GLRT for Cooperative Spectrum Sensing: Threshold Setting in Presence of Uncalibrated Receivers Andrea Mariani, Andrea Giorgetti, and Marco Chiani University of Bologna, Italy DEI - Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi” Cesena Campus 4th Int. Workshop of COST Action IC0902 Rome, Italy October 9–11, 2013 Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 1 / 18
Outline Study of GLRT in presence of uncalibrated receivers (SUs with different noise power) Statistical description of the test Approximated expressions for setting the decision threshold (Neyman-Pearson approach) Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 2 / 18
Cooperative sensing scenario PU � Networks . � . � . � . � . � . � Channel matrix . � . � . � SU N t SU � Network k Fusion � centre Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 3 / 18
System model Given n R cooperative SUs and n T PUs the output of the receiving antennas at the i -th time instant is y i = H x i + n i where n i ∈ C n R AWGN vector x i ∈ C n T PU transmitted symbol vector; x i ∼ CN ( 0 , R x ) H ∈ M n R × n T ( C ) channel gain matrix Observation matrix from n S snapshots Y = ( y 1 | · · · | y n S ) . Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 4 / 18
GLRT derivation Under the H j hypothesis, with j = 0 , 1, the likelihood function of Y is � � �� 1 − 1 S L ( Y | Σ j ) = π n R n S | Σ j | n S exp − n S tr Σ j n S YY H . 1 where the sample covariance matrix (SCM) is defined as S = The GLR to detect the hypothesis H 0 is � � � � �� Σ 1 � H 0 � � ≷ T = ξ � � �� H 1 Σ 0 � where 0 ≤ ξ ≤ 1. � Σ 1 and � Σ 0 are the ML estimates of Σ 0 and Σ 1 , respectively. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 5 / 18
Sphericity Test Assuming � � = σ 2 I n R , i.e. the SU have the same noise power y i y i H |H 0 H 0 : Σ 0 = E � � y i y i H |H 1 H 1 : No assumptions on Σ 1 = E The GLRT is H 0 | S | T (sph) = ≷ ξ ( tr { S } / n R ) n R H 1 n S YY H . 1 where S is the sample covariance matrix (SCM) S = Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 6 / 18
Test of Independence Assuming H 0 : Σ 0 is diagonal; ”unbalanced receivers” case H 1 : No assumptions on Σ 1 In this case the GLRT is H 0 | S | T (ind) = � n R ≷ ξ k =1 s k , k H 1 where s i , j is the ( i , j ) element of S . Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 7 / 18
Detection with uncalibrated receivers 1 0.8 0.6 P D T (ind) , ∆ = 0 dB T (ind) , ∆ = 0 . 5 dB T (ind) , ∆ = 1 dB 0.4 T (ind) , ∆ = 1 . 5 dB T (sph) , ∆ = 0 dB T (sph) , ∆ = 0 . 5 dB T (sph) , ∆ = 1 dB 0.2 T (sph) , ∆ = 1 . 5 dB 0 0 0.2 0.4 0.6 0.8 1 P FA Figure: ROC comparison between T (ind) and T (sph) in presence of 4 SUs and a single PU. We assume that the noise power levels in dB at the SU receivers equal � σ 2 ref , σ 2 ref + ∆ , σ 2 ref − ∆ , σ 2 � . The reference level σ 2 ref correspond to SNR = − 10 dB. ref n S = 500. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 8 / 18
Threshold setting To design the threshold for both tests we need to know the distribution of T (ind) and T (sph) under H 0 . Unfortunately, such statistical distributions are: known in closed-form for T (ind) , but expressed as the Meijer G-function multiplied by a normalizing constant 1 not known in closed-form for T (sph) , but expressed as an infinite sum 2 In both cases, such expressions cannot be easily inverted for threshold setting! Hence, we propose a moment matching approach . 1M. D. Springer and W. E. Thompson, “The distribution of products of beta, gamma and gaussian random variables,” SIAM Journal on Applied Mathematics, vol. 18, no. 4, pp. 712-737, Jun. 1970. 2B. N. Nagarsenker and M. M. Das, “Exact distibution of sphericity criterion in the complex case and its percentage points,” Communications in statistics, vol. 4, no. 4, pp. 363-374, 1975. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 9 / 18
