Filter design FIR filters Chebychev design linear phase filter - PowerPoint PPT Presentation

Filter design • FIR filters • Chebychev design • linear phase filter design • equalizer design • filter magnitude specifications 1

FIR filters finite impulse response (FIR) filter: n − 1 � y ( t ) = h τ u ( t − τ ) , t ∈ Z τ =0 • (sequence) u : Z → R is input signal • (sequence) y : Z → R is output signal • h i are called filter coefficients • n is filter order or length Filter design 2

filter frequency response: H : R → C h 0 + h 1 e − iω + · · · + h n − 1 e − i ( n − 1) ω H ( ω ) = n − 1 n − 1 � � = h t cos tω + i h t sin tω t =0 t =0 • (EE tradition uses j = √− 1 instead of i ) • H is periodic and conjugate symmetric, so only need to know/specify for 0 ≤ ω ≤ π FIR filter design problem: choose h so it and H satisfy/optimize specs Filter design 3

example: (lowpass) FIR filter, order n = 21 impulse response h : 0.2 0.1 h ( t ) 0 −0.1 −0.2 0 2 4 6 8 10 12 14 16 18 20 t Filter design 4

frequency response magnitude ( i.e. , | H ( ω ) | ): 1 10 0 10 | H ( ω ) | −1 10 −2 10 −3 10 0 0.5 1 1.5 2 2.5 3 ω frequency response phase ( i.e. , � H ( ω ) ): 3 2 � H ( ω ) 1 0 −1 −2 −3 0 0.5 1 1.5 2 2.5 3 ω Filter design 5

Chebychev design minimize max ω ∈ [0 ,π ] | H ( ω ) − H des ( ω ) | • h is optimization variable • H des : R → C is (given) desired transfer function • convex problem • can add constraints, e.g. , | h i | ≤ 1 sample (discretize) frequency: minimize k =1 ,...,m | H ( ω k ) − H des ( ω k ) | max • sample points 0 ≤ ω 1 < · · · < ω m ≤ π are fixed ( e.g. , ω k = kπ/m ) • m ≫ n (common rule-of-thumb: m = 15 n ) • yields approximation (relaxation) of problem above Filter design 6

Chebychev design via SOCP: minimize t � � A ( k ) h − b ( k ) � � ≤ t, subject to k = 1 , . . . , m where � 1 � cos ω k · · · cos( n − 1) ω k A ( k ) = 0 − sin ω k · · · − sin( n − 1) ω k � � ℜ H des ( ω k ) b ( k ) = ℑ H des ( ω k )   h 0 . .   h = . h n − 1 Filter design 7

Linear phase filters suppose • n = 2 N + 1 is odd • impulse response is symmetric about midpoint: h t = h n − 1 − t , t = 0 , . . . , n − 1 then H ( ω ) = h 0 + h 1 e − iω + · · · + h n 1 e − i ( n − 1) ω − e − iNω (2 h 0 cos Nω + 2 h 1 cos( N − 1) ω + · · · + h N ) = e − iNω � ∆ = H ( ω ) Filter design 8

• term e − iNω represents N -sample delay • � H ( ω ) is real • | H ( ω ) | = | � H ( ω ) | • called linear phase filter ( � H ( ω ) is linear except for jumps of ± π ) Filter design 9

Lowpass filter specifications δ 1 1 /δ 1 δ 2 ω p ω s π ω idea: • pass frequencies in passband [0 , ω p ] • block frequencies in stopband [ ω s , π ] Filter design 10

specifications: • maximum passband ripple ( ± 20 log 10 δ 1 in dB): 1 /δ 1 ≤ | H ( ω ) | ≤ δ 1 , 0 ≤ ω ≤ ω p • minimum stopband attenuation ( − 20 log 10 δ 2 in dB): | H ( ω ) | ≤ δ 2 , ω s ≤ ω ≤ π Filter design 11

Linear phase lowpass filter design • sample frequency • can assume wlog � H (0) > 0 , so ripple spec is 1 /δ 1 ≤ � H ( ω k ) ≤ δ 1 design for maximum stopband attenuation: minimize δ 2 1 /δ 1 ≤ � subject to H ( ω k ) ≤ δ 1 , 0 ≤ ω k ≤ ω p − δ 2 ≤ � H ( ω k ) ≤ δ 2 , ω s ≤ ω k ≤ π Filter design 12

• passband ripple δ 1 is given • an LP in variables h , δ 2 • known (and used) since 1960’s • can add other constraints, e.g. , | h i | ≤ α variations and extensions: • fix δ 2 , minimize δ 1 (convex, but not LP) • fix δ 1 and δ 2 , minimize ω s (quasiconvex) • fix δ 1 and δ 2 , minimize order n (quasiconvex) Filter design 13

example • linear phase filter, n = 21 • passband [0 , 0 . 12 π ] ; stopband [0 . 24 π, π ] • max ripple δ 1 = 1 . 012 ( ± 0 . 1 dB) • design for maximum stopband attenuation impulse response h : 0.2 0.1 h ( t ) 0 −0.1 −0.2 0 2 4 6 8 10 12 14 16 18 20 t Filter design 14

frequency response magnitude ( i.e. , | H ( ω ) | ): 1 10 0 10 | H ( ω ) | −1 10 −2 10 −3 10 0 0.5 1 1.5 2 2.5 3 ω frequency response phase ( i.e. , � H ( ω ) ): 3 2 1 � H ( ω ) 0 −1 −2 −3 0 0.5 1 1.5 2 2.5 3 ω Filter design 15

