III.B Linear Transformations: Matched filter [p. ] Definition [p. 3] Deterministic signal case [p. 5] White noise [p. 19] Colored noise [p. 4] Random signal case [p. 4] White noise [p. 4] Colored noise [p. 8] References 06/19/14 EC3410.SuFY14/MPF - Section IIIB 1
Definition: What is the Matched Filter used for? Used to detect the presence of signals in additive noise (radar, communications, etc ) Information signal structure or statistics are assumed to be knon To cases are considered: (1) Signal is deterministic; () Signal is random xn ( ) = sn ( ) + n ( ) Matched filter y( n) = ys ( n) + y( n) x(n) exists beteen [n 0, n p ] noisy signal y(n) is defined as y(n) = y s (n) + y (n) 06/19/14 EC3410.SuFY14/MPF - Section IIIB
Assume x(n) deterministic (additive noise) Goal: design the LTI filter so that: ( ) ( ) ( ) y s n p SNR = is maximized at n = n E y n { } yn = hn xn = h x n k k = 0 p T [ 0,, P 1] ; ( 0),, ( p) P 1 k ( ) h= h h x= x n x n s( n0 ) P 1 T ys( np) = hsn k ( p k) = h s here s= k = 0 s( np ) flipped n ( 0 ) P 1 T y ( ) ( ) np = hk np k = h here = k = 0 n ( p ) 06/19/14 EC3410.SuFY14/MPF - Section IIIB 3 p T
T T T ( ) ( ) y ( n ) = h s h s = h s s h s p T ( ) Ey np = E h h = h Rh SNR = T h ssh h Rh ( ) ( ) ( ) n np n p 0 T E E = n ( 0 ) ( 0 p) R ( 0) R n n = = R ( ) R ( 0 ) n0 np R 06/19/14 EC3410.SuFY14/MPF - Section IIIB 4
SNR = T h ssh h Rh A) Assume hite noise R = σ I SNR = T h ssh σ h normalizing constant Compute h by maximizing numerator and normalize resulting h so that σ = 1. h ( ) T ( T ) ( ) ( T ) T T T h s s h= s h s h = s h s h 06/19/14 EC3410.SuFY14/MPF - Section IIIB 5
Recall Cauchy Schartz inequality ( )( ) a b a a b b ith equality satisfied only hen a = Κ b ere e have: ( ) T ( T ) ( ) ( T ) T T T h s s h= s h s h = s h s h ere a= b= 06/19/14 EC3410.SuFY14/MPF - Section IIIB 6
SNR maximum is obtained hen h = Ks so that h Rh= 1 ( ) ( ) 1 1 Ks σ Ks = 1 K = = σ s σ s 1 h = s σ s Example: s(n) h(n) 0 M-1 n n 06/19/14 EC3410.SuFY14/MPF - Section IIIB 7
s(n) h(n) 0 M-1 n n 1) o can e express h(n) in terms of s(n)? Recall: h = 1 s σ s 06/19/14 EC3410.SuFY14/MPF - Section IIIB 8
s(n) h(n) 0 M-1 n n ) o can e compute y(n)? Assume no noise x(n)=s(n) Recall: yn ( ) = sn ( ) hn ( ) = skh ( ) n k k 06/19/14 EC3410.SuFY14/MPF - Section IIIB 9
3) o can e compute y(n) if s(n) is delayed? s (n) 0 T T+M-1 n Recall: y( n) = s ( n) h( n) = s ( k) h n k k 06/19/14 EC3410.SuFY14/MPF - Section IIIB 10
4) o do e compute y(n)? noisy case: Assume x(n)=s(n) +(n) Recall: yn ( ) = y( n) + y ( n) = sn ( ) hn ( ) + n ( ) hn ( ) = s 06/19/14 EC3410.SuFY14/MPF - Section IIIB 11
Example 1: sn ( ) = cos( π fn 0 ), n0 n n0 + M 1 Almost no Noise igh SNR level Estimated peak at sample 58 Estimated peak at sample 140 06/19/14 EC3410.SuFY14/MPF - Section IIIB 1
Example : sn ( ) = cos( π fn 0 ), n0 n n0 + M 1 Lo SNR level Estimated peak at sample 144 06/19/14 EC3410.SuFY14/MPF - Section IIIB 13
06/19/14 EC3410.SuFY14/MPF - Section IIIB 14
Example 3: Signals y 1igh (n), y 1Lo (n), and y 1VeryLo (n) each contain a finite time pulse sinusoidal s(n) distorted by additive hite noise in high, medium and lo SNR levels. The signal s(n) is defined as sn ( ) = cos( π fn), 0 n M 1. Assume M=50 and f s =1z. Verify that the cosine function has frequency 0.z by computing the PSD of the signal. Computing the PSD can be done using the functions periodogram.m or spectrum. If you use the spectrum function, follo the folloing script: figure, h = spectrum.elch; % Create a spectral estimator. psd(h,y,'fs',1); % generate PSD plot assuming the sampling frequency is 1z b) Compute the FIR matched filter output and extract the end point of the signal s(n) present in each noisy signals y1igh(n), y1lo(n), and y1verylo(n). Note the output of a signal input to a FIR LTI filter ith impulse response h(n) may be computed as Y_out=filter(h,1,y_in); here y_in is the filter input and h is the FIR filter impulse response. c) The signal y (n) contains multiple repetitions of s(n), as defined above. Identify the end of each signal occurrences. data contained in Pack3BData1.mat 0 06/19/14 EC3410.SuFY14/MPF - Section IIIB 15
Example 4: Detection/Identification via Matched filter Matched filters can be used to detect AND differentiate beteen different incoming symbols 06/19/14 EC3410.SuFY14/MPF - Section IIIB 16
06/19/14 EC3410.SuFY14/MPF - Section IIIB 17
