Optimal Filtering Optimal filtering is a means of adaptive extraction of a weak desired signal in the presence of noise and interfering signals Mathematically Given x n d n v n estimate and extract d n from the current and past values of x n EE 524 10 1 Optimal Filtering cont Let the filter coefficients be w0 w1 w wN 1 Filter output y n N 1 X b wk x n k wH x n d n k 0 EE 524 10 2 where x n x n 1 x n x n N 1 Omitting index n we can write db wH x x x0 x1 xN 1 EE 524 10 3 Optimal Filtering cont Minimize the Mean Square Error MSE MSE E e 2 E d wH x 2 E d wH x d xH w E d 2 wH E xd E dxH w wH E xxH w min MSE MSE w 0 E xxH w E xd 0 w R E xxH correlation matrix r E xd Wiener Hopf equation see Hayes 7 2 Rw r EE 524 10 wopt R 1r 4 Wiener Filter Three common filters 1 general non causal X H z hk z k k 2 general causal H z X hk z k k 0 3 finite impulse response FIR H z N 1 X hk z k k 0 EE 524 10 5 Wiener Filter Case 1 Non causal filter X MSE E e n 2 E d n hk x n k d n k rdd 0 X X h l rdx l l k X hlx n l l X hk rdx k X rxx l k hk h l k l Remark for causal and FIR filters only limits of sums differ Let hi i j i MSE i 0 MSE i 0 i rdx i X hk rxx i k i k EE 524 10 6 In Z domain Pdx z H z Pxx z which is the optimal non causal Wiener filter Example x n d n e n Pdd z 0 36 1 1 0 8z 1 0 8z Pee z 1 d n and e n uncorrelated EE 524 10 7 Optimal filter Pxx z Pdd z Pee z 0 36 1 1 0 8z 1 1 0 8z 1 0 5z 1 1 0 5z 1 6 1 1 0 8z 1 0 8z rdx z E d n k d n e n rdd k Pdx z Pdd z H z Pdx z 0 36 Pxx z 1 6 1 0 5z 1 1 0 5z h k 0 3 12 k EE 524 10 8 Case 2 Causal filter 7 3 2 in Hayes MSE E e n 2 E d n rdd 0 X h l rdx l l 0 rdx i X X hk x n k d n k 0 X X X k 0 k 0 l 0 hk rdx k X hlx n l l 0 rxx l k hk h l hk rxx i k i k 0 Let 1 1 B z B z Pxx z EE 524 10 9 Pick B z to be a stable causal minimum phase system Then 1 Pdx z H z B 1 z B z z causal H z 1 Pdx z B B z z where Y z X k yk z k X yk z k k 0 1 H z B z Pdx z B z EE 524 10 10 Causal Filter Example Same as before x n d n e n Pdd z 0 36 1 1 0 8z 1 0 8z Pee z 1 d n and e n uncorrelated EE 524 10 11 Optimal filter Pdx z Pdd z Pxx z Pdd z Pee z 1 0 5z 1 1 0 5z 1 6 1 1 0 8z 1 0 8z B z 1 Pdx z B z EE 524 10 1 1 0 8z 1 stable and causal 1 1 6 1 0 5z 0 36 1 1 0 8z 1 0 8z 1 1 0 8z 1 6 1 0 5z 0 36 1 1 6 1 0 8z 1 1 0 5z 5 5 i z 0 36 h 3 6 1 6 1 0 8z 1 1 0 5z 12 1 Pdx z B z 5 0 36 3 1 6 1 0 8z 1 5 1 1 1 1 0 8z 0 36 3 H z B z Pdx z B z 1 6 1 0 5z 1 1 6 1 0 8z 1 1 3 1 k 0 375 h k 2 k 0 1 2 1 0 5z 1 8 Case 3 FIR filter done before rdx i N 1 X hk rxx i k i k 0 EE 524 10 13 FIR Wiener Filter Generalization Theorem 1 Assume that x d N x d Cxx Cxd Cdx Cdd Then the posterior pdf of d given x fd x d x is also Gaussian with moments given by WH z 1 E d x d CdxCxx x x 1 Cd x Cdd CdxCxx Cxd EE 524 10 14 FIR Wiener Filter Generalization Consider d A y B where V11 V12 0 cov V21 V22 Also assume E 0 and cov We wish to predict d given y E EE 524 10 15 Theorem 2 Minimum Mean Square Error MMSE Solution b A C y B d 0 0 where C A B H V12 B B H V22 1 MMSE A AH V11 C B B H V22 C H A B H V12 C H C B AH V21 EE 524 10 16 Kalman Filter Consider a state space model x t Ax t 1 t y t Bx t t where t and t are independent sequences of zero mean random vectors with covariances and respectively t and x u are independent for t u and t and x u are independent for t u We wish to predict x t given y 1 y 2 y t EE 524 10 17 Idea Theorem 2 can be used to predict x s given y t We write such b s t and its minimum mean square error as P s t predictor as x How to construct a recursive procedure EE 524 10 18 Kalman Filter b t 1 t and its MMSE P t 1 t We wish to find Assume that we know x b t 1 t 1 Then we can write the following model x x t 1 y t 1 B b t 1 t and P t 1 t This implies where 0 x b t 1 t 1 x b t 1 t C t 1 y t 1 B x b t 1 t x where C t 1 P t 1 t B H BP t 1 t B H 1 Also P t 1 t 1 P t 1 t C t 1 BP t 1 t B H 1C t 1 H EE 524 10 19 Kalman Filter b t 1 t based on x b t t …