Eigenvalue Ratio Detection Based On Exact Moments of Smallest and Largest Eigenvalues

Transcription

1 Eigenale Ratio Detection Based On Exact Moments of mallest and Largest Eigenales Mhammad Zeeshan hakir, Wchen Tang, Anlei Rao, Mhammad Ali Imran, Mohamed-lim Aloini Diision of Physical ciences and Engineering, AUT Makkah Proince, Thwal 955-9, ingdom of adi Arabia Centre for Commnication ystems Research CCR, Uniersity of rrey Gildford GU 7XH, rrey, United ingdom Abstract Detection based on eigenales of receied signal coariance matrix is crrently one of the most effectie soltion for spectrm sensing problem in cognitie radios. Howeer, the reslts of these schemes always depend on asymptotic assmptions since the close-formed expression of exact eigenales ratio distribtion is exceptionally complex to compte in practice. In this paper, non-asymptotic spectrm sensing approach to approximate the extreme eigenales is introdced. In this context, the Gassian approximation approach based on exact analytical moments of extreme eigenales is presented. In this approach, the extreme eigenales are considered as dependent Gassian random ariables sch that the joint probability density fnction PDF is approximated by biariate Gassian distribtion fnction for any nmber of cooperating secondary sers and receied samples. In this context, the definition of Copla is cited to analyze the extent of the dependency between the extreme eigenales. Later, the decision threshold based on the ratio of dependent Gassian extreme eigenales is deried. The performance analysis of or newly proposed approach is compared with the already pblished asymptotic Tracy-Widom approximation approach. Index Terms pectrm sensing; eigenale ratio based detection; non-asymptotic Gassian approximation; Copla. I. INTRODUCTION In cognitie radio networks, the spectrm resorces are aailable for a secondary ser U only when they are not occpied by the primary ser PU, which aims at aoidance of intolerable interference. Ths, the U shold be able to detect the presence of the PU. In this context, high probability of accrate detection in spectrm sensing becomes extremely important in the implementation of cognitie radio networks. Eigenale ratio ER based detection has been considered as an effectie and efficient soltion to this problem [] [5]. The key adantage of this detection method lies in the fact that there is no need to obtain a prior information of the PU. The main idea is to decide whether the spectrm resorces are being sed by a PU or not by exploiting the extreme eigenales of the receied signal coariance matrix. Based on the resent reslts of random matrix theory, the ER detector exploit the ratio between the largest and the smallest eigenales of the coariance matrix as analytical statistics and determine the decision threshold. The decision threshold is precalclated, which is determined by the distribtion of the test statistics T N. Howeer, the exact distribtion of the test statistics of the ER detector is generally a mathematically intractable fnction. ome semi-analytical approaches for the distribtion are presented in [5], [] where the comptational complexity becomes intractable with the increase in nmber of Us and receied samples N. The exact expression for this ratio has also been deried in [], howeer the distribtion can only be ealated nmerically. Also, when and N are large, the complexity of the exact expression may become comptationally cmbersome. As a conseqence, a Gassian approximation is introdced in [7] to derie the analytical distribtion of T N sch that the decision threshold can be calclated in a closed-form. Despite the simplicity of the decision threshold, the proposed approximation is only alid nder the assmption that the distribtion of the largest and the smallest eigenales conerges to the Tracy-Widom distribtion of order two [8]. It has been also shown that sch conergence only occrs when, N and N c,. Howeer, the reslting cmlatie distribtion fnction CDF of the Tracy-Widom random ariable inole matrix determinants with fnction entries that are difficlt to ealate when and N are larger. The deried decision threshold is based on the asymptotic Gassianity of the extreme eigenale distribtion obtained by fitting asymptotic moments of Tracy-Widom distribtion of order [8]. Moreoer, the approximation is nder the assmption that the extreme eigenales are independent with each other which leads to inaccracy of considerable extent for moderate to small nmber of Us and samples. The contribtions of this paper are as follows: We introdce a simple non-asymptotic spectrm sensing approach based on non-asymptotic Gassian approximation for any nmber of collaborating Us and receied samples. The joint probability density fnction PDF of the largest and the smallest eigenales are approximated by a biariate Gassian distribtion fnction. The proposed approximation captres the non-asymptotic Gassianity of the extreme eigenales based on their exact analytical moments. The definition of Copla is cited to show that the largest and the smallest eigenales

