VIEWS: 253 PAGES: 18 CATEGORY: Technology POSTED ON: 11/9/2009 Public Domain
ECE642: Detection and Estimation Theory ECE642: Detection and Estimation Theory Dr. Sudharman K. Jayaweera Assistant Professor Department of Electrical and Computer Engineering University of New Mexico Lecture 10 - September 25th , Tuesday Fall 2007 Dr. S. K. Jayaweera, Fall 07 1 ECE642: Detection and Estimation Theory Detection of Coherent Signals with Unknown Amplitude in iid Noise • In many detection problems, we know the form of the received signal except its amplitude. • We can use following composite hypothesis testing problem to model this situation: H0 : Yk = Nk for k = 1, 2 . . . , n versus (1) H1 : Yk = θsk + Nk for k = 1, 2 . . . , n, θ > 0 • Here θ > 0 is a signal strength parameter Dr. S. K. Jayaweera, Fall 07 2 ECE642: Detection and Estimation Theory Detection of Coherent Signals with Unknown Amplitude in iid Noise: Vector Formulation • In vector notation: H0 : Y = N versus H1 : Y = θs + N, θ>0 (2) s1 N1 s N 2 2 where s = is known and N = is a continuous random . . sn Nn noise vector with iid components and marginal probability density function pNk Dr. S. K. Jayaweera, Fall 07 3 ECE642: Detection and Estimation Theory Coherent Signals with Unknown Amplitude in iid Noise: UMP Detection? • Since noise components are iid, given θ we can write the likelihood ratio between H0 and H1 as, n pNk (yk − θsk ) Lθ (y) = ∏ (3) k=1 pNk (yk ) • Recall that model (1) is in fact a composite hypothesis testing problem. • Then from previous discussions we know that, in general, the critical region Γθ = {y ∈ Rn | Lθ (y) > τ} will depend on θ (except for some special cases). • Hence, UMP tests for model (2) exist only for some special noise models (eg. Gaussian noise case considered in example A.1). Dr. S. K. Jayaweera, Fall 07 4 ECE642: Detection and Estimation Theory Locally Optimum (Locally Most Powerful) Detection of Coherent Signals with Unknown Amplitude in iid Noise • But of course we can easily ﬁnd LMP tests for model (2) • These LMP tests have a simple and intuitively reasonable structure that makes their investigation interesting. • Assuming sufﬁcient regularity, the locally optimum test for H0 versus H1 is (from (9.19)): 1 > ∂ δl0 (y) = ˜ γ if ∂θ Lθ (y) = τ (4) θ=0 0 < Dr. S. K. Jayaweera, Fall 07 5 ECE642: Detection and Estimation Theory Likelihood Ratio for Locally Optimum Detection of Coherent Signals in iid Noise • Differentiating (3), ∂ ∂ pNk (yk − θsk ) ∂θ (pN j (y j − θs j )) n n Lθ (y) = ∑ ∏ pN (yk ) ∂θ pN j (y j ) j=1 k=1 k k= j ∂ n n pN (yk ) s j p′ j (y j − θs j ) N Lθ (y) = − ∑ ∏ k θ=0 ∂θ pNk (yk ) pN j (y j ) θ=0 j=1 k=1 k= j n p′ j (y j )s j = −∑ N (5) j=1 pN j (y j ) ∂ where p′ j (x) = pN j (x) . (6) N ∂x Dr. S. K. Jayaweera, Fall 07 6 ECE642: Detection and Estimation Theory Locally Optimum Detection of Coherent Signals in iid Noise (ctd...) • Deﬁne the function: p′ 1 (x) N glo (x) = − . (7) pN1 (x) • Since we have assumed that noise samples are iid, pN j (y j ) = pN1 (y j ) for j = 1, 2, . . . , n. (8) • Hence, from (7) and (8) we can write (5) as, ∂ n ∂θ Lθ (y) = ∑ s j glo(y j ) (9) θ=0 j=1 Dr. S. K. Jayaweera, Fall 07 7 ECE642: Detection and Estimation Theory Locally Optimal Detector for Coherent Signals in iid Noise • Hence, from (4) and (5), the LMP test for H0 versus H1 is, 1 > δl0 (y) = ˜ γ if ∑n sk glo (yk ) = τ k=1 (10) 0 < Dr. S. K. Jayaweera, Fall 07 8 ECE642: Detection and Estimation Theory Locally Optimal Detector for Coherent Signals in iid Noise (ctd...) Figure 1: LMP Detector for Coherent Signals in iid Noise • Since this has a memoryless nonlinearity followed by a correlator, it is called a nonlinear correlator. • Just like the likelihood ratio, the locally optimum non linearity glo (.) shapes the observations yk ’s to reduce the detrimental effects of the noise as much as possible. Dr. S. K. Jayaweera, Fall 