Spectrum Opportunity Detection with Weak and Correlated Signals

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1 Specum Opportunity Detection with Weak and Correlated Signals Yao Xie Department of Elecical and Computer Engineering Duke University orth Carolina David Siegmund Department of Statistics Stanford University California Asact We present a novel score detector for temporal specum opportunity detection in cognitive radio y exploiting the differences in oth energy and correlation of the empty and and the occupied and Motivated y the challenge of detecting a weak primary user s signal without precise knowledge of the signal where the conventional energy detector faces the limit of SR wall we assume a simple model which captures a key difference etween noise and primary user s signal their correlation suctures Besides the merit of incorporating signal correlation our score detector also avoids the computational complexity of covariance maix inversion incurred y the corresponding imum likelihood statistic assuming signal correlation We provide a theoretical approximation to the false-alarm-rate of the score detector which can e used to determine the threshold efficiently We demonsate that our approximation is quite accurate and that our score detector has an advantage when the signal is weak and correlated I ITRODUCTIO Cognitive radio is an emerging technology which has a great potential to increase specum efficiency A key challenge in cognitive radio is specum opportunity detection Recently temporal specum opportunity detection via change-point detection approach has atacted much interest [] ecause the change-point formulation fits well with the temporal opportunity detection ojective Also there is a wealth of tools for changepoint detection from the statistics literature Existing methods for specum opportunity detection primarily use three signal processing techniques: matched filtering of the specum of the received signal with that of the primary user cyclostationary feature detection and energy detection exploiting the difference etween received signal power and noise power The matched filtering and feature detector require detailed knowledge of primary user s signal When such knowledge is not availale energy detector is a popular choice However there is a fundamental limit with the energy detector the SR wall when the signal-to-ratio SR is low error for energy detector is very high In wireless environment however due to fading SR of the primary user s signal can e quite low Hence in cognitive radio there is a need for developing weak signal detector which also requires little priori knowledge aout the primary user s signal Motivated y the challenge of detecting weak signals in this paper we propose a novel score detector y exploiting differences etween signal and noise in oth energy and correlation We assume a simple model for the primary user s signal which captures a key difference etween signal and noise: their correlation sucture Based on this model we develop a detector using the score statistic which avoids the computationally intensive inversion of covariance maix in the corresponding imum likelihood statistic assuming signal correlation In conast to the matched filtering detector our model does not require precise knowledge aout primary user s signal We derive a theoretical approximation to the false-alarm-rate of our score detector which can e used to determine the threshold efficiently We demonsate using numerical simulation that our approximation is quite accurate We also show that the score detector has etter performance than the energy detector and the imum likelihood detector [] that only assumes power change Sections are organized as follows Section II presents formulation Section III derives the score detector Section IV contains a theorem that approximates the proaility-of-falsealarm for the score detector Section V demonsates that our approximation is accurate and the score detector has good performance Finally VI concludes the paper II FORMULATIO Consider a cognitive radio system with a primary user and a secondary user A secondary user tunes to a frequency and and starts to take samples y n n = where is the numer of samples If the and is not occupied the samples are white noise If the and is occupied the samples consist