1. FIR Filter Design

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Clarissa Hensley
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1 ELEN E4810: Digital Signal Processing Topic 9: Filter Design: FIR 1. Windowed Impulse Response 2. Window Shapes 3. Design by Iterative Optimization 1 1. FIR Filter Design! FIR filters! no poles (just zeros)! no precedent in analog filter design! Approaches! windowing ideal impulse response! iterative (computer-aided) design 2

2 Least Integral-Squared Error! Given desired FR H d (e j! ), what is the best finite h t [n] to approximate it? best in what sense?! Can try to minimize Integral Squared Error (ISE) of frequency responses: & # " = 2# 1 H d e j$ %# ( ) % H t ( e j$ ) 2 d$ = DTFT{h t [n]} 3 Least Integral-Squared Error! Ideal IR is h d [n] = IDTFT{H d (e j! )}, (usually infinite-extent)! By Parseval, ISE " = h d n n=#$ [ ] # h t [ n] 2! But: h t [n] only exists for n = -M..M, M 4 $ % " # = % h d [ n] $ h t [ n] 2 + % h d n n=$m minimized by making h t [n] = h d [n], -M! n! M n<m, n>m [ ] 2 not altered by h t [n]

3 Least Integral-Squared Error! Thus, minimum mean-squared error approximation in 2M+1 point FIR is truncated IDFT: ' h t [ n] = 1 " 2" % H d ( e j# )e j#n d# $M & n & M ( $" ) 0 otherwise! Make causal by delaying by M points " h' t [n] = 0 for n < 0 5 Approximating Ideal Filters! From topic 6, ideal lowpass has: $ H LP e j" % & and: ( ) = 1 " < " c h LP [ n] = sin" c n #n (doubly infinite) #$ #! c! c $ 0 " c < " < #! 6

otherwise Causal shift 2M+1 points 7 wouldn t work if phases were nonzero! Freq. Resp. (FR) Arithmetic!

4 Approximating Ideal Filters! Thus, minimum ISE causal approximation to an ideal lowpass & sin " c n # M ( ( ) h ˆ 0 % n % 2M LP [ n] = ' $ ( n # M ) )( 0 otherwise Causal shift 2M+1 points 7 wouldn t work if phases were nonzero! Freq. Resp. (FR) Arithmetic! Ideal LPF has pure-real FR i.e. %(!) = 0, H(e j! ) = H(e j! ) " Can build piecewise-constant FRs by combining ideal responses, e.g. HPF: #$ #! c! c $ 8!!! &[n] h LP [n] = h HP [n] i.e. H(e j! ) = 1 H LP (e j! ) = 1 for! <! c = &[n] - (sin! c n)/$n

overshoot) " not optimal by minimax criterion 9 h t n Where Gibbs comes from!

5 Gibbs Phenomenon! Truncated ideal filters have Gibbs Ears: Increasing filter length " narrower ears (reduces ISE) but height the same (11% overshoot) " not optimal by minimax criterion 9 h t n Where Gibbs comes from! Truncation of h d [n] to 2M+1 points is multiplication by a rectangular window: [ ] = h d [ n] " w R [ n] $ [ ] = % 1 "M # n # M w R n & 0 otherwise w R [n]! Multiplication in time domain is convolution in frequency domain: 1 # g[n]" h[n] ' G(e j$ )H(e j(%&$ ) )d$ 2# &# 10

Rectangular window: $ [ ] = % 1 "M # n # M ( doesn t vary with length & 0 otherwise!

6 Where Gibbs comes from! Thus, FR of truncated response is convolution of ideal FR and FR of rectangual window (pdc.sinc): H d (e j! ) DTFT{w R [n]} periodic sinc... H t (e j! ) 11 w R n Where Gibbs comes from! Rectangular window: $ [ ] = % 1 "M # n # M ( doesn t vary with length & 0 otherwise! Mainlobe width (' 1/L) determines transition band! Sidelobe height determines ripples ( ) = $ e # j"n W R e j" n=#m 12 M ([ ] 2 " ) sin 2M +1 = sin 2 " periodic sinc

7 2. Window Shapes for Filters! Windowing (infinite) ideal response " FIR filter: h t n [ ] = h d [ n] " w[ n]! Rectangular window has best ISE error! Other tapered windows vary in:! mainlobe " transition band width! sidelobes " size of ripples near transition! Variety of classic windows Window Shapes for FIR Filters! Rectangular: [ ] =1 "M # n # M w n! Hann: cos(2" n 2M +1 )! Hamming: cos(2" n 2M +1 )! Blackman: cos(2" n ) 2M +1 ) 2M cos(2" 2n 14 big sidelobes! double width mainlobe reduced 1st sidelobe triple width mainlobe

8 Window Shapes for FIR Filters! Comparison on db scale: 2$ 2M+1 15 Adjustable Windows! Have discrete main-sidelobe tradeoffs...! Kaiser window = parametric, continuous tradeoff: w[ n] = I 0 (" 1 # ( M n ) )2 #M $ n $ M I 0 (")! Empirically, for min. SB atten. of ) db: & required 0.11(# $ 8.7) 50 < # " = 0.58(# $ 21) 0.4 order ( ' 21 % # % 50 N = " # 8 ( +0.08(# $ 21) 2.3$% ) 0 # < 21 modified zero-order Bessel function transition width 16

) 0.15 $ 600 Hz $ 4 khz! 2$ 8 khz 17 Windowed Filter Example 1. Get ideal filter impulse response: " c = 0.15# " h d n 2.