Equalizer design H ( ω ) G ( ω ) equalization: given • G (unequalized frequency response) • G des (desired frequency response) ∆ design (FIR equalizer) H so that � G = GH ≈ G des • common choice: G des ( ω ) = e − iDω (delay) i.e. , equalization is deconvolution (up to delay) • can add constraints on H , e.g. , limits on | h i | or max ω | H ( ω ) | Filter design 16

Chebychev equalizer design: � � � � � � minimize max G ( ω ) − G des ( ω ) � ω ∈ [0 ,π ] convex; SOCP after sampling frequency Filter design 17

time-domain equalization: optimize impulse response ˜ g of equalized system e.g. , with G des ( ω ) = e − iDω , � 1 t = D g des ( t ) = 0 t � = D sample design: minimize max t � = D | ˜ g ( t ) | subject to ˜ g ( D ) = 1 • an LP • can use � g ( t ) 2 or � t � = D ˜ t � = D | ˜ g ( t ) | Filter design 18

extensions: • can impose (convex) constraints • can mix time- and frequency-domain specifications • can equalize multiple systems, i.e. , choose H so G ( k ) H ≈ G des , k = 1 , . . . , K • can equalize multi-input multi-output systems ( i.e. , G and H are matrices) • extends to multidimensional systems, e.g. , image processing Filter design 19

Equalizer design example unequalized system G is 10 th 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 Filter design 20

1 10 | G ( ω ) | 0 10 −1 10 0 0.5 1 1.5 2 2.5 3 ω 3 2 � G ( ω ) 1 0 −1 −2 −3 0 0.5 1 1.5 2 2.5 3 ω design 30 th order FIR equalizer with � G ( ω ) ≈ e − i 10 ω Filter design 21

Chebychev equalizer design: � G ( ω ) − e − i 10 ω � � � � ˜ minimize max � ω 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 Filter design 22

� equalized frequency response magnitude | � G | 1 10 G ( ω ) | 0 10 | e −1 10 0 0.5 1 1.5 2 2.5 3 ω � equalized frequency response phase � G 3 2 G ( ω ) 1 0 e −1 −2 −3 0 0.5 1 1.5 2 2.5 3 ω Filter design 23

time-domain equalizer design: minimize max t � =10 | ˜ g ( t ) | equalized system impulse response ˜ g 1 0.8 0.6 g ( t ) 0.4 ˜ 0.2 0 −0.2 0 5 10 15 20 25 30 35 t Filter design 24

� equalized frequency response magnitude | � G | 1 10 0 10 G ( ω ) | | e −1 10 0 0.5 1 1.5 2 2.5 3 ω � equalized frequency response phase � G 3 2 1 0 G ( ω ) −1 e −2 −3 0 0.5 1 1.5 2 2.5 3 ω Filter design 25

Filter magnitude specifications transfer function magnitude spec has form L ( ω ) ≤ | H ( ω ) | ≤ U ( ω ) , ω ∈ [0 , π ] where L, U : R → R + are given • lower bound is not convex in filter coefficients h • arises in many applications, e.g. , audio, spectrum shaping • can change variables to solve via convex optimization Filter design 26

Autocorrelation coefficients autocorrelation coefficients associated with impulse response 1 ) ∈ R n are h = ( h 0 , . . . , h n − � r t = h τ h τ + t τ (we take h k = 0 for k < 0 or k ≥ n ) • r t = r − t ; r t = 0 for | t | ≥ n 1 ) ∈ R n • hence suffices to specify r = ( r 0 , . . . , r n − Filter design 27

Fourier transform of autocorrelation coefficients is n − 1 � � e − iωτ r τ = r 0 + 2 r t cos ωt = | H ( ω ) | 2 R ( ω ) = τ t =1 • always have R ( ω ) ≥ 0 for all ω • can express magnitude specification as L ( ω ) 2 ≤ R ( ω ) ≤ U ( ω ) 2 , ω ∈ [0 , π ] . . . convex in r Filter design 28

Spectral factorization question: when is r ∈ R n the autocorrelation coefficients of some h ∈ R n ? answer: ( spectral factorization theorem ) if and only if R ( ω ) ≥ 0 for all ω • spectral factorization condition is convex in r • many algorithms for spectral factorization, i.e. , finding an h s.t. R ( ω ) = | H ( ω ) | 2 magnitude design via autocorrelation coefficients: • use r as variable (instead of h ) • add spectral factorization condition R ( ω ) ≥ 0 for all ω • optimize over r • use spectral factorization to recover h Filter design 29

log-Chebychev magnitude design choose h to minimize max | 20 log 10 | H ( ω ) | − 20 log 10 D ( ω ) | ω • D is desired transfer function magnitude ( D ( ω ) > 0 for all ω ) • find minimax logarithmic (dB) fit reformulate as minimize t D ( ω ) 2 /t ≤ R ( ω ) ≤ tD ( ω ) 2 , subject to 0 ≤ ω ≤ π • convex in variables r , t • constraint includes spectral factorization condition Filter design 30

example: 1 /f (pink noise) filter ( i.e. , D ( ω ) = 1 / √ ω ), n = 50 , log-Chebychev design over 0 . 01 π ≤ ω ≤ π 1 10 0 10 | H ( ω ) | 2 −1 10 −2 10 −1 0 10 10 ω optimal fit: ± 0 . 5 dB Filter design 31