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Course: CAAM 236, Spring 2007
School: Monmouth IL
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(Fall EE236A 2007-08) Lecture 6 FIR lter design FIR lters linear phase lter design magnitude lter design equalizer design 61 FIR lters nite impulse response (FIR) lter: n1 y (t) = =0 h u(t ), tZ u : Z R is input signal ; y : Z R is output signal hi R are called lter coecients ; n is lter order or length lter frequency response: H : R C H ( ) = h0 + h1ej + + hn1ej (n1) n1 t=0 n1 t=0 = ht...

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(Fall EE236A 2007-08) Lecture 6 FIR lter design FIR lters linear phase lter design magnitude lter design equalizer design 61 FIR lters nite impulse response (FIR) lter: n1 y (t) = =0 h u(t ), tZ u : Z R is input signal ; y : Z R is output signal hi R are called lter coecients ; n is lter order or length lter frequency response: H : R C H ( ) = h0 + h1ej + + hn1ej (n1) n1 t=0 n1 t=0 = ht cos t j ht sin t (j = 1) periodic, conjugate symmetric, so only need to know/specify for 0 FIR lter design problem: choose h so H and h satisfy/optimize specs FIR lter design 62 example: (lowpass) FIR lter, order n = 21 impulse response h: 0.2 0.1 0 0.1 0.2 0 2 4 6 8 10 12 14 16 18 20 h(t) t frequency response magnitude |H ( )| and phase H ( ): 10 1 3 2 |H ( )| 10 H ( ) 0 1 0 1 2 3 0 10 1 10 2 10 3 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 FIR lter design 63 Linear phase lters suppose n = 2N + 1 is odd and impulse response is symmetric about midpoint: ht = hn1t, t = 0, . . . , n 1 then H ( ) = h0 + h1ej + + hn1ej (n1) = ejN H ( ) = ejN (2h0 cos N + 2h1 cos(N 1) + + hN ) term ejN represents N -sample delay H ( ) is real |H ( )| = |H ( )| called linear phase lter ( H ( ) is linear except for jumps of ) FIR lter design 64 Lowpass lter specications 1 1/1 2 p s specications: maximum passband ripple (20 log10 1 in dB): 1/1 |H ( )| 1, 0 p minimum stopband attenuation (20 log10 2 in dB): |H ( )| 2, FIR lter design s 65 Linear phase lowpass lter design sample frequency (k = k/K , k = 1, . . . , K ) 1/1 H (k ) 1 design for maximum stopband attenuation: minimize 2 subject to 1/1 H (k ) 1, 0 k p 2 H (k ) 2, s k passband ripple 1 is given an LP in variables h, 2 known (and used) since 1960s can assume wlog H (0) > 0, so ripple spec is can add other constraints, e.g., |hi| FIR lter design 66 example linear phase lter, n = 31 passband [0, 0.12 ]; stopband [0.24, ] max ripple 1 = 1.059 (0.5dB) design for maximum stopband attenuation impulse response h and frequency response magnitude |H ( )| 0.2 0.15 10 1 10 0 |H ( )| 0.1 h(t) 10 1 0.05 0 0.05 0.1 10 2 10 3 0 5 10 15 20 25 30 10 4 0 0.5 1 1.5 2 2.5 3 t FIR lter design 67 Some variations H ( ) = 2h0 cos N + 2h1 cos(N 1) + + hN minimize passband ripple (given 2, s, p, N ) minimize 1 subject to 1/1 H (k ) 1, 0 k p 2 H (k ) 2, s k minimize transition bandwidth (given 1, 2, p, N ) minimize s subject to 1/1 H (k ) 1, 0 k p 2 H (k ) 2, s k FIR lter design 68 minimize lter order (given 1, 2, s, p) minimize N subject to 1/1 H (k ) 1, 0 k p 2 H (k ) 2, s k can be solved using bisection each iteration is an LP feasibility problem FIR lter design 69 Filter magnitude specications transfer function magnitude spec has form L( ) |H ( )| U ( ), where L, U : R R+ are given and n1 n1 [0, ] H ( ) = t=0 ht cos t j ht sin t