ELEN E4810: Digital Signal Processing Topic 9: Filter Design: FIR. 1. Windowed Impulse Response 2. Window Shapes 3. Design by Iterative Optimization
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(2) 1. FIR Filter Design . FIR filters . . no poles (just zeros) no precedent in analog filter design. Approaches . Dan Ellis. windowing ideal impulse response iterative (computer-aided) design. 2012-11-19. 2.
(3) Least Integral-Squared Error . Given desired FR Hd(ej!), what is the best finite ht[n] to approximate it?. best in what sense? . Can try to minimize Integral Squared Error (ISE) of frequency responses:. =. 1 2. H d (e ) H t (e ) . j. j 2. d. = DTFT{ht[n]} Dan Ellis. 2012-11-19. 3.
(4) Least Integral-Squared Error . . Ideal IR is hd[n] = IDTFT{Hd(ej!)}, (usually infinite-extent) . hd [n] ht [n]. By Parseval, ISE =. 2. n= . But: ht[n] only exists for n = -M..M , =. M. hd [ n] ht [ n] n =M. minimized by making ht[n] = hd[n], -M ≤ n ≤ M Dan Ellis. 2012-11-19. 2. +. hd [n] n< M , n> M. 2. not altered by ht[n] 4.
(5) Least Integral-Squared Error . Thus, minimum mean-squared error approximation in 2M+1 point FIR is truncated IDTFT:. ht [n] = . 1 2. Hd (ej )ej. n. d. M n otherwise. 0. Make causal by delaying by M points → h′t[n] = 0 for n < 0 zero. ht[n]. h't[n]. n. n –M. Dan Ellis. M. 0. 2012-11-19. M. 2M. 5. M.
(6) Approximating Ideal Filters . From topic 6, ideal lowpass has:. 1 0. HLP (e ) = j. and: hLP [n] =. sin. −º. n. !c. º. | |< c c <| |<. h[n]. cn. −!c. !. 0.2. 0.15 0.1 0.05 0. (doubly infinite) Dan Ellis. -0.05 -20. -15. 2012-11-19. -10. -5. 0. 5. 10. 6. 15. n.
(7) Approximating Ideal Filters Thus, minimum ISE causal approximation to an ideal lowpass. ˆ LP [n] = h Causal shift. sin. c (n. M) (n M ). 0. 0 hLP[n]. . n. 2M. otherwise. 1.2 1 0.8 0.6 0.4 0.2 0 -0.2. 0. 5. 10. 15. 20. 25. 2M+1 points Dan Ellis. 2012-11-19. 7. n.
(8) Gibbs Phenomenon Truncated ideal filters have Gibbs’ Ears:. Increasing filter length → narrower ears (reduces ISE) but height the same. Ht(ejW). . 1.2. Gibbs ‘ears’. 1. (~9% overshoot). 0.8 0.6 0.4. → not optimal by minimax criterion. 0 -0.2. Dan Ellis. 129 point. 0.2. 0. 0.2. 2012-11-19. 25 point. 0.4. 0.6. 0.8. 8. WP.
(9) Where Gibbs comes from . Truncation of hd[n] to 2M+1 points is multiplication by a rectangular window:. ht [n] = hd [n] · wR [n] wR [n] = . 1 0. M n otherwise. h[n]. 0.2. 0.15. M. wR[n]. 0.1 0.05 0 -0.05 -20. -15. -10. -5. 0. 5. 10. Multiplication in time domain is convolution in frequency domain: g[n] · h[n]. Dan Ellis. 1 2. G(ej )H(ej( 2012-11-19. ). )d 9. 15. n.
(10) Where Gibbs comes from . Thus, FR of truncated response is convolution of ideal FR and FR of rectangular window (pd.sinc):. P. P. Hd(ej!). Dan Ellis. W. P. P. DTFT{wR[n]} periodic sinc... sin Nx sin x 2012-11-19. W. P. W. P. Ht(ej!). convanim.mp4. 10.
(11) Where Gibbs comes from . Rectangular window: 1 0. wR [n] = . . M n otherwise. WR (e ) =. M. =. Mainlobe width ( / 1/L) determines transition band Sidelobe height determines ripples. WR(ejW) Mainlobe pP 2M + 1. Sidelobes. 0. ≈ invariant with length. Dan Ellis. M j n e n= M sin(2M +1) 2 sin 2. j. -P. -0.5P. 0. 0.5P. “periodic sinc” 2012-11-19. 11. P. W.
