Plug-in Measure-Transformed Quasi Likelihood Ratio Test for Random Signal Detection Nir Halay and Koby Todros Dept. of ECE, Ben-Gurion University of the Negev, Beer-Sheva, Israel February 13, 2017 1 / 20
Outline Detection problem formulation Existing detectors Proposed detector Simulation studies Summary 2 / 20
Detection Problem Formulation H 0 : H 1 : { Xn = W n, n = 1,..., N Y m = W m (s), m = 1,..., M { Xn = S n a + W n, n = 1,..., N Y m = W m (s), m = 1,..., M, {X n C p }, {Y m C p }: Primary and secondary data {S n C}: i.i.d. zero-mean signal with unknown distribution a C p : Known steering vector {W n }, {W (s) m }: i.i.d. zero-mean symmetrically distributed homogeneous noise processes with unknown distribution {S n }, {W n } and {W (s) m } are mutually independent 3 / 20
Existing detectors Gauss-Gauss detector [Jin & Friedlander 2005] GLRT-based detector Assumes normally distributed signal and noise Simple implementation and tractable performance analysis Sensitive to deviation from normality (e.g., in the case of heavy-tailed noise that produces outliers) 4 / 20
Existing detectors CG-GLRT [Gerlach 1999, Shuai et. al. 2010] Assumes elliptical compound-gaussian noise Resilient against heavy-tailed noise outliers Iterative ML-estimation of the noise scatter matrix Converges under some regularity conditions Each iteration involves matrix inversion Does not reject large norm outliers Can be sensitive to non-elliptical noise 5 / 20
Proposed Detector General operation principle Selects a Gaussian model that best empirically fits a transformed probability distribution of the data Advantages Resilient to outliers (rejects large-norm outliers) Involves higher-order statistical moments Significant mitigation of the model mismatch effect Computational and implementation simplicity 6 / 20
Probability Measure Transform Definition Given a non-negative function u : X R + satisfying A transform T u : P X;H Q (u) X;H 0 < E [u (X) ; P X;H ] <. T u [P X;H ] (A) = Q (u) X;H (A) is defined as: ϕ u (x; H)dP X;H (x), A where ϕ u (x; H) u (x) E [u (X) ; P X;H ]. The function u ( ) is called the MT-function. 7 / 20
Probability Measure Transform The measure transformed mean and covariance µ (u) X;H = E [Xϕ u (X; H); P X;H ] where Σ (u) X;H = E [ XX H ] (u) ϕ u (X; H); P X;H µ X;Hµ (u)h X;H u (x) ϕ u (x; H) E [u (X) ; P X;H ] = dq(u) X;H dp X;H Conclusion The mean and covariance under Q (u) X;H only samples from P X;H. can be estimated using u (x) non-constant & analytic the mean and covariance under Q (u) X;H involve higher-order statistical moments of P X;H. 8 / 20
Probability Measure Transform Proposition (Consistent empirical MT mean and covariance) Let X n, n = 1,..., N denote a sequence of i.i.d. samples from P X;H, and define the empirical mean and covariance estimates: ˆΣ (u) X where ˆϕ u (X n ) ˆµ (u) X N n=1 N X n ˆϕ u (X n ) n=1 X n X H n ˆϕ u (X n ) ˆµ (u) x ˆµ (u)h x u(xn) N If n=1 u(xn). ] E [ X 2 u (X) ; P X;H <, then ˆµ (u) X w.p.1 µ (u) X;H and ˆΣ (u) X w.p.1 Σ (u) X;H as N. 9 / 20
Probability Measure Transform Proposition (Robustness to outliers) If the MT-function u(x) and u(x) x 2 are bounded, then the influence functions [Hampel, 1974] of ˆµ (u) X and ˆΣ (u) X Remark Condition is satisfied when u (x) Gaussian family. 1 0.9 are bounded. IF 0.8 0.7 0.6 0.5 0.4 IF Σ (u) x (y) IF µ (u) x (y) 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 y 10 / 20
