An Improved and More Accurate Expression for a PDF Related to Eigenvalue-Based Spectrum Sensing

An Impro v ed and More Accurate Expression for a PDF Related to Eigen v alue-Based Spectrum Sensing Fuhui Zhou, Member , IEEE , and Norman C. Beaulieu, F ellow , IEEE Abstract —Cooperative spectrum sensing based on the limiting eigen value ratio of the covariance matrix offers superior detection performance and o vercomes the noise uncertainty pr oblem. While an exact expression exists, it is complex and multiple useful approximate expressions hav e been published in the literature. An improved, more accurate, integral solution for the probability density function of the ratio is derived using order statistical analysis to r emove the simplifying, but incorrect, independence assumption. Thereby , the letter makes an advance in the rigorous theory of eigenv alue-based spectrum sensing. Index T erms —Cooperative spectrum sensing, eigen value ratio analysis, order statistics, probability distribution. I . I N T RO D U C T I O N T HE Eigen v alue-based detection schemes, using the eigen- values of the cov ariance matrix to construct a test statis- tic, are considered to be one of the most effecti ve methods to test for the presence of a primary user (PU) signal in cogni- tiv e radio systems [1]. The maximum-to-minimum eigenv alue (MME) detector is based on the ratio of the largest eigen v alue to the smallest eigenv alue of the cov ariance matrix. Ho wev er , theoretical results for eigenv alue ratio schemes usually depend on asymptotic assumptions, since the distribution of the ratio of two e xtreme eigen v alues is dif ficult to compute [1]-[5]. The probability of false alarm (PF A) is the probability that the PU is absent b ut is detected to be present. PF A is one of the most important performance metrics in spectrum sensing for cognitiv e radio. Accurate determination of the PF A improves the accuracy of the decision threshold of a detector , and the efficienc y of spectrum utilization. Note that the deriv ation of the PF A is dependent on the probability density function (PDF) of the test statistic of the detector under the hypothesis that the PU is absent. Meanwhile, this PDF is not known exactly in a tractable form suitable for further theoretical analysis, and the deriv ation of a popular approximation necessarily employs assumptions that are not rigorously correct mathematically . This letter makes a contribution to the theory of eigen value- based spectrum sensing in two ways, by removing in valid assumption, and by deriving a more accurate mathematically Fuhui Zhou is with the School of Information Engineering, Nanchang Univ ersity , P . R. China, 330031 (e-mail: zhoufuhui@ncu.edu.cn). Norman C. Beaulieu is with Beijing University of of Posts and T elecom- munications and the Beijing Key Laboratory of Network System Architecture and Con vergence, Beijing, China 100876 (e-mail: nborm@b upt.edu.cn). The research was supported by the National Natural Science Foundation of China (61701214), the Y oung Natural Science Foundation of Jiangxi Province (20171B AB212002), and the China Postdoctoral Science Foundation (2017M610400). tractable approximation to this important PDF after remo ving the in valid assumption. The distribution of the ratio of the two extreme eigenv alues for the MME scheme is commonly approximated by the T racy-W idom distribution based on the Trac y-W idom law [1], [2]. Howe ver , this approximation is based on the unrealistic assumption that the number of the recei ved signal samples as well as the number of the cooperating secondary users are infinite. It has been shown that this approximation is poor when the number of the signal samples is small [5], [6]. An improv ed approximation to the PDF of the ratio of the two extreme eigen values is deriv ed in [5] and [6] with the assumption that the two extreme eigen values are independent and Gaussian distributed. In [7] and [8], an e xact PDF for the ratio of the two extreme eigen values has been deriv ed. The deriv ed exact expression for the PDF is quite complex, and other authors have chosen to use the approximation ov er the exact solution in [5], [6], [9]-[11]. In this paper , an improved PDF approximation of the ratio of the two extreme eigen values is derived by using the order statistic theory and the Gaussian distribution assumption. It is sho wn that the derived PDF is more accurate than the commonly employed PDFs giv en in [2] and [5], and