(12) 2. Window Shapes for Filters . . Windowing (infinite) ideal response → FIR filter: ht [n] = hd [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.... Dan Ellis. 2012-11-19. 12.
(13) Window Shapes for FIR Filters Rectangular: w[n] = 1 M n M . Hann: 0.5 + 0.5 cos(2 2Mn +1) Hamming: 0.54 + 0.46 cos(2 2Mn +1) Blackman: 0.42 + 0.5 cos(2 2Mn +1) +0.08 cos(2 2M2n+1) . Dan Ellis. 20. 1. big sidelobes!. 15 10. 0.5. 5 0. -10. -5. 0. 5. 10. n. 0. 0. 8. 1. 0.4. 0.6. 0.8. PW. double width mainlobe. 6 4. 0.5. 0.2. 2 0. -10. -5. 0. 5. 10. n. 0. 0. 8. 1. 4 2. 0. -10. -5. 0. 5. 10. n. 0. 0. 8. 1. 0.6. 0.8. PW. 0.2. 0.4. 0.6. 0.8. PW. triple width mainlobe. 6 4. 0.5. 0.4. reduced 1st sidelobe. 6 0.5. 0.2. 2 0. -10. -5. 2012-11-19. 0. 5. 10. n. 0. 0. 0.2. 0.4. 13. 0.6. 0.8. PW.
(14) Window Shapes for FIR Filters Comparison on dB scale: \W(ejW)\ / dB. . 0. Blackman Hann. -10. Rectangular. -20. -30. -40. -50. Hamming -60. -70. -80. Dan Ellis. 2º 2M+1 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 2012-11-19. 0.7. 0.8. 0.9. W. P. 14.
(15) Adjustable Windows So far, discrete main-sidelobe tradeoffs.. Kaiser window = parametric, continuous tradeoff: I0 1 ( Mn )2 M n M modified zero-order w[n] = I0 ( ) Bessel function Empirically, for min. SB atten. of Æ dB: . (. =. 0.11( 0.58( 0 Dan Ellis. 8.7) 21)0.4 + 0.08(. 21). ). > 50 21 < 21. 2012-11-19. required order. 50. N=. 8 2.3. transition width 15.
(16) 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/ 600 2 = 0.15 sample: c = 8000 H(ej!) 0.15 º 600 Hz. Dan Ellis. º 4 kHz 2012-11-19. !. 2º 8 kHz (fs) 16.
(17) Windowed Filter Example Get ideal filter impulse response: n c = 0.15 hd [n] = sin 0.15 n 2. Get window: Hamming @ N = 25 → M = 12 (N = 2M+1) w[n] = 0.54 + 0.46 cos(2 25n ) 12 n 12 3. Apply window: h[n] 1.. 0.15. 0.1. h[n] = hd [n] w[n]. 0.05 0 -20. -10. 0. n. 10. n 0.54 + 0.46 cos 2 n = sin 0.15 12 n 12 ( n 25 ) Dan Ellis. 2012-11-19. 17. M.
(18) Freq. Resp. (FR) Arithmetic. wouldn’t work if phases were nonzero!. . Ideal LPF has pure-real FR i.e. µ(!) = 0, H(ej!) = |H(ej!)|. → Can build piecewise-constant FRs by combining ideal responses, e.g. HPF: 1. !. + 0 -1. ! =. 1. −º Dan Ellis. −!c. !c. º. ! 2012-11-19. ±[n] +. i.e. H(ej!) = 1 HLP(ej!) = 1 for |!| < !c. -hLP[n] = hHP[n] = ±[n] - (sin!cn)/ºn 18.
(19) 3. Iterative FIR Filter Design . . Can derive filter coefficients by iterative optimization: Filter coefs h[n]. Goodness of fit criterion → error ". Update filter to reduce ". Estimate derivatives ∂"/∂h[n]. Desired response H(ej!). Gradient descent / nonlinear optimiz’n. Dan Ellis. 2012-11-19. 19.
(20) Error Criteria p j j = R W ( ) [D(e ) H (e )] d error measurement error actual desired region weighting response response. exponent: 2 → least sq ∞ → minimax. R H(ejW) W1 W(W) E(W) Dan Ellis. W2 W3. D(ejW) W4. W W W = W(!)·[D(ej!) – H(ej!)]. 2012-11-19. 20.
