III.C - Linear Transformations: Optimal Filtering
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1 1 III.C - Linear Transformations: Optimal Filtering FIR Wiener Filter [p. 3] Mean square signal estimation principles [p. 4] Orthogonality principle [p. 7] FIR Wiener filtering concepts [p. 8] Filter coefficients computation [p. 10] MMSE computation [p. 14] Examples [p. 38] FIR Weiner filter and error surfaces [p. 44] Applications to channel equalization [p. 49] Applications to noise cancellation [p. 54] Applications to spatial filtering (antennas) [p. 60] Appendices [p. 64] References
2 Optimal Filtering Problem: How to estimate one signal from another. In many applications desired signal is not observable directly (convolved with another signal or distorted by noise). Examples: Information signal transmitted over channel gets corrupted with noise. Image recorded by system is subject to distortions. Directional antenna array is vulnerable to string jammers in other directions due to sidelobe leakage, etc. In this course: Emphasis on least square techniques to estimate/recover signal (i.e., case 2 ). Other types of norms could be used to solve problems 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 2
3 3 Mean Square Signal Estimation Transmitted signal Possible procedure: s Mean square estimation, i.e., Distorted received signal x { 2 ( ) } minimize ξ = E s d x d ( x) = E[ s x] leads to (proof given in Appendix A) Optimal processor best estimate of transmitted signal s (as a function of received signal x) conditional mean!, usually nonlinear in x [exception when x and s are jointly normal Gauss Markov theorem] Complicated to solve, Restriction to Linear Mean Square Estimator (LMS), estimator of s is forced to be a linear function of measurements x: H d = h x Solution via Wiener Hopf equations using orthogonality principle d
4 Orthogonality Principle Use LMS Criterion: estimate s by where weights {h i } minimize MS error: σ = { 2 ( ) } 2 e E s d x Theorem: Let error e = s - d h { *} { * } d = 2 e H h x minimizes the MSE quantity σ if h is chosen such that E ex = E x e = 0, = 1,, N i i i i.e., the error e is orthogonal to the observations x, i = 1..., N used to compute the filter output. i Corollary: minimum MSE obtained: e min the minimum error obtained for the optimum filter vector. (Proof given in Appendix B) σ = { } E se 2 * where e is 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 4
5 P = 2 dn ( ) = hxn ( ) + hxn ( 1) 0 1 s x(n) x(n-1) 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 5
6 6 signal s(n) + + noise w(n) x(n) dn ( ) filter n 0 x n d(n) d(n) Typical Wiener Filtering Problems Form of Desired Problem Observations Signal Filtering of x(n) = s(n)+w(n) d(n) = s(n) signal in noise n 0 n 0 n 0 d(n) n q n p q n 1 n+p n d(n) n+1 Prediction of x(n) = s(n)+w(n) d(n) = s(n+p); signal in noise p > 0 Smoothing of x(n) = s(n)+w(n) d(n) = s(n q); signal in noise q > 0 Linear x(n) = s(n 1) d(n) = s(n) prediction