white noise and a primary user s signal Assume that the primary user s signal emerges at an unknown time k k Using these samples our goal is to detect whether or not the and is occupied This prolem can e formulated as the following hypothesis test H : y n = w n n = yn = w H : n n = k; y n = x n + w n n = k + where w n σ ie it is iid Gaussian white noise with zero mean and variance σ and x n is the primary user s signal seen y the secondary user Typically x n is temporally correlated due to dispersion of wireless channel and modulation scheme used y the ansmitter Let denote the anspose of a vector or maix We assume the covariance maix of the signal vector x k [x k+ x ] is Ex k x k V kθ

2 where is the signal power and V kθ R k k is the normalized covariance maix parameterized y θ Θ R d For example when x n has weak temporal correlation we can model it using the first-order autoregressive process AR- which corresponds to d = Hence under H the covariance maix of the received signal vector y k [y k+ y ] is E H y k y k Σ = σ I k + V kθ 3 We assume the signal correlation model eg AR- is known ut parameters k and θ are unknown The energy detector detects the primary user y comparing the receive energy with a threshold > yn n= In [] the imum likelihood ML detector exploiting power difference without assuming signal correlation sucture is studied For fixed sample size it corresponds to the following detector: + σ k + σ y T σ k y k k log σ [ ] for some threshold where the unknown is assumed to e within the range of [ ] and is a parameter that specifies the minimum numer of samples needed detection The log likelihood ratio of hypothesis H versus H for given k and θ is : L k θ = I y k Σ y k log Σ 4 σ σ k The imum likelihood detector assuming signal correlation sucture detects when: L k θ 5 k θ Θ for some threshold > It has a major drawack that it requires computing a maix inversion Σ for each and all possile values of k θ and III SCORE DETECTOR We present a imum score detector that can avoid the covariance maix inversion in the imum likelihood formulation 5 Maximum score is an alternative to imum likelihood that is frequently used in deriving efficient test especially when the parameter for H only contains one single point In our case H corresponds to = in 3 and hence we can derive an efficient score detector The score detector is derived as follows Consider the derivative of L in 4 with respect to and evaluated at = : L = [V kθ y k y k σ I] 6 = σ 4 where A denotes the ace of a square maix A In the following E and var denote the proaility mean and variance under the null hypothesis We can verify the mean of the derivative under the null hypothesis is zero: E L = = The covariance of 6 under null hypothesis can e shown to e L var = = σ 4 V kθ V kθ 7 The score statistics is L = normalized y its mean and variance: Zk θ = V kθ yk+ y k+ V kθ V σ kθ I k 8 ote that Zk θ is a two dimensional random field in k and θ Θ with zero mean and unit variance The second expression shows that Zk θ can e interpreted as a matched-filter : matching the theoretical covariance maix with the sample covariance maix The score detector detects a signal when: Zk θ 9 k θ Θ for some threshold > We consider two performance meics for the detectors presented aove: the proaility-of-false-detection e H T and the proaility-of-detection under H : d H T f α where T is the statistic of the corresponding detector Assume that e is related to the threshold y some function f: e = f We will otain this function in Theorem roaility of Detection Energy ML ower Score False Alarm Rate Fig : roaility-of-detection versus proaility-of-false-detection for the energy detector ML power detector and the score detector for θ = 7 and σ s = 7 The noise variance is σ = IV AROXIMATIO OF ROBABILITY-OF-FALSE-DETECTIO We prove the following theorem that approximates the proaility-of-false-detection e of the score detector for a

