Nov '04CS3291: Section 61 Baseado no material da: UNIVERSITY of MANCHESTER Department of Computer Science CS3291: Digital Signal Processing Section 6 IIR discrete time filter design

Nov '04CS3291: Section Introduction: Many design techniques for IIR discrete time filters have adopted ideas of analogue filters Transform H a (s) for analogue ‘prototype’ filter into H(z) for discrete time filter. Begin with a reminder about analogue filters Analogue filters

Nov '04CS3291: Section 63 Analogue filters have transfer functions: a 0 + a 1 s + a 2 s a N s N H a (s) =  b 0 + b 1 s + b 2 s b M s M Replace s by j  for frequency-response. For RE(s)  0, H a (s ) is Laplace Transform of h a (t). In terms of poles and zeros: (s - z 1 ) ( z - z 2 )... ( s - z N ) H a (s) = K  (s - p 1 ) ( z - p 2 )... ( s - p M ) Many known techniques for deriving H a (s); e.g. Butterworth low-pass approximation Analogue filters have infinite impulse-responses.

Nov '04CS3291: Section 64 Analogue Butterworth low-pass filter of order n [n/2] is integer part of n/2 & P = 0 /1 if n is even / odd

Nov '04CS3291: Section 65

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Nov '04CS3291: Section 68 Intro to Bilinear Transformation method Most common method for transforming H a (s) to H(z) for IIR discrete time filter. Consider derivative approximation technique: D[n] = dy(t) /dt at t=nT  ( y[n] - y[n-1]) / T. dx(t) /dt at t=nT  (x[n] - x[n-1]) / T. d 2 y(t)/dt 2 at t=nT  (y[n] - 2y[n-1]+y[n-2])/T 2 d 3 y(t)/dt 3 at t=nT  (y[n]-3y[n-1]+3y[n-2]-y[n-3])/T 3 “Backward difference” approximation introduces delay which becomes greater for higher orders. Try "forward differences" : D[n]  [y[n+1] - y[n]] / T, etc. But this does not make matters any better.

Nov '04CS3291: Section 69 Bilinear approximation: 0.5( D[n] + D[n-1])  (y[n] - y[n-1]) / T & similarly for dx(t)/dt at t=nT. Similar formulae may be derived for d 2 y(t)/dt 2, and so on. If D(z) is z-transform of D[n] : 0.5( D(z) + z -1 D(z) ) = ( Y(z) - z -1 Y(z) ) / T

Nov '04CS3291: Section 610  replacing s by [ (2/T) (z-1)/(z+1)] is bilinear approximation. If LT of y(t) is Y(s) LT of dy(t)/dt is sY(s). s y(t) dy(t)/dt (2/T) (z-1)/(z+1) {y[n]}  {dy(t)/dt / t=nT }

Nov '04CS3291: Section Bilinear transformation: Most common transform from H a (s) to H(z). Replace s by (2 / T) (z-1) / (z+1) to obtain H(z). For convenience take T=1. Example 1 1 H a (s) =  then H(z) =  1 + RC s 1 + RC z + 1 =  (1 + 2RC)z + (1 - 2RC)

Nov '04CS3291: Section 612 Re-express as: 1 + z - 1 H(z) = K  1 + b 1 z - 1 where K = 1 / (1 + 2RC) & b 1 = (1 - 2RC) / (1 + 2RC) Properties: (z - 1) (i) H(z) = H a (s) where s = 2  (z + 1) (ii) Order of H(z) = order of H a (s) (iii) If H a (s) is causal & stable, so is H(z). (iv) H(e j  ) = H a (j  ) where  = 2 tan(  /2)

Nov '04CS3291: Section 613 Proof of (iv): When z = e j , then e j  - 1 2(e j  / 2 - e - j  / 2 ) s = 2  =  e j  + 1 e j  / 2 + e -j  / 2 2( 2 j sin (  / 2) =  2 cos (  / 2) = 2 j tan(  / 2)

Nov '04CS3291: Section 614 Frequency warping: By (iv), H(e j  ) = H a (j  ) with  = 2 tan(  /2).  from -  to  mapped to  in the range -  to .

Nov '04CS3291: Section 615 Mapping approx linear for  in the range -2 to 2. As  increases above 2, a given increase in  produces smaller and smaller increases in .

Nov '04CS3291: Section 616 Comparing (a) with (b) below, (b) becomes more and more compressed as    . Frequency warping must be taken into account with this method

Nov '04CS3291: Section Design of IIR low-pass filter by bilinear transfm Given cut-off frequency  C in radians/sample:- (i) Calculate  C = 2 tan(  C /2) radians/sec. (  C is "pre-warped" cut-off frequency) (ii) Find H a (s) for analogue low-pass filter with 1 radian/s cut-off. (iii) Scale cut-off frequency of H a (s) to  C (iv) Replace s by 2(z - 1) / (z+1) to obtain H(z). (v) Rearrange the expression for H(z) (vi) Realise by biquadratic sections.

Nov '04CS3291: Section 618 Example Design 2 nd order Butterworth-type IIR low-pass filter with  C =  / 4. Solution: Prewarped frequency  C = 2 tan(  / 8) = Analogue Butterworth low-pass filter with c/o 1 radian/second: 1 H a (s) =  1 +  2 s + s 2 Scale c/o to 0.828, 1 H a (s) =  1 +  2 s/ (s/0.828) 2 then replace s by 2 (z+1) / (z-1) to obtain: z 2 + 2z + 1 H(z) =  z z + 3.4

Nov '04CS3291: Section 619 which may be realised by the signal flow graph:-

Nov '04CS3291: Section Higher order IIR digital filters: Normally cascaded biquad sections. Example 6.3: Design 4th order Butterwth-type IIR low-pass digital filter with 3 dB c/o at f S / 16.. Solution: (a) Relative cut-off frequency is  /8. Prewarped cut-off :  C = 2 tan((  /8)/2)  0.4 radians/s. Formula for 1 radian/s cut-off is: Replace s by s/0.4 then replace s by 2 (z-1) / (z+1) to obtain:

Nov '04CS3291: Section 621 H(z) may be realised as:

Nov '04CS3291: Section 622

Nov '04CS3291: Section 623 Compare gain-response of 4th order Butt low-pass transfer function used as a prototype, with that of derived digital filter. Both are 1 at zero frequency. Both are at the cut-off frequency. Analogue gain approaches 0 as    whereas digital filter gain becomes exactly zero at  = . Shape of Butt gain response is "warped" by bilinear transfn. For digital filter, cut-off rate becomes sharper as    because of the compression as   .