7 FIR Wiener Filtering Concepts Filter criterion used: minimization of mean square error between d(n) and dˆ ( n). d(n) What are we doing here? x(n) d a f n filter - s(n) signal noise w(n) We want to design a filter (in the generic sense can be: filter, smoother, predictor) so that: a f a f a f P 1 * k= 0 d n = h k x n k How d(n) is defined specifies the operation done: filtering: d(n)=s(n) predicting: d(n)=s(n+p) smoothing: d(n)=s(n-p) 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 7 e(n)
8 8 How to find h k? Minimize the MSE: { 2 ( ) ˆ ( ) } d n E d n P 1 k = 0 ( k) hx n = h T [, ], ( ), x( n P 1) h= h0 hp 1 x= x n + * k Wiener filter is a linear filter orthogonality principle applies * ( ) ( ) { } E x n i e n = 0, i = 0,..., P 1 * P 1 * E x( n i) d( n) hk x( n k) = 0, i = 0,..., P 1 k = 0 P 1 ( ) ( ) r i hr k i = 0, i= 0,..., P 1 xd k x k = 0 H x T
9 i = 0 i = 1 Matrix form: r 0 R 0 R 1 R P 1 h ( ) P 1 * dx k x k = 0 ( ) ( ) r i = hr k i, i= 0,..., P 1 ( ) ( ) ( ) * dx x x x 0 * r ( 1) R ( 1) ( 0) ( 2) dx x Rx Rx P h 1 = R ( 1) ( 0) x P+ Rx hp 1 Note: different notation than in [Therrien, section 7.3]! 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 9
10 10 Minimum MSE (MMSE) obtained when h is obtained from solving WH equations. For best h obtained: σ min { 2 } { } min ( ( ) ( )) min = E e = E dn dn e 2 * e = ( ) { * } min ( n) E d n e P 1 = E d n d n hk x n k k = 0 d * ( ) ( ) ( ) P 1 ( 0) ( ) = R h r k k = 0 σ = R 0 h r 2 e min d ( ) T dx k dx *
11 11 Summary: FIR Wiener Filter Equations s(n) signal + noise w(n) + x(n) filter FIR Wiener filter is a FIR filter such that: d a f n d(n) + - e(n) where P 1 dˆ n = hx k n k ( ) * ( ) k = 0 2 σ ˆ e = E d n d n ( ) ( ) 2 is minimum.
12 12 How d(n) is defined specifies the specific type of Wiener filter designed: filtering: smoothing: predicting: W-H eqs: MMSE: min * 1 * Rh x = rdx => hopt = Rx rdx 2 T T σ e = Rd(0) hopt rdx = Rd(0) rdx h opt
13 13 Example 1: Wiener filter (filter case: d(n) = s(n) & white noise) Assume x(n) is defined by s(n) signal w(n) s(n), w(n) uncorrelated w(n) white noise, zero mean s(n) + noise + x(n) filter R w (n) = 2δ(n) R s (n) = 2 (0.8) n d a f n d(n) + - e(n)
14 14
15 15 Filter length Filter coefficients 2 [0.405, 0.238] [0.382, 0.2, 0.118] 0.76 MMSE 4 [0.377, 0.191, 0.01, 0.06] [0.375, 0.188, 0.095, 0.049, 0.029] 6 [0.3751, , , , , ] 7 [0.3750, , , , , , ] 8 [0.3750, , 0.038, 0.049, , , ,
16 16 Example 2: Application to Wiener filter (filter case: d(n) = s(n) & colored noise) s(n), w(n) uncorrelated, and zero-mean w(n) noise with R w (n) = 2 (0.5) n s(n) signal with R s (n) = 2 (0.8) n
17 17
18 Filter length Filter coefficients MMSE 2 [0.4156, ] [0.4122, , ] [0.4106, , ] 5 [0.4099, , , , ] 6 [0.4095, , , , , ] 7 [0.4094, , , , , , ] 8 [0.4093, , , , , , , ] /20/14 EC3410.SuFY14MPF - Section IIIC - FIR 18
19 19 Example 3: Application of Wiener filter to one-step linear prediction tracking of moving series forecasting of system behavior data compression telephone transmission W-H equations where h opt * ( ) = hx( n ) dˆ n ( ) d n = R r 1 x P 1 = 0 =? l * dx
20 20 Geometric interpretation: Assume a filter of length 2 x(n+1) x(n-1) e(n) ˆd ( n) x(n) ( ) ( ) ˆd n e n = = e(n) is the error between true value x(n+1) and predicted value for x(n+1) based on x(n) and x(n-1) represents the new information in x(n+1) which is not contained in x(n) and x(n-1) e(n) is called the innovation process corresponding to x(n)