3 given threshold which can e used to determine the threshold efficiently Theorem : When under H the proaility-offalse-detection of the score detector 9 is e H Zk θ where = π d k θ Θ k= gk θ Hk θ Ṽ = θ Θ [ξ k θ] d ξ k θ µk θ k ν V kθ µk θ k V kθ V kθ ξ k θ > is the solution to [ hξ I k ξ Ṽ Ṽ var ξ Z = ] Ṽ dθ + o = ] [ I k ξ Ṽ Ṽ I ξ Ṽ Ṽ ψξ = ξṽ log I ξṽ gk θ = exp ξ k θ + ψξ k θ πvarξ Z the Hessian of the covariance maix Fisher information maix is Hk θ = E Zk θzk s s s=θ µk θ = k V k+θ V k+θ V kθ V T kθ and special function Corollary 844 of [] or in [3]: [ x Φ x ] νx x Φ x + φ x 3 where φx and Φx are the density and disiution function of respectively The proof of this theorem is provided in Appendix V UMERICAL EXAMLES In this section we first verify that our approximation in Theorem is very accurate and then demonsate that the score detect compares favoraly with other detectors that do not exploit signal correlation A Approximation accuracy of Theorem for e Consider the first-order autoregressive process AR with d = θ [ 5] = = 3 The AR process evolves according to x n+ = θx n + ε n 4 where θ < and the process noise ε n are iid The covariance maix for AR is θ θ θ k V kθ = θ θ θ k 5 To demonsate that our approximation in Theorem is also accurate for d > we also consider a first-order autoregressive moving-average process ARMA with d = parameters θ and φ The AR process evolves according x n+ + φx n = θε n + ε n+ where ε n The corresponding covariance maix is V kθ = + θ φθ φ θ φθ φφ θ φθ φ θ φθ + θ φθ φ θ φθ φφ θ φθ In Tale I for AR and Tale II for AR the Monte Carlo results are all otained from ials ote that our approximation in Theorem is very close to the corresponding Monte Carlo result in oth examples TABLE I: Approximate of e for AR Monte Carlo Theorem TABLE II: Approximate e for ARMA Monte Carlo Theorem B Comparison of proaility-of-detection d We compare our score detector with the energy detector and the imum likelihood ML power detector [] We simulate the signal as an AR process 4 with θ = θ and process noise with variance σs The signal power of the AR process can e estimated using the variance of the steady state = σs/ θ Assume = the primary user emerges at k = 5 with = and consider various ue signal correlation θ The proaility-of-detection is simulated via 5 Monte Carlo ials For the score statistic we set θ [3 8] Fig demonsates that the score detector has the highest proaility-of-detection among the three detectors over a wide range of proaility-of-false-detection false-alarm rate when θ = 7 and σs = 7 Tale III further demonsates that the score detector has the est performance among the three detectors different θ and σs values 3

4 TABLE III: d for a fixed e = α = Each cell of the tale corresponds to score detector / ML power detector / energy detector The old numers correspond to score detector θ σs /68/6 9/76/6 9/75/ /79/7 95/83/73 98/86/79 VI COCLUSIOS We have present a novel score detector to detect weak signal exploiting oth the signal temporal correlation and power Here we consider cognitive radio specum detection evertheless our formulation of the score detector is general and it can e used for similar prolems We provide a theoretical approximation to the false-alarm-rate of the score detector which can e used to determine the threshold efficiently We demonsate that our approximation is quite accurate and that the score detector has an advantage when signal is weak and correlated AEDIX In the following we prove Theorem for d = Last Hitting Time Formula: We discretize the parameter θ θ θ k y rectangular mesh grid of size of times where > is a small numer The size of the mesh is chosen to alance the difference in the order of the variance in these two coordinates Then e of the score detector can e approximated as Z ij D i j 6 where the index set D = i j : i θ j θ 7 covers the parameter space Let the index set Ji j denotes everything to the future of the current index i j upper and to the right of i j in the random field Ji j = i j D : j j or j = j and i i Using the last hitting time decomposition [5] we can rewrite 6 as Z i j ij D Z i j i j i j D = i j D Z ij Ji j ij Ji j Z Z i j = + x dx i j Z i j = + x 8 ext we will find approximations for the proaility Z i j = + x dx 9 and the conditional proaility Z i j ij Ji j Z i j = + x respectively Skewness Correction: ote that Zk θ is a quadratic function in a Gaussian random vector so Gaussian disiution is not likely to e a good approximation for Zk θ Moreover since is smaller than 9 in magnitude getting an accurate approximate for is important On the other hand Gaussian approximation is etter for the mean than for the tail of the disiution To otain a etter approximation we use the change-of-measure