2 are dependent and their dependency increases with moderate to small ales of and N. The rest of this paper is organized as follows. ection II defines the system model to explain the detection problem of PU; ection III introdces the Gassian approximation to the extreme eigenales. In this section, we recast the Gassianity of the extreme eigenale distribtion in the framework of their exact analytical moments. Dependency analysis based on Gassian Copla between the extreme eigenales is also presented; Finally, we calclate an accrate optimal decision threshold for the false alarm probability. Conclsions are presented in IV. II. YTEM MODEL Consider there are collaborating Us sch that each ser collects N samples dring the sensing time to detect a PU. The Us may be considered as a receie antennas in one secondary terminal or secondary terminals each with single antenna, or any combinations of these. The collected samples from collaborating Us will be forwarded to a fsion center for combined processing. The aim of the U cognitie phase is to constrct and analyze tests associated with the following hypothesis testing problem: H : yn = wn H : yn = hn sn + wn where yn = [y n,, y n] T is the obsered complex time series, wn represents a complex circlar Gassian white noise process with nknown ariance σw. In, the ector hn C typically represents the propagation channel between the PU and collaborating Us and the signal sn denotes a standard scalar i.i.d circlar complex Gassian process w.r.t samples n =,,, N and stands for the sorce signal to be detected. We stack the obsered data into N data matrix Y which may be expressed as y y y N y y y N Y = y y y N The receied sampled coariance matrix is R = Y Y H with ordered eigenales λ > λ > > λ >, where H is the Hermitian conjgate operator. These eigenales behae differently depending on the presence or absence of the PU. The spectral properties of the coariance matrix is the reason of sing the ratio between the largest and the smallest eigenales of this matrix as a test statistics to infer the presence or absence of the PU. The test statistics for ER detector is T N = λ /λ and denoting as the decision threshold employed by the detector sch that T N H H to decide if the target spectrm resorce is occpied or not. III. NON-AYMPTOTIC GAUIAN APPROXIMATION BAED ON EXACT MOMENT OF EXTREME EIGENVALUE In this section, we first derie the moments of the extreme eigenales. Later, we fit their statistical mean and ariance to a Gassian distribtion fnction to approximate the joint PDF of extreme eigenales and calclate a decision threshold. A. Moments of Largest and mallest Eigenales Let s denote deta, B as the cross-determinant of two matrices A = {a i,j } m m and B = {b i,j } n n with m n sch that m m deta, B = i+j a i,j B i,j i= j= where B i,j is the minor of matrix B with i-th row and j-th colmn deleted. The PDF of the largest λ and the smallest λ eigenales are gien by [9] f λ λ = C deta, B 5 where the constant C = Γ NΓ with Γ m n = m i= n i!; the element of matrix A is a i,j = e λ λ N +i+j ; the elements of matrix B for λ and λ are b i,j = N + i + j, λ and b i,j = ΓN + i + j, λ respectiely, with, and Γ, as the lower and the pper incomplete gamma fnctions respectiely. Considering the p-th moment of λ and λ, we hae { } E λ p =C deta p, B dλ m m i+j a i,j B i,j dλ, 7 =C i= j= where the element of A p are a i,j = e λ λ p+n +i+j. Moments of the mallest Eigenale λ : For the smallest eigenale λ, we hae the minors of matrix B as B i,j = sgnα ΓN + L αk,k, λ, 8 α where L k can be determined by k + α k if α k < i and k < j L αk,k = k + α k + if α k i and k j k + α k otherwise Note that Γn, λ = e λ n! n l= λl l!, then B i,j may be expressed as B i,j = λ l sgnα N +L αk,k! l α l!, where L l = N +L α, l = N +L α, l = 9 N +L α, l = ;

3 E {λ p } = C = C E {λ p } = C i,j= i+j α i,j= i+j α i,j= i+j α sgnα N + L αk,k! l l! e +λ λ l +N +i+j+p dλ Γ l sgnα N + L αk,k! + N + i + j + p l l! + l +N +i+j+p sgnα N + L αk,k! Γ + N + i + j + p! + +N +i+j+p l = l + l + + l and l! = l! l! l!. By sbstitting into 7, we hae which is showing at top of the page. The p-th moment of the smallest eigenale λ can be calclated sing. Moments of the Largest Eigenale λ : imilarly, for the largest eigenale λ, we hae B i,j = α sgnα N + L αk,k, λ, where L αk,k is determined by 9. Note that n, λ = n! e λ n l= λl l!, we may express B i,j as B i,j = sgnα N + L αk,k! α N +L αk,k e λ λ l k l k! = α sgnα l k = N + L αk,k! e λ λ, 5! where is any sbset of the set {l, l,, l } with l k from to N + L αk,k, is the sm oer all the elements in, is the cardinality of sbset, is the sm of all the elements in the sbset, and! is the prodct of the factoring of each element in. For example, if = {l i, l i,, l ik }, then we hae = k, = li + l i + + l ik,! = l i! l i! l ik! and = N +L αi,i l i = N +L αi,i l i = N +L αik,i k l ik = For the special case, when is empty, we hae =, = and! =. With B i,j aboe, the p-th moment of the largest eigenale λ can be calclated as gien in. Choose p = and p = to calclate the first and the second moments respectiely of λ and λ. The nmerical ales of mean and ariance of the extreme eigenales are shown in Table I. The CDFs of the largest and the smallest ; Fig.. F λ x F λ y imlaiton TW approx. Analytical = N = = N = 5 λ a.8... imlaiton TW Approx. Analytical = N = = N = λ b CDFs of :a the largest eigenale; b the smallest eigenale. eigenales for selected ales of and N are shown in Fig. a and Fig. b respectiely. The figres illstrate that the proposed non-asymptotic Gassian approximation based on exact moments of extreme eigenales otperforms the approximation based on the moments of the Tracy-Widom distribtion. Moreoer, or reslts are in perfect agreement with the empirical reslts. B. Joint PDF and Copla of Extreme Eigenales We employ the exact analytical moments of extreme eigenales to recast the biariate Gassian distribtion fnction as a joint PDF of extreme eigenales.