Moments of T (ind) Theorem (On the distribution of the independence test) Consider the test statistic T ( ind ) = | S | / � n R k =1 s k , k , where S = { s i , j } is the SCM of a n R -variate complex Gaussian population with zero mean and diagonal covariance matrix. Then T ( ind ) can be expressed as � n R T ( ind ) = T n R T n R − 1 · · · T 2 = T k . k =2 where { T k } k =2 ,..., n R are independent beta distributed r.v.s with p.d.f. � B ( n S − k +1 , k − 1) t n S − k (1 − t ) k − 2 , 1 0 ≤ t ≤ 1 0 , otherwise . Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 10 / 18
Moments under H 0 Moments of T (ind) Based on the previous theorem the moments of T (ind) can be derived as n R m (ind) � T p � � = E p k k =2 n R − 1 n R − 1 � Γ( n S ) � Γ( n S − k + p ) m (ind) � = . p Γ( n S + p ) Γ( n S − k ) k =1 Moments of T (sph) are given by [Nagarsenker and Das, 1975] n n R p n R Γ( n S n R ) Γ( n S − i + 1 + p ) m (sph) � R = . p Γ( n S n R + n R p ) Γ( n S − i + 1) i =1 Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 11 / 18
MM-based approximation (1) We approximate T (ind) and T (sph) to beta distributed r.v.s. Thus the approximated p.d.f. is given by � B (a , b) t a − 1 (1 − t ) b − 1 , 1 0 ≤ t ≤ 1 f T ( t ) ≃ 0 , otherwise � 1 0 x a − 1 (1 − x ) b − 1 dx is the beta function with where B (a , b) = parameters a and b : a = m 1 (m 2 − m 1 ) , m 2 1 − m 2 b = (1 − m 1 ) (m 2 − m 1 ) . m 2 1 − m 2 Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 12 / 18
MM-based approximation (2) Thus P FA � Pr { T < ξ |H 0 } can be expressed as � ξ 1 B (a , b) t a − 1 (1 − t ) b − 1 dt = � P FA ≃ B (a , b , ξ ) 0 � ξ 0 x a − 1 (1 − x ) b − 1 dx is the regularized beta where � 1 B (a , b , ξ ) = B (a , b) function. The decision threshold can be easily calculated as B − 1 � � ξ = � a , b , P DES . FA Note that � B − 1 ( · , · , · ) can be easily computed using standard mathematical software. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 13 / 18
MM-based approximation - T (ind) 1 0.8 CDF ( x ) 0.6 n R =8 n R =6 n R =4 0.4 0.2 simulated β approx. 0 0 0.2 0.4 0.6 0.8 1 x Figure: Comparison between the CDF based on the moment matching strategy and numerically simulated curve for T (ind) under H 0 with n S = 20. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 14 / 18
MM-based approximation - T (sph) 1 0.8 CDF ( x ) 0.6 n R =8 n R =6 n R =4 0.4 0.2 simulated β approx. 0 0 0.2 0.4 0.6 0.8 1 x Figure: Comparison between the CDF based on the moment matching strategy and numerically simulated curve for T (sph) under H 0 with n S = 20. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 15 / 18
Chi squared approximation comparison 1 1 0.8 0.8 CDF ( x ) CDF ( x ) 0.6 0.6 n S =100 n S =500 n S =1000 n S =200 0.4 0.4 β approx. β approx. 0.2 0.2 χ 2 approx. χ 2 approx. estimated estimated 0 0 0 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 0.9 1 x x (a) (b) Figure: Comparison among the CDF and the empirically estimated curve for T (ind) under H 0 . n R = 20. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 16 / 18
Conclusions We studied the generalized likelihood ratio test (GLRT) for cooperative spectrum sensing The most proper assumption is that every SU experience a different noise power level ( noise unbalances ) T (ind) is robust to noise unbalances ⇒ T (ind) should be adopted in place of T (sph) Both tests can be very well approximated as beta r.v.s. (under null hyp.) We provided easy-to-use expressions for setting the decision threshold under the Neyman-Pearson framework. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 17 / 18
Thank you! andrea.giorgetti@unibo.it A. Mariani, A. Giorgetti, and M. Chiani, “Test of Independence for Cooperative Spectrum Sensing with Uncalibrated Receivers,” in Proc. IEEE Global Commun. Conf. (GLOBECOM 2012), Anaheim, CA, USA, Dec. 2012. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 18 / 18
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