t=0 arises in many applications, e.g., audio, spectrum shaping not equivalent to a set of linear inequalities in h (lower bound is not even convex) can change variables and convert to set of linear inequalities FIR lter design 610 Autocorrelation coecients autocorrelation coecients associated impulse with response h = (h0, . . . , hn1) Rn are n1t rt = =0 h h +t (with hk = 0 for k < 0 or k n) rt = rt and rt = 0 for |t| n; hence suces to specify r = (r0, . . . , rn1) Fourier transform of autocorrelation coecients is n1 R( ) = e j r = r0 + t=1 2rt cos t = |H ( )|2 can express magnitude specication as L( )2 R( ) U ( )2, . . . linear inequalities in r FIR lter design 611 [0, ] Spectral factorization question: when is r Rn the autocorrelation coecients of some h Rn? answer (spectral factorization theorem): if and only if R( ) 0 for all spectral factorization condition is convex in r (a linear inequality for each ) many algorithms for spectral factorization, i.e., nding an h such that R( ) = |H ( )|2 magnitude design via autocorrelation coecients: use r as variable (instead of h) add spectral factorization condition R( ) 0 for all optimize over r use spectral factorization to recover h FIR lter design 612 Magnitude lowpass lter design maximum stopband attenuation design with variables r becomes minimize 2 subject to 1/1 R( ) 1, [0, p] R( ) 2, [s, ] R( ) 0, [0, ] 2 (i corresponds to i in original problem) now discretize frequency: minimize 2 subject to 1/1 R(k ) 1, 0 k p R(k ) 2, s k R(k ) 0, 0 k . . . an LP in r, 2 FIR lter design 613 Equalizer design g (t) h(t) (time-domain) equalization: given g (unequalized impulse response) gdes (desired impulse response) design (FIR equalizer) h so that g = h g gdes common choice: pure delay D: gdes(t) = as an LP: 1 t=D 0 t=D minimize maxt=D |g (t)| subject to g (D) = 1 614 FIR lter design example unequalized system G is 10th order FIR: 1 0.8 0.6 g (t) 0.4 0.2 0 0.2 0.4 0 1 2 3 4 5 6 7 8 9 t 10 1 3 2 | G( ) | G( ) 1 0 1 2 10 0 10 1 0 0.5 1 1.5 2 2.5 3 3 0 0.5 1 1.5 2 2.5 3 FIR lter design 615 design 30th order FIR equalizer with G( ) ej 10 minimize maxt=10 |g (t)| equalized system impulse response g 1 0.8 g (t) 0.6 0.4 0.2 0 0.2 0 5 10 15 20 25 30 35 t equalized frequency response magnitude |G| and phase G 10 1 3 2 e | G( ) | e G( ) 1 0 1 2 10 0 10 1 0 0.5 1 1.5 2 2.5 3 3 0 0.5 1 1.5 2 2.5 3 FIR lter design 616 Magnitude equalizer design H ( ) G( ) given system frequency response G : [0, ] C design FIR equalizer H so that |G( )H ( )| 1: minimize max[0,] |G( )H ( )|2 1 use autocorrelation coecients as variables: minimize subject to |G( )|2R( ) 1 , R( ) 0, [0, ] when discretized, an LP in r, , . . . FIR lter design 617 [0, ] Multi-system magnitude equalization given M frequency responses Gk : [0, ] C design FIR equalizer H so that |Gk ( )H ( )| constant: minimize maxk=1,...,M max[0,] |Gk ( )H ( )|2 k subject to k 1, k = 1, . . . , M use autocorrelation coecients as variables: minimize subject to |Gk ( )|2R( ) k , R( ) 0, [0, ] k 1, k = 1, . . . , M . . . when discretized, an LP in k , r, FIR lter design 618 [0, ], k = 1, . . . , M example. M = 2, n = 25, k 1 unequalized and equalized frequency responses 2.5 2.5 2 2 1.5 Gk ( ) H ( ) | 2 0.5 1 1.5 2 2.5 3 Gk ( ) | 2 1.5 1 1 0.5 0.5 0 0 0 0 0.5 1 1.5 2 2.5 3 FIR lter design 619 FIR lter design 620
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