Page 1 of 15

Chapter 10: Design of Digital Filters - IIR

With an understanding of sampling, rudimentary filter operations in the z domain, the Fourier series and transform the topic of filter design can be addresses.

Filters to discuss

finite impulse response infinite impulse response

Important characteristics: order/length/complexity, magnitude response, phase response, group delay, filter region or type, passband/transition band/stopband,

Matlab Support: http://www.mathworks.com/help/signal/ug/iir-filter-design.html

Infinite Impulse Response Filters

zeros and poles significantly lower order than FIR feedback implementation methods to translate from analog Laplace-domain to digital z-domain filters can be used

N

k

kk

M

k

kk

za

zb

zH

0

0

NN

NN

MM

MM

zazazaa

zbzbzbb

zA

zBzH

1

11

10

11

110

Typically, 10 a

N

kk

jw

M

kk

jw

N

kk

M

kk

MN

ez

e

e

d

p

z

zz

d

pzH jw

1

1

0

0

1

1

0

0

k

N

1kk

jwM

1k0

0ez zeMNw

d

pzH jw

jwez

g dz

zHdz

dw

wdw

lnRe

Page 2 of 15

10.3 Design of IIR Filters from Analog Filters

There are multiple methods to “approximate” the performance of analog filters in the digital domain.

A

A

BB

sasasasa

sbsbsbbsH

3

32

21

2210

1

D

D

P

zdzdzdzd

zpzpzppzH

3

32

21

1

32

21

10

1

IIR Filter Design Approach

(1) Convert the digital filter specification into an analog filter specification

(2) Determine the analog filter transfer function

(3) Transform the analog filter into a digital filter

How to accomplish step (3)

Convert from Laplace Domain to Z Domain using a zero order hold (Dorf). Proakis describes this as “Design by Approximation of the Derivatives (10.3.1)

Impulse Invariance Method: (Proakis 10.3.2) Make the digital filter matched the time sampled impulse response http://en.wikipedia.org/wiki/Impulse_invariance

Bilinear Transformation: (Proakis 10.3.3) Use an approximate mapping from the s-plane to the z-plane using a complex variable transformation http://en.wikipedia.org/wiki/Bilinear_transform

Page 3 of 15

10.3.1 IIR Filter Design by Approximation of Derivatives

Approximate derivatives by first order approximations As the sampling rates become significantly higher than Nyquist, the approximation

greatly improves.

The first order derivative approximation

T

TnyTny

dt

dy

Tnt

1

With “s” a derivative operator, set

T

zzYsYs

11

or T

zs

11

In addition the second derivative can be approximated as

TT

TnyTny

T

TnyTny

dt

yd

Tnt

211

2

2

22

2 212

T

TnyTnyTny

dt

yd

Tnt

or

21

2

212 121

T

z

T

zzs

To generalize, we substitute k

k

T

zs

11

Therefore we set

T

zs

a sHzH 11

Page 4 of 15

Mapping a value from the s-plane into the z-plane would be performed as 11 zTs

Tsz

1

1

For a sinusoidal signal in the s-plane where js

22 11

1

1

1

T

Tj

TTjz

This is a different mapping than previously discussed where Tsz exp In this mapping.

0z

T

1

2

1

2

1

11

1

11

122

j

TT

TTj

TT

z

0 1z

T

1

2

1

2

1

11

1

11

122

j

TT

TTj

TT

z

0z

The mapping places the s-domain LHP into a circle centered at ½ in the z-domain!

See example 10.3.1 in text for math/Matlab example.

Page 5 of 15

10.3.2 IIR Filter Design by Impulse Invariance

The impulse response is one method of characterizing the response of a system transfer function or filter. If it is an accurate representation, the digitization of the impulse response should generate a possible solution to the analog to digital conversion process. That is

,1,0, nTnhnh Based on sampling the continuous time signal xa(t) with spectrim Xa(F) at a sample rate Fs, we previously defined (section 6.1)

sk

sas F

FfwhereFkfXFfX

,

Setting the digital filters frequency response as

k

sas FkfHFfH

or in terms of T

k

a T

kH

TTH

21

If we generalize this formula based on the s to z domain mapping Tsz exp

k

aTsz T

kjsH

TzH

21exp

This maintains the mapping previously discussed where wjrTjTTsz expexpexp

and the mapping shown below

Page 6 of 15

Under the assumption that the transfer function in the s-domain can be described as

N

k k

ka ps

csH

1

the partial fraction expansion defines the impulse response

0,exp1

tfortpcthN

kkka

Discretizing

0,exp1

nforTnpcTnhnhN

kkka

The z-domain filter response is then

0 1

expn

nN

kkk zTnpczH

Manipulating, as always,

N

k n

n

kk zTpczH1

1exp

and for a stable system (poles less than zero)

N

k kk zTp

czH1

1exp1

1

the poles in the z-domain are Tpz kk exp

See example 10.3.3 in text for math example.

Page 7 of 15

10.3.3 IIR Filter Design by Bilinear Transformation

The most popular method of conversion is the bilinear transform

1

1

1

12

z

z

Ts

And

1

1

1

11

z

z

Tsa sHzH

This works well for w=0 and w=, but is warped in between these frequencies. To observe the “warping”,

wjrz exp and js

For

1

12

z

z

Ts

1exp

1exp2

wjr

wjr

Ts

wrr

wrj

wrr

r

Twjs

cos21

sin2

cos21

1222

2

Therefore,

wrr

r

T cos21

122

2

wrr

wr

T cos21

sin222

When taken on the “jw” axis, r=1 and

2

tan2

cos1

sin2 w

Tw

w

T

or

2

atan2T

w

Page 8 of 15

Graphically the warping appears as

If desired, the digital filter specification can be “pre-warped” before defining the analog filter. This would allow the bilinear transformed filter to more closely match the desired characteristics.