4 TABLE I NUMERICAL REULT OF LARGET AND MALLET EIGENVALUE λ λ, N imlated Tracy Widom Approx. Analytical imlated Tracy Widom Approx. Analytical mean ariance mean ariance mean ariance mean ariance mean ariance mean ariance, , , , Let λ N µ λ, σλ, λ N µ λ, σλ be a biariate Gassian random ariables with µλ σ µ = and Σ = λ ρσ λ σ λ µ λ ρσ λ σ λ σλ are the mean and the coariance of biariate Gassian distribtion fnction respectiely, and where ρ is the correlation coefficient between λ and λ. If f λ,λ x, y is the joint PDF of the extreme eigenales then by exchanging the Gassian moments with the exact analytical moments of extreme eigenales, the joint PDF can be approximated as z f λ,λ x, y = πσ λ σ λ ρ exp.5 ρ 7 where z x µ λ σ λ ρx µ λ y µ λ σ λ σ λ + y µ λ σ λ Now, we analyze the dependency between the random ariables λ, λ to approximate ρ by plotting their Copla. A Copla is a mltiariate distribtion fnction with known marginal cmlatie distribtion fnctions CDFs []. More specifically, a biariate joint distribtion fnction F λ,λ x, y = P r{λ x, λ y} of two random ariables λ and λ, may be represented by a Copla C as a fnction of their marginal CDFs F λ x = P r{λ x} and F λ y = P r{λ y} and therefore may be expressed as [] F λ,λ x, y = CF λ x, F λ y C, 8 where = F λ x and = F λ y; C, is the associated Copla distribtion fnction. Ths, we hae C, = F λ,λ F λ, F λ 9 By exploiting the chain rle, the corresponding joint PDF f λ,λ x, y may be decomposed as f λ,λ x, y = F λ,λ x, y = CF λ x, F λ y x y x y = C, F λ x F λ y c, f λ xf λ y x y It is obios that the joint PDF is the prodct of the marginal PDFs f λ x and f λ y and Copla density fnction c,. The definition of Copla identifies a strong relationship which proides link between the marginal PDFs/CDFs and the respectie joint PDF/CDF. Also, c, =, for independent random ariables [] a = ;N = c = ;N = b = ;N = ; ρ = d = ;N = 5; ρ =. Fig.. Plot of Copla between random ariables λ and λ based on empirical ble scatter plot and Gassian red scatter plot marginal distribtion fnctions. Let Φ λ x and Φ λ y are the marginal distribtion fnctions of the approximated Gassian random ariables λ and λ respectiely. Using the statistical ales of µ λ, µ λ, σ λ, σ λ, the Copla distribtion of 7 is gien by [] C, = Φ λ Φ λ f λ,λ x, y dxdy where Φ is the inerse of the standard niariate Gassian distribtion fnction. The plots of Copla between λ and λ based on the empirical and Gassian marginal distribtion fnctions are showing in Fig. a and Fig. b respectiely for, N =,. It can be seen that the strctre of Copla based on the empirical distribtion fnctions appears same as the Copla strctre based on the Gassian distribtion fnctions for ρ =.. It can also be noticed that the Copla strctre is distinctie for small ale of and N i.e. the extreme eigenales are dependent with each other. Howeer, with the increase in nmber of and N, the dependency between λ and λ decreases. The same is shown in Fig. c-d for, N =, 5 where the correlation between the extreme

5 = µ λ µ λ τ ρσ λ σ τ µ λ λ σλ + µ λ σλ + ρ τ σλ σλ µ λ µ λ ρσ λ σ λ µ λ τ σλ + µ λ τ σλ eigenales decreases to ρ =.. Therefore, the dependency between λ and λ can not be ignored if accrate spectrm sensing is reqired. Howeer, some analytical approaches may be applied to calclate the exact ale of ρ. C. Optimal Decision Threshold We hae already shown that the joint PDF of λ, λ can be well approximated by a biariate Gassian distribtion fnction knowing their exact moments and ρ. If the joint PDF of λ, λ N µ, Σ is gien as 7, then the CDF of the ratio of two dependent Gassian random ariables may be expressed as [] { } µλ µ λ F = Φ σ λ σ λ a.5 where a = ρ σλ σ λ σ λ + and