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ELEN E4810: Digital Signal Processing Topic 8: Filter Design: IIR

ELEN E4810: Digital Signal Processing Topic 8: Filter Design: IIR. Filter Design Specifications Analog Filter Design Digital Filters from Analog Prototypes. 1. Filter Design Specifications. The filter design process:. performance constraints. Analysis. Design. Implement. G ( z )

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ELEN E4810: Digital Signal Processing Topic 8: Filter Design: IIR

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  1. ELEN E4810: Digital Signal ProcessingTopic 8: Filter Design: IIR • Filter Design Specifications • Analog Filter Design • Digital Filters from Analog Prototypes Dan Ellis

  2. 1. Filter Design Specifications • The filter design process: performance constraints Analysis Design Implement G(z) transferfunction Problem Solution • magnitude response • phase response • cost/complexity • FIR/IIR • subtype • order • platform • structure • ... Dan Ellis

  3. Performance Constraints • .. in terms of magnitude response: Dan Ellis

  4. Performance Constraints • “Best” filter: • improving one usually worsens others • But: increasing filter order (i.e. cost)improves all three measures smallestPassband Ripple narrowest Transition Band greatest Minimum SB Attenuation Dan Ellis

  5. Passband Ripple • Assume peak passband gain = 1then minimum passband gain = • Or, ripple PB rippleparameter Dan Ellis

  6. Stopband Ripple • Peak passband gain is A larger than peak stopband gain • Hence, minimum stopband attenuation SB rippleparameter Dan Ellis

  7. Filter Type Choice: FIR vs. IIR IIR • Feedback(poles & zeros) • May be unstable • Difficult to control phase • Typ. < 1/10th order of FIR (4-20) • Derive from analog prototype FIR • No feedback(just zeros) • Always stable • Can be linear phase • High order (20-2000) • Unrelated to continuous-time filtering BUT Dan Ellis

  8. FIR vs. IIR • If you care about computational cost use low-complexity IIR (computation no object  Lin Phs FIR) • If you care about phase response use linear-phase FIR (phase unimportant  go with simple IIR) Dan Ellis

  9. IIR Filter Design • IIR filters are directly related to analog filters (continuous time) • via a mapping of H(s) (CT) to H(z) (DT) that preserves many properties • Analog filter design is sophisticated • signal processing research since 1940s  Design IIR filters via analog prototype • hence, need to learn some CT filter design Dan Ellis

  10. 2. Analog Filter Design • Decades of analysis of transistor-based filters – sophisticated, well understood • Basic choices: • ripples vs. flatness in stop and/or passband • more ripples narrower transition band Dan Ellis

  11. Im{z} Im{s} ejw jW Re{z} Re{s} 1 stablepoles stablepoles s-plane z-plane CT Transfer Functions • Analog systems: s-transform (Laplace) Continuous-time Discrete-time Transform Frequencyresponse Pole/zerodiagram Dan Ellis

  12. filterorder N Butterworth Filters Maximally flat in pass and stop bands • Magnituderesponse (LP): • W<<Wc, |Ha(jW)|2 1 • W = Wc, |Ha(jW)|2 = 1/2 3dB point Dan Ellis

  13. Butterworth Filters 6N dB/octrolloff • W>>Wc, |Ha(jW)|2 (Wc/W)2N • flat @ W = 0 for n = 1 .. 2N-1 Log-logmagnituderesponse Dan Ellis

  14. Butterworth Filters • How to meet design specifications? •  DesignEquation =“selectivity”, < 1 =“discrimination”, <<1 Dan Ellis

  15. Butterworth Filters • but what is Ha(s)? • Traditionally, look it up in a table • calculate N normalized filter with Wc = 1 • scale all coefficients for desired Wc • In fact,where Im{s}  Wc  Re{s}   s-plane Dan Ellis

  16. Butterworth Example Design a Butterworth filter with 1 dB cutoff at 1kHz and a minimum attenuation of 40 dB at 5 kHz Dan Ellis

  17. Butterworth Example • Order N = 4 will satisfy constraints;What are Wc and filter coefficients? • from a table, W-1dB = 0.845 whenWc = 1Wc = 1000/0.845 = 1.184 kHz • from a table, get normalized coefficients for N = 4, scale by 1184 • Or, use Matlab: [b,a] = butter(N,Wc,’s’); M Dan Ellis

  18. Chebyshev I Filter • Equiripple in passband (flat in stopband) minimize maximum error Chebyshev polynomial of order N Dan Ellis

  19. Chebyshev I Filter • Design procedure: • desired passband ripple e • min. stopband atten., Wp, WsN : 1/k1, discrimination 1/k, selectivity Dan Ellis

  20. Chebyshev I Filter • What is Ha(s)? • complicated, get from a table • .. or from Matlab cheby1(N,r,Wp,’s’) • all-pole; can inspect them: ..like squashed-in Butterworth Dan Ellis

  21. Chebyshev II Filter • Flat in passband, equiripple in stopband • Filter has poles and zeros (some ) • Complicated pole/zero pattern constant ~1/TN(1/W) zeros on imaginary axis Dan Ellis

  22. Elliptical (Cauer) Filters • Ripples in both passband and stopband • Complicated; not even closed form for N function; satisfiesRN(W-1) = RN(W)-1zeros for W>1 poles for W<1 very narrow transition band Dan Ellis