is simpler than the exact PDF giv en in [7]. The rest of this paper is organized as follows. Section II presents the system model and MME spectrum sensing. An improv ed solution for the PDF of the ratio of two extreme eigen values is presented in Section III. Section IV presents simulation results. The paper concludes with Section V . I I . S Y S T E M M O D E L A N D M M E S P E C T R U M S E N S I N G As sho wn in Fig. 1, a cognitiv e radio network with M SUs that cooperatively detect one PU is considered. During the sensing time, each SU collects N samples of receiv ed signal, denoted by x i ( n ) , where n = 1 , 2 , · · · , N and i = 1 , 2 , · · · , M . Note that all the SUs receiv e the signal at the same time. T o achiev e synchronous sampling, each SU has the center frequency deriv ed from the local oscillator and the same digital clock [1], [2], [5], [6]. This system model for cooperativ e spectrum sensing has been widely applied in these works. Then, the samples are transmitted to the fusion center . The aim of cooperativ e spectrum sensing is to construct a test statistic and make a decision between the two hypotheses ( H 0 and H 1 ) based on those collected samples, where H 0 denotes the absence of the PU, and H 1 represents the presence    Fusion Center (FC) PU SU SU 1 SU 2 SU M PU Fig. 1. The system model. of the PU. Thus, the samples from each SU under the two hypotheses are gi ven as H 0 : x i ( n ) = w i ( n ) (1a) H 1 : x i ( n ) = h i ( n ) √ P s s ( n ) + w i ( n ) (1b) where w i ( n ) ∼ C N  0 , σ 2 w  is complex Gaussian noise and C N  0 , σ 2 w  denotes the complex Gaussian normal distribution with mean zero and variance σ 2 w . s ( n ) is the primary user signal and h i ( n ) are the channel coefficients. P s is the transmitted power of the primary user . The distribution of the PU signal is unknown and independent of the noise. Based on the collected samples from M SUs, a data matrix is defined as X =  X T 1 , X T 2 , · · · , X T M  , where X m = [ x m (1) , x m (2) , · · · , x m ( N )] with m = 1 , 2 , · · · , M . The sample cov ariance matrix is defined as R x = (1 / N ) XX H , where ( · ) H represents the Hermitian transpose operator . Let λ 1 ≥ λ 2 ≥ · · · ≥ λ M denote the ordered eigen values of the matrix R x . The test statistic for the MME spectrum sensing scheme was formulated in [5]. It is denoted by T ξ , gi ven as T ξ = λ 1 λ M H 0 < − > H 1 γ ξ (2) where γ ξ is the decision threshold of the MME spectrum sensing scheme. I I I . I M P RO V ED S O L U T I O N F O R T H E P D F O F T H E R A T I O O F T W O E X T R E M E E I G E N V A L U E S U N D E R H Y P OT H E S I S H 0 In [1], [2], the PDF of the ratio of two limiting eigenv alues under the hypothesis H 0 is approximated by the PDF of the T racy-W idom distribution. This approximation is poor when the number of samples is small or moderate. The PDF of the ratio is deri ved based on the assumption that the two limiting eigen values are independent normal random variables [5]. The assumption that the two extreme eigen values are independent is not correct and is remo ved in this paper . In [5], when there only exist Gaussian noises, the largest eigenv alue and the smallest eigen value of the co variance matrix are assumed to be normal random v ariables, namely λ 1 ∼ N  u λ 1 , σ 2 λ 1  (3a) λ M ∼ N  u λ M , σ 2 λ M  (3b) where u λ 1 and u λ M are the means of the largest eigenv alue and of the smallest eigen value, respecti vely . The variances of the largest eigen value and of the smallest eigen value are denoted by σ 2 λ 1 and σ 2 λ M , respectively . According to [5], the means and variances of the largest eigen value and of the smallest eigen value are gi ven as u λ K = E ( λ K ) (4a) σ 2 λ K = E  λ 2 K  − u 2 λ K (4b) E ( λ p 1 ) = C − 1 0 β λ 1 ( p ) (4c) E ( λ p M ) = C − 1 0 β λ M ( p ) (4d) where C 0 = M Π i =1 ( N − i )! M Π j =1 ( M − j )! ; E [ · ] denotes the expectation operator; K = { 1 , M } ; β λ 1 ( p ) and β λ M ( p ) are giv en by eq. (5) at the top of the ne xt page. In eq. (5) , sgn ( · ) denotes the Signum function; α m is the m th element of the α and α is the permutation of { 1 , 2 , · · · , M − 1 } ; p i,j = p + N − M + i + j ; P l M − 1 1 = M − 1 P i =1 l i ; l M − 1 1 ! = M − 1 Q i =1 l i ! ; P L 1 ∼ M − 1 l 1 ∼ M − 1 = P L 1 l 1 =0 P L 2 l 2 =0 · · · P L M − 1 l M − 1 =0 ; S is any subset of the set { l 1 , l 2 , · · · , l M − 1 } and l m is from 0 to L α m ,m − 1 ; |S | represents the cardinality of