Example 5: 1) Assume you ish to recover a bit sequence generated using FSK. Bit 0 is generated by cos(πf 0 n), n=0, 49, Bit 1 is generated by cos(πf 1 n), n=0, 49. Assume the sampling frequency f s =1z and f 0 <f 1 The received noisy signal y(n) contains a succession of bits imbedded in hite noise. Reconstruct the bit sequence. data contained in Pack3BData.mat 06/19/14 EC3410.SuFY14/MPF - Section IIIB 18
B) Assume colored noise Assume R = LL (Choleski decomposition) SNR T h ssh = = h R h h s h L L h Again pick h so that numerator is maximized & normalize so that h R h = 1. Recall Cauchy Schartz inequality ( )( ) a b a a b b Equality reached hen a = Κ b 1 1 = = ( ) ( ) h s h L L s L h L s a maximum reached hen b 1 ( ) L h= K L s 06/19/14 EC3410.SuFY14/MPF - Section IIIB 19
Maximum reached hen K selected so that: L h = KL s 1 ( ) 1 1 => h= K LL s => h = KR s 1 1 ( ) ( ) h R h= K R s R R s = 1 = T K s R T K s R R = s = 1 K = 1 s R 1 1 R s s = 1 06/19/14 EC3410.SuFY14/MPF - Section IIIB 0
1 1 h= R s s R s 1 Note hen ( ) 1 σ s 1 σ s σ R = σ I h= = s s 06/19/14 EC3410.SuFY14/MPF - Section IIIB 1
Example 6: sn ( ) = a n, 0 n M 1 a < 1 Find matched filter coefficients 06/19/14 EC3410.SuFY14/MPF - Section IIIB
s[n] [Manolakis] [ ] hn [ ] hn Comments: Consider the finite-duration deterministic signal s(n)=(0.6) n u(n), corrupted by additive noise (n) ith autocorrelation sequence r (k) = σ ρ k /(1 ρ ), σ = 0.5. The impulse response of the 8 th -order matched filter for ρ = 0.1 and ρ= -0.8 is computed. The flipped versions h of the optimum matched filters h are shon above. Note: for ρ = 0.1 the matched filter looks like the signal because the correlation beteen the samples of the interference is very small; that is, the additive noise is close to hite, for ρ = -0.8 the correlation increases, and the shape of the optimum filter differs more from the shape of the signal. oever, as a result of the increased noise correlation, the optimum SNR increases. [Manolakis] 06/19/14 EC3410.SuFY14/MPF - Section IIIB 3
Assume x(n) is random sequence (additive noise) SNR { ( ) } s p { ( ) } p E y n = = E y n h Rh s h Rh SNR is maximized hen: h is selected so that it is the eigenvector associated ith the maximum eigenvalue of White noise case: Rh= λr h, s and normalize the eigenvector so that h R h = 1.(see next page) R = σ I regular eigenvector problem Colored noise case: R is a full matrix generalized eigenvector problem eig.m No simple closed form solution for h 06/19/14 EC3410.SuFY14/MPF - Section IIIB 4
o to implement the normalization step so that h R h = 1. Assume h h Rh Kh Rh = Kh, find K so that = 1 ( Kh) R ( Kh) = 1 = 1 K= 1/ h Rh 06/19/14 EC3410.SuFY14/MPF - Section IIIB 5
Example 7: 1) The signal y high (n) and y lo (n) contain a sequence of finite time cosine function s(n) distorted by additive noise. The signal s(n) is defined as sn ( ) = cos( π fn+ ϕ), 0 n M 1, 0 here M=50, f 0 = 0.1z (assume f s =1z), and ϕ ~U[0,π]. a) Assume you kno the first 000 points of the received data contains noise only. Evaluate hether the noise appears to be hite or not. b) Compute the noise-only and signal-only correlation sequence values needed to compute the matched filter impulse responses. c) Compute the matched filter impulse response h(n) d) Extract the end points of the occurrences of s(n) located ithin y(n). (igh-snr scenario) e) Extract the end points of the occurrences of s(n) located ithin y (n) (Lo SNR scenario) Data contained in Pack3BData3.mat 06/19/14 EC3410.SuFY14/MPF - Section IIIB 6
Example 8: Assume you ish to recover a bit sequence generated using FSK. Bit 0 is generated by cos(πf 0 n+θ 0 ), n=0, 49, Bit 1 is generated by cos(πf 1 n+θ 1 ), n=0, 49. Assume f 0 <f 1, the sampling frequency f s =1z, & θ 0, θ 1 independent and U[0,π[. Assume you kno the first 000 points of the received data contains noise only. The received noisy signal y(n) contains a succession of bits imbedded in additive ss noise. a) Evaluate hether the noise appears to be hite or not. b) Compute the noise only and signal only correlation sequence values needed to compute the matched filter impulse responses. c) Compute the matched filter impulse response h(n) d) Reconstruct the bit sequence. Data contained in Pack3BData4.mat 06/19/14 EC3410.SuFY14/MPF - Section IIIB 7
References [1]: Discrete Random Signals and Statistical Signal Processing, C. Therrien, Prentice all, 199 []: Statistical and Adaptive Signal Processing, D. Manolakis, V. Ingle & S. Kogon, Artech ouse, 005 [3] Adaptive Filters, nd Ed., S. aykin, Prentice all, 1991 06/19/14 EC3410.SuFY14/MPF - Section IIIB 8