Prewarping allows the adjustment of the frequency relationship at one frequency point. At this point, the desired mapping will be 1.

See Example 10.3.4 for a numerical example without warping

Page 9 of 15

Prewarping

If we want the “translation” to be correct at a specific reference frequencies we can take.

2tan

2

2tan

2

refref w

T

w

T

Then the mapping becomes

2tan

2tan

w

wref

ref

And the transformation from s to z becomes

1

1

2tan

z

z

ws

ref

ref

See Example 10.3.4 for a numerical example with warping

Selecting a generic Warping Factor

As a consequence, the filter generation is often performed based on a fixed sample rate for T at T=2 and then the filter is frequency translated based on the desired sample rate. Notice that at T=2, the original mapping becomes

1

1

1

12

z

z

z

z

Ts

Page 10 of 15

10.3.4 Characteristics of Commonly Used Analog Filters

Butterworth (no ripple, all poles)

Chebyshev type 1 (ripple in passband, all poles)

Chebyshev type 2 (ripple in stop band, poles and zeros)

Elliptic or Cauer (ripple in passband and stopband, poles and zeros)

10.3.5 Examples of Filters Using Bilinear Transforms

So, what algorithm do you think Matlab uses when generating the digital version of a filter?

A prewarped bilinear transform makes sense. The prewarping must occur at the passband frequency for the low-pass segment of the filter.

Page 11 of 15

10.4 Frequency Transformation

10.4.1 Analog Transformation

For a transfer function in s, there are multiple “transformations” that can transform the defined function into various alternate forms.

For a prototype low pass filter, all the following can be performed:

Convert a prototype lowpass filter to a new lowpass filter with a different bandedge.

ssp

p

'

Transformation

sHsHsH

p

ppLPFp '

Convert a prototype lowpass filter to a highpass filter.

ss pp '

Transformation

sHsHsH pp

pHPFp

'

Convert a prototype lowpass filter to a bandpass filter.

1','

p

p

p forss

uu

uu

s

ss

2

Transformation

sHsHsH

p

ppLPFp '

uu

uuLPFBPFLPF s

sHsHsH

2

or

Page 12 of 15

Transformation

uu

uuppBPFp s

sHsHsH

2

1

Convert a prototype lowpass filter to a bandstop filter.

1','

p

p

p forss

uu

uu

s

ss

2

Transformation

sHsHsH

p

ppLPFp '

uu

uuLPFBSFLPF s

sHsHsH

2

or

Transformation

uu

uuppBSFp s

sHsHsH

21

Page 13 of 15

MATLAB Implementations

lp2lp.m

Lowpass to lowpass analog filter transformation. [NUMT,DENT] = lp2lp(NUM,DEN,Wo) transforms the lowpass filter prototype NUM(s)/DEN(s) with unity cutoff frequency of 1 rad/sec to a lowpass filter with cutoff frequency Wo (rad/sec). [AT,BT,CT,DT] = lp2lp(A,B,C,D,Wo) does the same when the filter is described in state-space form.

lp2hp.m

lp2hp Lowpass to highpass analog filter transformation. [NUMT,DENT] = lp2hp(NUM,DEN,Wo) transforms the lowpass filter prototype NUM(s)/DEN(s) with unity cutoff frequency to a highpass filter with cutoff frequency Wo. [AT,BT,CT,DT] = lp2hp(A,B,C,D,Wo) does the same when the filter is described in state-space form.

lp2bp.m

lp2bp Lowpass to bandpass analog filter transformation. [NUMT,DENT] = lp2bp(NUM,DEN,Wo,Bw) transforms the lowpass filter prototype NUM(s)/DEN(s) with unity cutoff frequency to a bandpass filter with center frequency Wo and bandwidth Bw. [AT,BT,CT,DT] = lp2bp(A,B,C,D,Wo,Bw) does the same when the filter is described in state-space form.

lp2bs.m

lp2bs Lowpass to bandstop analog filter transformation. [NUMT,DENT] = lp2bs(NUM,DEN,Wo,Bw) transforms the lowpass filter prototype NUM(s)/DEN(s) with unity cutoff frequency to a bandstop filter with center frequency Wo and bandwidth Bw. [AT,BT,CT,DT] = lp2bs(A,B,C,D,Wo,Bw) does the same when the filter is described in state-space form.

Page 14 of 15

10.4.2 Digital Transformations

See Table 10.8 on p. 733.

While possible, it may be easier to “reconstruct” the digital filter using the appropriate MATLAB application!

Page 15 of 15

MATLAB Implementations

firlp2hp Convert FIR lowpass filter to Type I FIR highpass filter

firlp2lp Convert FIR Type I lowpass to FIR Type 1 lowpass with inverse bandwidth

iirbpc2bpc

Transform IIR complex bandpass filter to IIR complex bandpass filter with different characteristics

iirlp2bp Transform IIR lowpass filter to IIR bandpass filter

iirlp2bpc IIR lowpass to complex bandpass transformation

iirlp2bs Transform IIR lowpass filter to IIR bandstop filter

iirlp2bsc Transform IIR lowpass filter to IIR complex bandstop filter

iirlp2hp Transform lowpass IIR filter to highpass filter

iirlp2lp Transform lowpass IIR filter to different lowpass filter

iirlp2mb Transform IIR lowpass filter to IIR M-band filter

iirlp2mbc Transform IIR lowpass filter to IIR complex M-band filter

iirlp2xc Transform IIR lowpass filter to IIR complex N-point filter

iirlp2xn Transform IIR lowpass filter to IIR real N-point filter

iirpowcomp Power complementary IIR filter