21 21 Geometric interpretation: Assume x(n+1) only has NO new information (i.e., information in x(n+1) is that already contained in x(n) and x(n-1). Filter of length 2. ˆ ( ) ( ) Plot xn ( + 1), d n, e n x(n-1) x(n) ( ) ( ) ˆd n e n = =
22 22 Geometric interpretation: Assume x(n+1) only has new information (i.e., information in x(n+1) is NOT contained at all in x(n) and x(n-1) ). Filter of length 2. ˆ ( ) ( ) Plot xn ( + 1), d n, e n x(n-1) x(n) ( ) ( ) ˆd n e n = =
23 23 1-step ahead predictor scenario RP x(n) defined as x(n) =x(n-1) +v(n) a < 1 v(n) is white noise. 1-step predictor of length 2. AR (1) process Predictor a = 0.5 x ( n) =+ a x( n 1) + a x( n 2) 1 2 seed = 1024 L a = + NM O QP
24 24
25 25 Link between Predictor behavior & input signal behavior 1) Case 1: s(n) = process with correlation R ( k) = δ( k) δ( k 1) δ( k+ 1) s Investigate performances of N-step predictor as a function of changes N 2) Case 2: s(n) = process with correlation R k a a k s ( ) =, < 1 Investigate performances of predictor as a function of changes in a
26 26 1) Case 1: s(n) = wss process with R ( k) = δ( k) δ( k 1) δ( k+ 1) s
27 27 2) Case 2: s(n) = wss process with k s ( ) =, < 1 R k a a % EC MPF % Compute FIR filter coefficients for % a 1-step ahead predictor of length 2 % for correlation sequence of type R(k)=a^{ k } % Compute and plot resulting MMSE value A=[-0.9:0.1:0.9]; for k0=1:length(a) a=a(k0); for k=1:3 rs(k)=a^(k-1); end Rs=toeplitz(rs(1:2)); rdx=[rs(2);rs(3)]; h(:,k0)=rs\rdx; mmse(k0)=rs(1)-h(:,k0)'*rdx; end stem(a,mmse) xlabel('value of a') ylabel('mmse(a)') title('mmse(a)for1-step predictor of length 2, for R_s(k)=a^{ k }')
28 28 Example 4: s(n) = process with w(n) = white noise, zero mean s(n), w(n) uncorrelated R n s Design the 1-step ahead predictor of length 2. Compute MMSE. a f a f = n wa f a f = 2δ R n n
29 29 1- step ahead Predictor Filter Length Coefficients MMSE Filter MMSE 2 [0.3238, ] [0.3059, 01.6, ] [0.3015, , , ] 5 [0.3004, , , , 0.04, ] 6 [0.3001, , , , , ] 7 [0.3, 0.15, , , , 0.001, ] 8 [0.3, 0.15, 0.075, , , , , 0.003]
30 Length N-step ahead Predictor 1-step ahead MMSE 2-step ahead MMSE Filter Filter MMSE /20/14 EC3410.SuFY14MPF - Section IIIC - FIR 30
31 31 Example 5: s(n) = process with w(n) = wss noise, zero mean R n s s(n), w(n) uncorrelated Design the 1-step ahead predictor of length 2 Design 1-step back smoother of length 2 R w a f a f = n a f a f n = n
32 32
33 33 1-step ahead predictor (Col. Noise) Length Coefficients MMSE 2 [0.3325, ] [0.3297, 0.06, ] [0.3285, , , ] [0.3279, , 0.04, , ] 6 [0.3276, , , , , ] 7 [0.3275, , , , , , ] 8 [0.3275, , , , 0.018, , , ]
34 34 Length MMSE (Col. Noise) N-step ahead Predictor N-step back Smoother Filter 1-step 2-step 3-step 4-step 1-step 2-step 3-step 4-step
35 Comments 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 35
36 36 Example 6: s(n) = cos(ω 0 n+φ), φ~u[0,2π] Design the 1-step ahead predictor of length 2, and associated MMSE Design 1-step back smoother of length 2, and associated MMSE
37 37
