Baseado no material da: UNIVERSITY of MANCHESTER Department of Computer Science

Nov '04 CS3291: Section 6 1

Baseado no material da:

UNIVERSITY of MANCHESTER

Department of Computer Science

CS3291: Digital Signal Processing

Section 6

IIR discrete time filter design


6.1. Introduction:

• Many design techniques for IIR discrete time filters have adopted ideas of analogue filters

• Transform Ha(s) for analogue ‘prototype’ filter into H(z) for discrete time filter.

• Begin with a reminder about analogue filters.

6.2. Analogue filters


• Analogue filters have transfer functions: a0 + a1s + a2s2+ ... + aNsN

H a (s) = b0 + b1s + b2s2 + ... + bMsM

• Replace s by j for frequency-response.

• For RE(s) 0, Ha(s ) is Laplace Transform of h a (t).• In terms of poles and zeros:

(s - z 1 ) ( z - z 2 ) ... ( s - z N) Ha(s) = K (s - p 1 ) ( z - p 2 ) ... ( s - p M)

• Many known techniques for deriving Ha(s); e.g. Butterworth low-pass approximation•Analogue filters have infinite impulse-responses.


Analogue Butterworth low-pass filter of order n

nC

G2)/(1

1)(

[n/2]

1k

2

212sin21 1

1

cc

p

c

a

ssnk

ssH

[n/2] is integer part of n/2 & P = 0 /1 if n is even / odd


[n/2]

1k

2

212sin21 1

1

cc

p

c

a

ssnk

ssH

Ca

Cs

sHGn

/1

1)(

)/(1

1)( : 1

2

24 )/(/21

1)(

)/(1

1)( : 2

CC

a

Css

sHGn

1 when )1)(1(

1 )(:3

2

Ca sss

sHn


1 when )...1)(..1(

1 )(

:4

22

Ca sssssH

n

)...1)(..1)(1(

1 )(

:5

22 ssssssH

n

a


radian/s

Butterworth low-pass

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6

Ga

in

n=2

n=1

n=4


Intro to Bilinear Transformation method

• Most common method for transforming Ha (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. d2y(t)/dt2 at t=nT (y[n] - 2y[n-1]+y[n-2])/T2

d3y(t)/dt3 at t=nT (y[n]-3y[n-1]+3y[n-2]-y[n-3])/T3

“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.


• 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 d2y(t)/dt2, and so on.

• If D(z) is z-transform of D[n] :

0.5( D(z) + z-1D(z) ) = ( Y(z) - z-1Y(z) ) / T

)(1

12 )(

1

1

5.0

1 )(

1

1

zYz

z

TzY

z

z

TzD


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).

sy(t) dy(t)/dt

(2/T) (z-1)/(z+1){y[n]} {dy(t)/dt / t=nT}


6.7. Bilinear transformation:

•Most common transform from Ha (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)

1

12

z

z

T


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) = Ha(s) where s = 2 (z + 1)(ii) Order of H(z) = order of Ha(s)

(iii) If Ha(s) is causal & stable, so is H(z).

(iv) H(ej) = Ha(j ) where = 2 tan(/2)


Proof of (iv):

When z = ej, then

ej - 1 2(e j / 2 - e - j / 2 ) s = 2 = ej + 1 e j / 2 + e -j / 2

2( 2 j sin ( / 2) = 2 cos ( / 2)

= 2 j tan( / 2)


Frequency warping:

By (iv), H(ej) = Ha(j) with = 2 tan(/2). from - to mapped to in the range - to .

-3.14

-2.355

-1.57

-0.785

0

0.785

1.57

2.355

3.14

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

Radians/secondRa

dia

ns/s

am

ple


• Mapping approx linear for in the range -2 to 2.

• As increases above 2 , a given increase in produces smaller and smaller increases in .


Comparing (a) with (b) below, (b) becomes more and more compressed as .

|Ha(j )| |H(exp(j )|

(a): Analogue gain response (b): Effect of bilinear transformation

Frequency warping must be taken into account with this method


6.8. Design of IIR low-pass filter by bilinear transfmGiven 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 Ha(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.


•Example Design 2nd order Butterworth-type

IIR low-pass filter with C = / 4.

•Solution: Prewarped frequency C = 2 tan( / 8) = 0.828

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/0.828 + (s/0.828) 2

then replace s by 2 (z+1) / (z-1) to obtain: z 2 + 2z + 1 H(z) = z 2 - 9.7 z + 3.4


which may be realised by the signal flow graph:-

x[n] y[n] 0.093

2 0.94

-0.33

21

21

33.094.01

21093.0

zz

zzzH


6.9 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 fS / 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:

22 85.11

1

77.01

1

sssssH a

Replace s by s/0.4 then replace s by 2 (z-1) / (z+1) to obtain:

48.0365.1

12028.0

74.06.1

12033.0

2

2

2

2

zz

zz

zz

zzzH


H(z) may be realised as:

x[n] 0.033

2 1.6

-0.74

0.028 y[n]

2 1.36

-0.48

Fourth order IIR Butterworth filter with cut-off fs/16


Analogue gain resp.

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Radians/second

Ga

in

Gain response of 4th order IIR filter

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

0 0.785 1.57 2.355 3.14

Radians/sample

Ga

in


•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 0.707 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 .

Baseado no material da: UNIVERSITY of MANCHESTER Department of Computer Science

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