subset S . Since the largest eigenv alue and the smallest eigenv alue are order statistics, the joint PDF , f r,s ( x, y ) , of X r and X s , 1 ≤ r < s ≤ M , X r ≤ X s , for x ≤ y , is [12], f r,s ( x, y ) = M ! [ F r ( x )] r − 1 f r ( x ) f s ( y ) ( r − 1)! ( s − r − 1)! ( M − s )! × [ F s ( y ) − F r ( x )] s − r − 1 [1 − F s ( y )] M − s (6) where F r ( x ) and F s ( y ) are the cumulati ve marginal distri- bution functions (CMDFs) for X r and X s , respectively , and f r ( x ) and f s ( y ) denote the mar ginal PDFs for X r and X s . Thus, the joint PDF for the largest eigen value and the smallest eigen value, e f λ 1 ,λ M ( x, y ) , is giv en by e f λ 1 ,λ M ( x, y ) = M ( M − 1) f λ 1 ( x ) f λ M ( y ) × [ F λ 1 ( x ) − F λ M ( y )] M − 2 (7) where F λ 1 ( x ) and F λ M ( y ) are the cumulativ e distribution functions (CDF) of the two extreme eigenv alues, respecti vely , and f λ 1 ( x ) and f λ M ( y ) are their corresponding marginal PDFs. M is the number of secondary users. Therefore, the improv ed PDF for the ratio of the two extreme eigen values, e f Z ( z ) is deri ved as e f Z ( z ) = Z ∞ −∞ e f λ 1 ,λ M ( y z , y ) | y | dy (8) β λ M ( p ) = M X i,j ( − 1) i + j X α sgn ( α ) M − 1 Y m =1 Γ ( L α m ,m ) X L 1 ∼ M − 1 l 1 ∼ M − 1 Γ  P l M − 1 1 + p i,j − 1  l M − 1 1 ! M P l M − 1 1 + p i,j − 1 ! (5a) β λ 1 ( p ) = M X i,j ( − 1) i + j X α sgn ( α ) M − 1 Y m =1 Γ ( L α m ,m ) X S ( − 1) |S | Γ ( P S + p i,j − 1) Q l M − 1 1 ! M P l M − 1 1 + p i,j − 1 ! (5b) L α m ,m =    N − M + m + α m − 1 if α m < i and m < j N − M + m + α m + 1 if α m ≥ i and m ≥ j N − M + m + α m otherwise (5c) where |·| is the magnitude operator . After substituting eq. (7) into eq. (8) and some algebraic manipulations, the improved PDF , e f Z ( z ) , is gi ven by e f Z ( z ) = M ( M − 1) M − 2 X i =1  M − 2 i  ( − 1) M − 2 − i × Z ∞ −∞ [ F λ 1 ( y z )] i [ F λ M ( y )] M − 2 − i f λ 1 ( y z ) f λ M ( y ) | y | dy . (9) Therefore, the improved PDF for the ratio of the extreme eigen values is deriv ed, based on the assumption that the two extreme eigen values follow normal distrib utions [5], as e f Z ( z ) = M ( M − 1) M − 2 X i =1  M − 2 i  ( − 1) M − 2 − i × Z ∞ −∞ (  Φ  y z − u 1 σ 1  i  Φ  z − u 2 σ 2  M − 2 − i × | y | e −  ( yz − u 1 ) 2 2 σ 1 + ( y − u 2 ) 2 2 σ 2  2 π σ 1 σ 2 ) dy (10) where Φ ( x ) is the CDF of the standard normal distribution. Although the solution for the ratio of the two limiting eigen- values given in eq. (10) is in integral form, the integral is well behav ed, having a stictly positi ve integrand, and it can be e valuated readily by using standard numerical computation or commonly av ailable mathematical softwares. It is seen that the proposed solution for the PDF of the ratio of the two limiting eigen values is simpler than the solution gi ven in [8, eq. (7) ]. It is seen from eq. (10) and eq. (7) giv en in [8] that the complexity of these two expressions mainly depends on the multiple integration. Moreov er , double inte grations are required. In [8], there are O  N 2  additions due to the permutation operation and O  N M 4  multiplications in the integrals, where O is the big O notation [13]. In eq. (10) , M − 1 additions and O ( M ) multiplications in the integrals are required. In the simulation, a comparison of the required time for these two expressions is given to further clarify the superiority of our proposed PDF in term of the complexity . I V . S I M U L A T I O N E V A L UAT I O NS A N D D I S C U S S I O N In this section, simulation results are giv en to contrast the proposed simple form for the PDF of the ratio of the two 0 2 4 6 8 10 12 14 16 18 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 The ratio of the two limiting eigenvalues Probability density function (PDF) Empirical PDF PDF [7] Eq.(10) Eq.(30) [5] TW PDF [2] 4 5 6 0.36 0.38 0.4 0.42 5 6 7 0.14 0.16 0.18 0.2 0.22 Zoom in Zoom in Fig. 2. Comparison of the improv ed PDF approximation for the ratio of the two limiting eigen values of the cov ariance matrix with the known approximate PDFs for N = 50 and M = 10 . limiting eigen values and the expressions for the PDF of that ratio gi ven in [2], [5] and [7]. W e also present some example results that compare the accuracies of the new and previous theoretical approximations for the PDF of the ratio of the eigen values to the exact PDF obtained by simulation. The noises are independent identically distributed Gaussian real noises with mean zero and unit variance. All the