Measure Transformed Gaussian Quasi LRT MT-GQLRT [Todros & Hero, 2016] Compares the empirical KLDs between Q (u) X;H and two Gaussian measures Φ(µ (u), X;H0 Σ(u) ) and Φ(µ(u), X;H0 X;H1 Σ(u) ) X;H1 [ ] T u = D ˆΣ(u) LD X Σ (u) X;H0 [ ] D ˆΣ(u) LD X Σ (u) X;H1 + ˆµ (u) X ˆµ (u) X µ (u) X;H0 µ (u) X;H1 2 (Σ (u) x;h 0 ) 1 2 (Σ (u) x;h 1 ) 1 H 1 H 0 τ, 11 / 20
Measure Transformed Gaussian Quasi LRT MT-GQLRT for the considered detection problem Class of MT-functions: { ( ) } u (x) = v P a x, v : C p R +, v(x) = v( x) MT-mean and MT-covariance under H 0 and H 1 : Σ (u) = X;H0 Σ(u) W Equivalent test statistic: µ (u) = 0, k = 0, 1 X;Hk T u = a H ( Σ (u) W and Σ (u) = X;H1 σ2 Saa H + Σ (u) W ) 1 Ĉ(u) X ( Σ (u) W ) 1 a 12 / 20
Plug-in Measure Transformed Gaussian Quasi LRT Plug-in MT-GQLRT Plug-in the empirical MT-covariance of the noise obtained from noise-only secondary data {Y m } ( T u a H ˆΣ(u) Y ) 1 Ĉ(u) X ( ) 1 H 1 ˆΣ(u) Y a t, H 0 Theorem (Asymptotic normality) Assume that 1. σs 2 > 0, Σ(u) W is non-singular 2. E [ u 2 ] [ ] (X) ; P X;H, E X 4 u 2 (X) ; P X;H are finite Then ( ) ( ) N T u η (u) D H N 0, λ (u) H N for H = H 0, H 1 13 / 20
Plug-in Measure Transformed Gaussian Quasi LRT Threshold determination t = ˆη (u) H0 + Q 1 (α) ˆλ (u), H0 Guarantees asymptotic CFAR= α ˆη (u) H0 ( ) 1 ah ˆΣ(u) Y a ˆλ(u) H0 M N M m=1 ˆϕ 2 u (Y m ) ( ( ah ˆΣ(u) Y ) 1 Ym 2 ˆη (u) H0 ) 2 14 / 20
Plug-in Measure Transformed Gaussian Quasi LRT Selection of the MT-function to induce outlier resilience Gaussian MT-function: u G (x; ω) = exp ( P a x 2 /ω 2) Define the Asymptotic Relative Local Power Sensitivity to change in σs 2 (under nominal Gaussian distribution): R (ω, Σ W ) β u G σs 2 / β LRT σs 2 = σ 2 S =0 NG (ω, ΣW ) + Q 1 (α) N + Q 1 (α) Select the lowest ω such that R (ω, Σ W ) r, 0 << r < 1 In practice Σ W is replaced by: ˆΣ W (ω) = ( ˆΣ(u G ) Y (ω) P a /ω 2 ) 1 15 / 20
Simulation Studies Setup BPSK signal with variance σ 2 S Steering vector: a 1 p [ 1, e iπ sin(θ),..., e iπ(p 1) sin(θ)] T, p = 8, θ = 60 Sample size in primary and secondary data: N = 1000, M = 5000 False alarm rate α = 0.05 Asymptotic Relative Local Power Sensitivity: r = 0.9 16 / 20
Simulation Studies Detection in Gaussian noise Zero-mean Gaussian noise with Toeplitz structured covariance Σ: { σ 2 [Σ] i,j Wb j i, i j ( ) b i j, b < 1,, i > j σ 2 W 1 0.8 Power 0.6 0.4 Omniscient LRT Gauss-Gauss detector 0.2 CG-GLRT Plug-in MT-GQLRT (empirical) Plug-in MT-GQLRT (asymptotic) 0-20 -18-16 -14-12 -10-8 SNR [db] 17 / 20
Simulation Studies Detection in elliptical CG noise Elliptical t-distributed noise with λ = 0.2 degrees of freedom and dispersion matrix Σ 1 0.8 Power 0.6 0.4 0.2 Omniscient LRT Gauss-Gauss detector CG-GLRT Plug-in MT-GQLRT (empirical) Plug-in MT-GQLRT (asymptotic) 0-20 -18-16 -14-12 -10-8 -6-4 SNR [db] 18 / 20
Simulation Studies Detection in non-elliptical noise Elliptical t-distributed noise + BPSK interference (SIR = 5 [db]) 1 0.8 Power 0.6 0.4 0.2 Omniscient LRT Gauss-Gauss detector CG-GLRT Plug-in MT-GQLRT (empirical) Plug-in MT-GQLRT (asymptotic) 0-20 -18-16 -14-12 -10-8 -6-4 SNR [db] 19 / 20
Conclusions Plug-in MT-GQLRT for detection of a random signal that lies on a known rank-one subspace Simple implementation and tractable performance analysis Less sensitive to model mismatch as compared to other model based detectors Extension to non-homogeneous noise environment 20 / 20