9 Windowed Filter Example! Design a 25 point FIR low-pass filter with a cutoff of 600 Hz (SR = 8 khz)! No specific transition/ripple req s " compromise: use Hamming window! Convert the frequency to radians/sample: " c = # 2$ = 0.15$ H(e j! ) 0.15 $ 600 Hz $ 4 khz! 2$ 8 khz 17 Windowed Filter Example 1. Get ideal filter impulse response: " c = 0.15# " h d n 2. Get window: N = 25 " M = 12 (N = 2M+1) " w n 3. Apply window: [ ] = sin0.15#n [ ] = cos 2# n 25 h[ n] = h d [ n] " w[ n] = sin0.15#n 18 #n cos #n 2#n 25 ( ) $12 % n %12 ( ) $12 % n %12 M

3. Iterative FIR Filter Design! Can derive filter coefficients by iterative optimization: Filter coefs h[n] Goodness of fit criterion " error * desired response H(e j!

10 3. Iterative FIR Filter Design! Can derive filter coefficients by iterative optimization: Filter coefs h[n] Goodness of fit criterion " error * desired response H(e j! ) Update filter to reduce * Estimate derivatives +*/+h[n]! Gradient descent / nonlinear optimiz n 19 Error Criteria error measurement region ' #&R " = W # error weighting [ ( ) % H( e j# )] ( ) $ D e j# desired response actual response p d# exponent: 2 " least sq s, " minimax = W(!) [D(e j! ) H(e j! )] 20

11 Minimax FIR Filters! Iterative design of FIR filters with:! equiripple (minimax criterion)! linear-phase " symmetric IR h[n] = ( )h[-n]! Recall, symmetric FIR filters have FR H e j" H " with pure-real ( ) = e # j"m H " # M ( ) = a k k=0 ( ) [ ]cos k" ( ) a[0] = h[m] a[k] = 2h[M - k] i.e. combo of cosines of multiples of! M (type I) n 21 Minimax FIR Filters! Now, cos(k!) can be expressed as a polynomial in cos(!) k and lower powers! e.g. cos(2!) = 2(cos!) 2-1! Thus, we can find ) s such that M H " $ ( ) = #[ k] ( cos" ) k k=0! M th order polynomial in cos!! )[k]s are simply related to a[k]s M th order polynomial in cos! 22

12 5th order polynomial in cos! Minimax FIR Filters M H (") = $ #[ k] ( cos" ) k k=0 M th order polynomial in cos!! An M th order polynomial has at most M - 1 maxima and minima: H (") " H (#) has at most M-1 min/max (ripples) 23 Alternation Theorum! Key ingredient to Parks-McClellan: H (") is the unique, best, weightedminimax order 2M approx. to D(e j! ) -! H (") has at least M+2 extremal freqs " 0 < " 1 <...< " M < " M +1 over! subset R! error magnitude is equal at each extremal: ( ) = " $i " # i! peak error alternates in sign: ( ) = $" (# i+1 ) " # i 24

Alternation Theorum! Hence, for a frequency response: (10th order filter, M = 5)! If *(!) reaches a peak error magnitude * at some set of extremal frequences! i!

13 Alternation Theorum! Hence, for a frequency response: (10th order filter, M = 5)! If *(!) reaches a peak error magnitude * at some set of extremal frequences! i! And the sign of the peak error alternates! And we have at least M+2 of them! Then optimal minimax 25 Alternation Theorum! By Alternation Theorum, M+2 extrema of alternating signs ( optimal minimax filter! But H (") has at most M-1 extrema ( need at least 3 more from band edges! 2 bands give 4 band edges ( can afford to miss only one! Alternation rules out transition band edges, thus have 1 or 2 outer edges 26

14 Alternation Theorum! For M = 5 (10 th order):! 8 extrema (M+3, 4 band edges) - great!! 7 extrema (M+2, 3 band edges) - OK!! 6 extrema (M+1, only 2 transition band edges) " NOT OPTIMAL 27 Parks-McClellan Algorithm! To recap:! FIR CAD constraints D(e j! ), W(!) " *(!)! Zero-phase FIR ~ H(!) =. k ) k cos k! " M-1 min/max! Alternation theorum optimal " "M+2 pk errs, alter ng sign! Hence, can spot best filter when we see it but how to find it? 28

Parks-McClellan Algorithm ~ ~! Alternation " [H(!)-D(!)]/W(!) must = ±* at M+2 (unknown) frequencies {! i }...! Iteratively update h[n] with Remez exchange algorithm:! estimate/guess M+2 extremals {!

15 Parks-McClellan Algorithm ~ ~! Alternation " [H(!)-D(!)]/W(!) must = ±* at M+2 (unknown) frequencies {! i }...! Iteratively update h[n] with Remez exchange algorithm:! estimate/guess M+2 extremals {! i }! solve for )[n], * ( " h[n] )! find actual min/max in *(!) " new {! i }! repeat until *(! i ) is constant! Converges rapidly! 29 Parks-McClellan Algorithm! In Matlab, filter order (2M) band edges $ >> h=remez(10, [ ], desired magnitude [ ], at band edges [1 2]); error weights per band 30

Filter Design Problem

Filter Design Problem Design of frequency-selective filters usually starts with a specification of their frequency response function. Practical filters have passband and stopband ripples, while exhibiting