Φ is the σ λ CDF of a standard Gassian random ariable. Using, we may calclate an accrate decision threshold analytically in closed form based on exact moments. For a gien target false alarm probability and fixed ρ, the decision threshold is obtained by soling = F The soltion is shown at the top of the page as, where τ = Φ. In Fig., we plot the decision threshold as a fnction of for small and moderate nmber of and N to compare the Gassianity of λ and λ based on exact moments with their Gassianity based on the moments of Tracy-Widom distribtion. It can be seen that the proposed non-asymptotic Gassian approximation performs extremely well in comparison with the Tracy-Widom approximation approach and or reslts match the empirical reslts. The proposed approximation is eqialently good for any nmber of and N, howeer the reslts are significantly accrate for reasonably moderate to small nmber of and N. IV. CONCLUION In this paper, a non-asymptotic Gassian approximation based on the exact analytical moments of the extreme eigenales has been employed to approximate the decision threshold for ER detector. Copla plot showed that the largest and the smallest eigenales are dependent and their dependency shold not be ignored specially for moderate to small nmber of collaborating Us and samples. The accracy of or proposed decision threshold has been ealated in comparison with decision threshold calclated sing Tracy-Widom approximation approach. Or proposed approximation is accrate enogh for any nmber of collaborating Us and samples a = ;N = 8 imlation TW approx.; ρ = 7 Analytical; ρ =. 5 imlation TW approx.; ρ = Analytical; ρ = c = ;N = 8 8 imlation TW approx.; ρ = Analytical; ρ = b = ;N = imlation TW approx.; ρ =.5.5 Analytical; ρ = d = ;N = 5 Fig.. Threshold as a fnction of for arios ales of and N. REFERENCE [] Y. Zeng and Y. C. Liang, Eigenale based spectrm sensing algorithms for cognitie radios, in IEEE Trans. Commns., ol. 57, no., pp , Jn. 9. [] F. Penna, R. Garello and M. A. pirito, Cooperatie spectrm sensing based on the limiting eigenale ratio distribtion in Wishart matrices, in IEEE Commns. Letters, ol., no. 7, pp , Jl. 9. [] F. Penna, R. Garello, D. Figlioli and M. A. pirito, Exact nonasymptotic threshold for eigenale-based spectrm sensing, in Proc. ICT Conf. Cognitie Radio Oriented Wireless Networks and Commns., CrownCom 9, Hannoer, Germany, Jn. 9. [] Y. Zeng, Y.-C. Liang, A. T. Hoang, and R. Zhang, A reiew on spectrm sensing for cognitie radio: challenges and soltions, in EURAIP Jor. Adances in ignal Processing, ol., Article ID 85,. [5] Y. Zeng and Y. Liang, Maximm-minimm eigenale detection for cognitie radio, in IEEE Intl. Conf. Personal, Indoor and Mobile Radio Commns., PIMRC 7, Athens, Greece, ep. 7. [] L.. Cardoso, M. Debbah, P. Bianchi and J. Najim, Cooperatie spectrm sensing sing random matrix theory, in Proc. Intl. ymp. Wireless Perasie Compting, IWPC 8, pp. 8, antorini, Greece, Jl. 8. [7] L. Wei and O. Tirkkonen, pectrm sensing with Gassian approximated eigenale ratio based detection, in Proc. IEEE Intl. ymp. Wireless Commn. ystems, IWC, pp. 9 95, York, U, ep.. [8] C. Tracy and H. Widom, On orthogonal and symplectic matrix ensembles, in pringer Jor. Commns. in Mathematical Physics., ol. 77, no., pp , 99. [9] M. C. A. Zanella and M. Z. Win, On the marginal distribtion of the eigenales of Wishart matrices, in IEEE Trans. Commns., ol. 57, no., pp. 5, Apr. 9. [] T.. Drrani and X. Zeng, Copla for biariate probability distribtion, in IET Electronics Letters, ol., no., pp. 8 9, Feb. 7. [] D. V. Hinkley, On the ratio of two correlated normal random ariables, in Oxford Jor. Biometrika, ol. 5, no., pp. 5 9, Dec. 99.