38 38 Weiner Filters and Error Surfaces 2 e Recall h opt computed from { 2 ( ) } ( ) d Rh x opt = r * dx {( )( ( ) ) * } H H H σ = E d n h x = E d n h x d n h x H ( ) { H 0 } 2 Real( T ) dx = R + h E xx h h r for real signals d(n), x(n) 2 ( ) T e d x σ = R 0 + hrh 2h r T dx using the fact that H ( h x) * H x h H H ( ) = ( ) d n h x = h d n x
39 39 2 ( ) T e d x σ = R 0 + hrh 2h r T dx For filter length 1 h = h 0 x = x(n) σ = R 0 + h R 0 2hr 0 ( ) ( ) ( ) 2 2 e d 0 x 0 dx 2 σ e 2 σ emin h opt h 0
40 40 For filter length P = 2 2 ( ) T e d x σ = R 0 + hrh 2h r T [ ] ( ) ( ) h= h0, h1 ; x= x n, x n 1 T ( ) 2 T e d x σ = R 0 + hrh 2h r ( 0 ) [ h, h ] [ h h ] ( 0) ( 1) ( 0) x( 1) ( 1) ( 0) x 0 = Rd Rx R x h 1 r dx 2 0, 1 rdx T dx R R h σ = A h + Ah + A h + Ah e Ahh + ( ) Rd 0 T dx
41 2 σ e h opt,0 h opt,0 h 0 h 0 h opt,1 h opt,1 h 1 h opt depends on r dx and R x h 1 moves down ( ) 2 T e d x 2 shape of σ e 2 σ up and e σ = R 0 + hrh 2h r R x : specifies shape of ( ) 2 σ e h r dx : specifies where the bowl is in the 3-d plane but doesn t change the shape of the bowl R d (0): moves bowl up and down in 3-d plane but doesn t change shape or location of bowl depends on R x information: λ(r x ) & eigenvectors 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 41 T dx σ e h Rh x = 2Rh 2r = x r dx dx
42 Correlation matrix Eigenvalue Spread Impact on Error Surface Shape see plots Eigenvector Directions for 2 2 Toeplitz Correlation Matrix R x ( ) x( ) ( ) ( ) Rx 0 R 1 normalize 1 a = Rx 1 Rx 0 correlation a 1 eigenvalues of R x ( ) λ a 0 λ a = = 1 + a eigenvectors 1 λ a u11 = 0 a 1 λ u 12 ( λ ) u 11 au = 0 ( + a) u 11 + au 12 = 0 λ 1 = 1 a u11 = u u 12 1 = 1 1 λ2 = 1+ a u2 = 1 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 42
43 43 Error surface shape and eigenvalue ratios a=0.1 a=-0.99 a=-0.5
44 44 Application to Channel Equalization s(n) W(z) Goal: Implement the equalization filter H(z) as a stable causal FIR filter Goal: Recover s(n) by estimating channel distortion (applications in communications, control, etc.) Information Available: x n = z n + v n ( ) ( ) ( ) channel output Delay D v(n) d(n) = s(n-d) z(n) x(n) ˆd ( n) + H(z) e(n) additive noise due to sensors d( n) = s( n D) s(n): original data samples Assumptions: 1) v(n) is stationary, zero-mean, uncorrelated with s(n). 2) v(n) = 0 & D=0
45 45 Assume: W( z) = z z Questions: 1 2 1) Assume v(n) = 0 & D=0. Identify the type of filter (FIR/IIR) needed to cancel channel distortions. Identify resulting H(z) 2) Identify whether the equalization filter is causal and stable. 3) Assume v(n) = 0 & D 0. Identify resulting H 2 (z) in terms of H(z). s(n) W(z) Delay D z(n) H(z) ( ) ˆd n + d(n) = s(n-d) e(n)
46 46
47 47
48
49 49 Assume D 0 s(n) W(z) Delay D z(n) H 2 (z) ( ) ˆd n + d(n) = s(n-d) e(n)
50
51 51 Application to Noise Cancellation Goal: Recover s(n) by compensating for the noise distortion while having only access to the related noise distortion signal v(n) (applications in communications, control, etc.) Signal source s(n) d(n)=s(n)+w(n) + - en ( ) = dn ( ) dn ( ) w(n) Information Available: xn ( ) = vn ( ) s(n)+w(n) & v(n) White noise source w(n) Transformation Of white noise into correlated noise A(z) Wiener filter ( ) dn Assumption: w(n) is stationary, zero-mean, uncorrelated with s(n).