(21) Minimax FIR Filters . Iterative design of FIR filters with: . . equiripple (minimax criterion) linear-phase → symmetric IR h[n] = (–)h[-n]. (type I order 2M). n. M. Recall: symmetric FIR filters have FR j jM ˜ H e =e H ( ) with pure-real M a[0] = h[M] ˜ H ( ) = a[k ] cos(k ). ( ). k=0. a[k] = 2h[M - k]. i.e. combo of cosines of multiples of ! Dan Ellis. 2012-11-19. 21.
(22) Minimax FIR Filters . k=0. Now, cos(k!) can be expressed as a polynomial in cos(!)k and lower powers . . M ˜ H ( ) = a[k ] cos(k ). e.g. cos(2!) = 2(cos!)2 - 1. Thus, we can find Æs such that. ˜ H(⇥) =. M. M. a[k]cos(k⇥) = k=0. . Dan Ellis. [k](cos⇥). k. k=0. Mth order polynomial in cos!. Æ[k]s easily lead to a[k]s 2012-11-19. 22.
(23) Minimax FIR Filters M k M order ˜ H ( ) = [k ](cos ) polynomial in cos! k=0 th. . An Mth order polynomial has at most M - 1 maxima and minima: M. ˜ ) H(. [k](cos )k. cosW. k=0 0.2. 5th order 0.1 polynomial 0 in cos! -0.1. 0.2 0.1. 3. 3. 2. 4. -1. W. 0. cosW. 2. 4. -0.1. -0.2. W P. 1. 1. -0.2 -0.5. 0. 0.5. 1. W . 0. 0.2. 0.4. 0.6. 0.8. P. H˜ ( ) has at most M-1 min/max (ripples) Dan Ellis. 2012-11-19. 23.
(24) Alternation Theorem . Key ingredient to Parks-McClellan: H˜ ( ) is the unique, best, weightedminimax order 2M approx. to D(ej!) ˜ ( ) has at least M+2 “extremal” freqs H 0 < 1 < ... < M < M +1 over ! subset R . error magnitude is equal at each extremal:. . peak error alternates in sign:. ( i ) = i. ( i ) = ( i+1 ). Dan Ellis. 2012-11-19. 24.
(25) Alternation Theorem . Hence, for a frequency response: D(ejW) desired frq.resp.. 1 0.8. . 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. H(ejW) response mag.. 0.6 0.4. E scaled error. W(W). 0.2. bound. W. 0. Wp. 0. Wc. P. band-edge extrema. E. E(W) error 0. . -E. W0. W1 W2. W3W4 W5. W6. local min/max extrema. . (10th order filter, M = 5) Dan Ellis. 2012-11-19. 25.
(26) Alternation Theorem . . By Alternation Theorem, 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. Dan Ellis. 2012-11-19. 26.
(27) Alternation Theorem . For M = 5 (10th order): . . . Dan Ellis. 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 2012-11-19. 1. ~ H1W 0.5. 0 0. 0.2. 0.4. 0.6. 0.8. P. 0.4. 0.6. 0.8. P. 0.4. 0.6. 0.8. P. W. 1. ~ H2W 0.5. 0 0. 0.2. W. 1. ~ H3W 0.5. 0 0. 0.2. 27. W.
(28) Parks-McClellan Algorithm . To recap: . . . . FIR CAD constraints D(ej!), W(!) → "(!) Zero-phase FIR ~ H(!) = ßkÆkcosk! → M-1 min/max Alternation theorem optimal → ≥ M+2 pk errs, alter’ng sign. Hence, can spot ‘best’ filter when we see it – but how to find it?. Dan Ellis. 2012-11-19. 28.
(29) 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 Æ[n], " ( → h[n] ) find actual min/max in "(!) → new {!i} repeat until |"(!i)| is constant. Converges rapidly!. Dan Ellis. 2012-11-19. 29. M.
(30) Parks-McClellan Algorithm . In Matlab, band edges ÷ º. filter order (2M). >> h=firpm(10, [0 0.4 0.6 1],. . [1 1 0 0], desired magnitude. at band. edges [1 2]); error weights . per band. 0.6 0.4. h[n]. ~ H(W). 1. 1 0.5. 0.2. 0. 0.5. -0.5. 0 -0.2. Dan Ellis. -5. 0. 5. n. 0 0. 0.5. 2012-11-19. W 1. z-plane. -1 -1. 0. 1. 2. 30.
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