  23. Analog Filter Types Summary N = 6 r = 3 dB A = 40 dB Dan Ellis

  24. Analog Filter Transformations • All filters types shown as lowpass;other types (highpass, bandpass..)derived via transformations • i.e. • General mapping of s-planeBUT keep jW jW; frequency response just ‘shuffled’ Desired alternateresponse; still arational polynomial lowpass prototype ^ Dan Ellis

  25. Lowpass-to-Highpass • Example transformation: • take prototype HLP(s) polynomial • replace s with • simplify and rearrange  new polynomial HHP(s) ^ Dan Ellis

  26. Lowpass-to-Highpass • What happens to frequency response? • Frequency axes inverted imaginary axisstays on self... ...freq.freq. LP passband HP passband LP stopband HP stopband Dan Ellis

  27. Transformation Example Design a Butterworth highpass filter with PB edge -0.1dB @ 4 kHz (Wp)and SB edge -40 dB @ 1 kHz (Ws) • Lowpass prototype: make Wp = 1 • Butterworth -0.1dB @ Wp=1, -40dB @ Ws=4 ^ ^ Dan Ellis

  28. Im{s}  Wc  Re{s}   Transformation Example • LPF proto has • Map to HPF: N zeros @ s = 0 ^ ^ ^ new poles @ s = WpWp/pl Dan Ellis

  29. Transformation Example • In Matlab: [N,Wc]=buttord(1,4,0.1,40,'s'); [B,A] = butter(N, Wc, 's'); [n,d] = lp2hp(B,A,2*pi*4000); WpWs Rp Rs Dan Ellis

  30. 3. Analog Protos  IIR Filters • Can we map high-performance CT filters to DT domain? • Approach: transformationHa(s)G(z) i.e. where s = F(z) maps s-plane  z-plane: Im{s} Im{z} Ha(s0) s = F(z) G(z0) Every value of G(z)is a value of Ha(s) somewhere on the s-plane & vice-versa Re{z} Re{s} 1 z-plane s-plane Dan Ellis

  31. CT to DT Transformation • Desired properties for s = F(z): • s-plane jW axis  z-plane unit circle  preserves frequency response values • s-plane LHHP  z-plane unit circle interior preserves stability of poles Im{s} Im{z} Imu.c. ejw jW Re{z} Re{s} 1 s-plane z-plane LHHPUCI Dan Ellis

  32. Bilinear Transformation • Solution: • Hence inverse: • Freq. axis? • Poles? Bilinear Transform unique, 1:1 mapping |z| = 1 i.e.on unit circle • < 0 |z| < 1 Dan Ellis

  33. Bilinear Transformation • How can entire half-plane fit inside u.c.? • Highly nonuniform warping! s-plane z-plane Dan Ellis

  34. Bilinear Transformation • What is CTDT freq. relation Ww ? • i.e. • infinite range of CT frequency maps to finite DT freq. range • nonlinear; as w  p im.axis u.circle pack it all in! Dan Ellis

  35. Frequency Warping • Bilinear transform makes for all w, W • Same gain & phase (e, A...), in same ‘order’, but with warped frequency axis Dan Ellis

  36. Design Procedure • Obtain DT filter specs: • general form (LP, HP...), • ‘Warp’ frequencies to CT: • Design analog filter for •  Ha(s), CT filter polynomial • Convert to DT domain: •  G(z), rational polynomial in z • Implement digital filter! Old- style Dan Ellis

  37. Bilinear Transform Example • DT domain requirements:Lowpass, 1 dB ripple in PB, wp = 0.4p,SB attenuation ≥ 40 dB @ ws = 0.5p,attenuation increases with frequency SB ripples,PB monotonic Chebyshev I Dan Ellis

  38. Bilinear Transform Example • Warp to CT domain: • Magnitude specs:1 dB PB ripple 40 dB SB atten. Dan Ellis

  39. Bilinear Transform Example • Chebyshev I design criteria: • Design analog filter, map to DT, check: i.e. need N = 8 >> N=8; >> wp=0.7265; >> [B,A]=cheby1(N,1,wp,'s'); >> [b,a] = bilinear(B,A,.5); M Dan Ellis

  40. Other Filter Shapes • Example was IIR LPF from LP prototype • For other shapes (HPF, bandpass,...): • Transform LPX in CT or DT domain... CTtrans Bilinear transform HD(s) Bilinear warp Analog design DTspecs CTspecs HLP(s) GD(z) Bilinear transform DTtrans GLP(z) Dan Ellis

  41. DT Spectral Transformations • Same idea as CT LPFHPF mapping, but in z-domain: • To behave well, should: • map u.c.  u.c. (preserve G(ejw) values) • map u.c. interior  u.c. interior (stability) • i.e. • in fact, matches the definition of an allpass filter ... replace delays with Dan Ellis

  42. DT Frequency Warping • Simplest mappinghas effect of warping frequency axis: a > 0 :expand HF a < 0 :expand LF Dan Ellis

  43. Another Design Example • Spec: • Bandpass, from 800-1600 Hz (SR = 8kHz) • Ripple = 1dB, min. stopband atten. = 60 dB • 8th order, best transition band • Use elliptical for best performance • Full design path: • design analog LPF prototype • analog LPF  BPF • CT BPF  DT BPF (Bilinear) Dan Ellis

  44. Another Design Example • Or, do it all in one step in Matlab: [b,a] = ellip(8,1,60, [800 1600]/(8000/2)); Dan Ellis

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