simulation results are achiev ed by using 10 6 Monte Carlo simulations. The number of samples and the number of secondary users are set as N = 50 and M = 10 or N = 50 and M = 20 , respectiv ely . Fig. 2 shows the commonly employed approximations to the PDF of the ratio of the two limiting eigen values of the cov ariance matrix obtained by using existing methods and our proposed approximation which is obtained without the assumption that the two eigen values are independent. In Fig. 2, the empirical PDF curve is the empirical PDF of the ratio of the two liming eigenv alues while the TW PDF is the PDF approximation obtained by using the T racy-W idom distribution of order 2 [2]. The PDF curve labeled eq. (30) [5] is the approximate solution given in [5, eq. (30) ]. The PDF curve labeled PDF [7] is the exact PDF giv en in [7]. It is observed that the PDF of the ratio test statistic obtained 0 2 4 6 8 10 12 14 16 18 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 The ratio of the two limiting eigenvalues Probability of density function (PDF) Empirical PDF Eq.(10) Eq.(30) [5] 4 4.5 5 5.5 6 6.5 7 0.48 0.5 0.52 0.54 0.56 0.58 0.6 4 5 6 7 0.36 0.38 0.4 0.42 0.44 0.46 0.48 Zoom in Zoom in N=50, M=10 N=50, M=20 Fig. 3. Comparison of the improv ed PDF approximation for the ratio of the two limiting eigen values of the cov ariance matrix with the known approximate PDFs, for N = 50 and M = 10 or N = 50 and M = 20 . by using the new solution matches better with the empirical PDF than the PDFs obtained from the other two methods. It is also seen that the PDF giv en by the exact solution in [7] matches well with the empirical PDF . The result consists with the result obtained in [7]. The PDF from [5, eq. (30) ] and the PDF used to approximate the T racy-W idom PDF in [2] are both approximate PDFs, and both are inferior approximations to the new approximation. The T racy-W idom PDF of order 2 is not a good approximation to the precise PDF obtained by simulation. The reason is that the T racy-W idom PDF of order 2 for the ratio of the two limiting eigenv alues is valid when lim N →∞ M N = c , where c is a constant. Howe ver , in the simulation, N and M are set as 50 and 10 , respectiv ely , and these values do not satisfy the limiting condition. It is seen that the ne w solution provides very high accuracy . The reason is that our new solution is derived without the independence assumption that the samples are independent, and the only source of discrepancy is the Gaussian distribution assumption, which causes only small discrepancies when the number of samples is e ven moderately large. These results sho w that the major source of error in the previous approximations is the independence assumption, and not the Gaussian assumption. Fig. 3 sho ws the comparison of the improved PDF approx- imation for the ratio of the two limiting eigen values of the cov ariance matrix with the kno wn approximate PDF gi ven by eq. (30) [5] with different M . It is seen that the accuracy of both our proposed solution and the form gi ven by [5] increases with M . The reason is that the accuracy of the mean and variance of the two limiting eigen values increases with M . In order to compare the complexity of our proposed PDF form with that of the form given by eq. (7) in [8], the computation times for different parameters ( N , M ) are given in T able 1. The results are obtained by using a computer with 64-bit Intel(R) Core(TM) i7-4790 CPU, 8 GB RAM. It is seen from T able 1 that the required time for calculating the PDF T ABLE I C O MPA R IS O N O F T H E R E Q UI R E D C O M P UTA T I ON TI M E S ( S ) ` ` ` ` ` ` ` ` ` Schemes ( N , M ) (50 , 5) (100 , 5) (100 , 10) (100 , 20) Eq. (7) in [8] 10.248 14.835 27.482 43.498 Eq. (10) 6.529 6.572 10.593 22.179 giv en by eq. (7) in [8] is lar ger than that for our proposed PDF . This indicates the complexity of our proposed expression is lower than that presented in [8]. It further verifies that our proposed PDF is simpler than that proposed in [8]. V . C O N C L U S I O N A new approximation for the PDF of the ratio of the two limiting eigen values of the cov ariance matrix in eigenv alue- based spectrum sensing was deriv ed based on order statistic analysis. The new approximate solution is the most accurate approximation kno wn, and its deri vation does not rely on an in valid independence assumption used to deriv e a popular previous approximation. 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