07 9 ECE642: Detection and Estimation Theory Example A.3: Locally Optimum Detection of Coherent Signals in iid Gaussian Noise • Suppose noise samples are all N (0, σ2 ). Then, for k = 1, . . . , n y2 1 − k2 pN1 (yk ) = √ e2σ 2πσ2 ∂ yk ∴ [pN1 (yk )] = −pN1 (yk ) 2 (11) ∂yk σ • Hence from (7) and (11), yk gl0 (yk ) = (12) σ2 • Hence, in this case gl0 (yk ) is in fact a linear function of the observation yk Dr. S. K. Jayaweera, Fall 07 10 ECE642: Detection and Estimation Theory Example A.3: Locally Optimum Detection of Coherent Signals in iid Gaussian Noise (ctd...) • Hence, in this special case Fig. 1 in fact is the same correlator that we obtained in example A.1 (Fig. 9.3) • But, of course from the discussion on UMP tests we know that for Gaussian noise case the detector in Fig. 9.3 is in fact UMP for H0 versus H1 • Since it is UMP, then of course it is also LMP!. Dr. S. K. Jayaweera, Fall 07 11 ECE642: Detection and Estimation Theory Example A.4: Locally Optimum Detection of Coherent Signals in iid Laplacian Noise • Suppose now that noise samples are iid Laplacian with α −α|yk | pN1 (yk ) = e for yk ∈ R 2 • Then, ∂ ∂ α −α√y2 α −α√y2 1 2 −1 pN1 (yk )) = e k = − e k α y 2 2yk ∂yk ∂yk 2 2 2 k α −α|yk | αyk α2 −α|yk | yk = − e = − e 2 y2 2 | yk | k α2 −α|yk | = − e sgn(yk ) (13) 2 = −αpN1 (yk )sgn(yk ) (14) Dr. S. K. Jayaweera, Fall 07 12 ECE642: Detection and Estimation Theory Example A.4: Locally Optimum Detection of Coherent Signals in iid Laplacian Noise (ctd...) • Hence from (7) and (14): glo (yk ) = α sgn(yk ) (15) • Thus, LMP test for H0 versus H1 in Laplacian noise (from (10)) is: 1 > δl0 (y) = ˜ γ if ∑n α sk sgn(yk ) = τ k=1 (16) 0 < Dr. S. K. Jayaweera, Fall 07 13 ECE642: Detection and Estimation Theory Locally Optimum Detector for Coherent Signals in iid Laplacian Noise Figure 2: LMP Detector for Coherent Signals in iid Laplacian Noise • Thus, in this case the LMP test correlates a scaled version of the signal with the sequence of signs of the observations. • The function glo (yk ) as given by (15) is called a hard limiter – Compare this with the soft limiter we encountered in example A.2 as the optimal detector for the same Laplacian noise (when hypotheses are simple). Dr. S. K. Jayaweera, Fall 07 14 ECE642: Detection and Estimation Theory Example A.5: Locally Optimum Detection of Coherent Signals in iid Cauchy Noise • A noise model that has an even heavier tail than the Laplacian noise is the Cauchy Noise. • Cauchy Noise has the pdf: 1 pN1 (yk ) = for yk ∈ R (17) π(1 + y2 ) k • Then ∂ −π 2yk 2yk pN1 (yk ) = = −pN1 (yk ) (18) ∂yk π(1 + y2 ) 2 1 + y2 k k • Hence, from (7) and (18): 2yk glo (yk ) = 1 + y2 k Dr. S. K. Jayaweera, Fall 07 15 ECE642: Detection and Estimation Theory Example A.5: Locally Optimum Detector for Coherent Signals in iid Cauchy Noise • Then, the LMP detector for Cauchy noise is (from (10)) 1 > δl0 (y) = ˜ γ if ∑n 1+y2 sk = τ k=1 2yk (19) k 0 < Figure 3: LMP Detector for Coherent Signals in iid Cauchy Noise Dr. S. K. Jayaweera, Fall 07 16 ECE642: Detection and Estimation Theory Example A.5: Locally Optimum Detection of Coherent Signals in iid Cauchy Noise (ctd...) • Note that, function glo (yk ) in this case, is approximately linear near yk = 0. • Hence, for small values of observations, the detector approximately correlates the observation yk with the signal sk . • However, for the observations with large magnitudes, the function glo (yk ) asymptotically goes to zero. – Hence, the detector makes the contribution from large observations to the accumulated sum very small (because there is a high probability that those large values are due to (Cauchy distributed) noise.) Dr. S. K. Jayaweera, Fall 07 17 ECE642: Detection and Estimation Theory Next Time Detection of Deterministic Signals in Correlated Gaussian Noise (Section III.B) Dr. S. K. Jayaweera, Fall 07 18