52 52 Assume: vn ( ) = avn ( 1) + wn ( ), a= 0.6, wn ( ) white noise with variance σ sn ( ) = sin( ω n+ φ), φ ~ U[0, 2 π] 0 2 w Compute the FIR Wiener filter of length 2 and evaluate filter performances
53 Results Interpretation 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 53
54 54 Application to Noise Cancellation (with information leakage) Signal source s(n) White noise source w(n) w(n) K Transformation Of white noise into correlated noise A(z) d(n)=s(n)+w(n) v(n) + xn ( ) = vn ( ) + Ksn ( ) Wiener filter - en ( ) = dn ( ) dn ( ) Information Available: ( ) dn s(n)+w(n) & v(n)+ks(n) Assumption: w(n) is stationary, zero-mean, uncorrelated with s(n).
55 Results Interpretation 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 55
56 Application to Spatial Filtering Reference signal d(n)=s(n) e(n) s(n) ± 1 BPSK v 1 (n) v 2 (n) x 1 (n ) x 2 (n ) Noisy received signals x 1 (n) x 2 (n) Filtering ( ) dn Information Available: snapshot in time of received signal retrieved at two antennas & reference signal Assumption: v 1 (n), v 2 (n) zero mean wss white noise RPs independent of each other and of s(n). Goal: Denoise received signal 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 56
57 Application to Spatial Filtering, cont 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 57
58 58
59 59 Application to Spatial Filtering, cont Did we gain anything by using multiple receivers? s(n) ± 1 BPSK x(n) v(n) Filtering Process Assumption: v(n) zero mean RP independent of s(n). Goal: Compute filter coefficients, filter output, and MMSE.
60 60
61 61 Example: Gain Pattern at filter output Example: N-element array, - desired signal at θ 0 =30, - interferences at θ 1 = -20 θ 2 = 40 - Noise power: 0.1 Array steering vector A w ( θ ) = H w a( θ ) H w w 2
62 Appendices 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 62
63 Appendix A: Derivation of proof for Mean Square estimate derivation (p. 3) Proof: ξ = Problem: find ξ = ( ) d x { 2 ( ) } d x ( ( )) ( ) E s { T ( ) } = E s d x s d x L( ss, Define ( x) ) M ( ( )) ( ) ( ) ( ( )) ( ) ( ) = M Ls, d x f s x ds f x d x 0 so that K is minimum : loss function L s, d x f s x f x d x ds, Bayes Rule K ξ is minimized if K is minimized for each value of x K d ( ) ( ) 2 ( s d ( x) ) f ( s x) ds ( s d ( x) ) f ( s x) ds K 2 = s d ( x) f s x ds d d = = 0 = 0 Es [ x] = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 sf s x ds = d x f s x ds s f s x ds = d x f s x ds s f s x ds = d x d ( x) 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 63
64 64 2 K For a minimum we need: > 0 2 d Note: 2 K 2 d ( ( s d ( x) ) f ( s x) ds) d = ( 2) / = ( 2) f =(2) 1 > 0 ( ) s x ds
65 Appendix B: Proof of corollary of orthogonality principle Proof: H e= s h x= s ( h+ h h ) H x h where is the weight vector defined so that the orthogonality principle holds. Resulting error is called e = s h H x σ 2 e 2 H H { } ( ) ( ) H = E e = E s h x+ h h x H H = E e ( h h) x + ( ) 2 H H 2 H H E e = + ( h h) xe + ( h h) x + ex ( h h) { } 2 H ( ) { H } { } 2 { H } { * } E se H 2 H ( ) { }( ) = E e + h h E xe + E h h x + E ex h h { H H } { H } H { H } ( ) = E e = E s h x e = E se h E xe = E se = 06/20/14 EC3410.SuFY14MPF - Section IIIC - FIR 65
66 66 References [1]: Discrete Random Signals and Statistical Signal Processing, C. Therrien, Prentice Hall, 1992 [2]: Statistical and Adaptive Signal Processing, D. Manolakis, V. Ingle & S. Kogon, Artech House, 2005 [3]: Adaptive Filters, 2 nd Ed., S. Haykin, Prentice Hall, 1991
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