technique to shift the mean of the random field to the threshold and use complete cumulative generating function ote that 9 can e written as Z i j = + x dx g i j exp ξ dx x where g and ξ are defined in and respectively 3 Local Analysis of Covariance: Consider the covariance E Zn θ Zm θ for scores at change-points n and m and with parameters θ and θ respectively Assume the covariance maix associated with y n is V nθ and that associated with y m is V mθ respectively Also assume n > m so the dimension of the covariance maix for y m is larger than that for y n Also note that y n and y m have overlapping samples so V nθ is a su-lock maix of V mθ Based on these oservations after some derivations we find the covariance of Zk θ under H to e E Zn θ Zm θ V nθ V nθ = [ ] / V nθ V nθ V mθ V mθ A special case is when θ = θ n = m then ecomes E Zn θ = which is consistent with the unit variance of Zk θ To study the local covariance of the random field set θ = θ θ = θ + δ n = k and m = k i i = k in We have to run the index k for the change point ackwards ecause the smaller the k the more post-change samples we have and hence the larger the dimension of the covariance maix V nθ Assume δ and i are small relative to θ and k respectively We will expand everything in terms of θ k δ and i Using the first order approximation keeping only the first order terms the local covariance is give y E Zk θzk i θ + δ γ k θδ µk θ k i + oδ + oi 3 When d = the Hessian maix Hk θ is a scalar and is 4

5 γk θ = V kθv kθ V kθ V kθ 4 where the prime f denotes fθ θ ote that γk θ is independent of i The µθ k defined in is the ratio of the average sum-of-squares of the additional terms when we decrease k y a small amount over the average sum-of-squares of the original terms in the covariance maix V kθ 4 Local Random Field Decomposition: ote that the first order expansion of the covariance does not have cross product terms This implies that if we assume Zk θ to e Gaussian it can e decomposed as a sum of two independent one dimensional random processes Using the local covariance we just found we have the following Lemma: Lemma : Assume ξ n with ξ and d where d > is some constant The discretized [ ] process Z k i θ + j ξ i Z j conditioned on Z k θ = ξ can e written as sum of two independent processes: [ Z k i θ + j ξ] Z k θ = ξ = S i + V j where S i = i l= a l with µk θ µk θ a l k k and V j = γ jv γ j with V By Lemma using the techniques in [5] and [4] we have the conditional proaility can e written in terms of decomposed random processes: [ Z i j ij Ji j Z i j = + x S i x i Z ] i j x i S i + j V j x 5 A similar argument for 5 can e found in [5] and [4] 5 Limit y Shrinking : ut this together with approximation for 9 in the approximate significant level ecomes Z i j ij D g i j i j D exp ξ x S i + V j x i j S i x i dx 6 The following Lemma which is an extension of Lemma in [4] enales us to find an expression for integration over x in 8: Lemma 3: Assume x x iid µ σ with µ > Define the random walk S = S i = i l= x l i = and the smooth varying random process V j = βjv β j for some constants > β > As for some constant α > we have = e αx S i x i S i + V j x i j β µ µ ν π σ σ dx 7 where νx is defined in 3 Finally using Lemma 3 for 6 with α = ξ β = γ µ = µkθ k and σ = µkθ k we have the approximate significance level g i j µi j π i i j D ν µ i j i γ i j 8 As the Riemann sum 8 converges to the approximation in Theorem The proof is more complex when the numer of parameters d > and we have to use different techniques such as computing a Mill s ratio type of expression y decomposing the process as a random walk plus a smooth Gaussian field The complete proof can e found in [6] REFERECES [] L Lai Y Fan and H V oor Quickest detection in cognitive radio: a sequential change detection framework IEEE GLOBECOM 8 [] D Siegmund Sequential Analysis: Tests and Confidence Intervals Springer Series in Statistics Springer 985 [3] D Siegmund and B Yakir The statistics of gene mapping Springer 7 [4] H-J Kim and D Siegmund The likelihood ratio test for a change-point in simple linear regression Biomeica vol 76 no 3 pp [5] D Siegmund Approximate tail proailities for the ima of some random fields The Annuals of roaility vol 6 no pp [6] Y Xie and D Siegmund Change-point detection of weak signals: How to use signal correlation? tech rep 5