文献以及部分1907 年以前的文献，包括有期刊、专利、会议录、论文、技术报告、书等，涵盖化学、生化、化学工程以及相关学科，还有尚未完全编目收录的最新文献。（目前＞2,430 万条参考书目记录，每天更新3000 条以上） MEDLINE?:包含来自70 多个国家、3900 多种期刊的生物医学文献，覆盖1951 到现在的所有文献，以及尚未完全编目收录的最新文献。（目前＞1300 万参考书目记录，每周更新4 次） REGISTRYSM: 涵盖从1957 年到现在的特定的化学物质，包括有机化合物、生物序列、配位化合物、聚合物、合金、片状无机物。REGISTRY 包括了在CASM中引用的物质以及特定的注册。例如：管制化学品列表如TSCA 和EINECS 中的注册。（目前＞7400 万

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