ch9(1)handouts_3e

11Copyright 2005, S. K. Mitra

Digital Filter DesignDigital Filter Design

Objective - Determination of a realizable transfer function G(z) approximating a given frequency response specification is an important step in the development of a digital filter

If an IIR filter is desired, G(z) should be a stable real rational function

Digital filter design is the process of deriving the transfer function G(z)


Digital Filter SpecificationsDigital Filter Specifications Usually, either the magnitude and/or the

phase (delay) response is specified for the design of digital filter for most applications

In some situations, the unit sample responseor the step response may be specified

In most practical applications, the problem of interest is the development of a realizable approximation to a given magnitude response specification


Digital Filter SpecificationsDigital Filter Specifications We discuss in this course only the

magnitude approximation problem There are four basic types of ideal filters

with magnitude responses as shown below

1

0 c c

HLP(e j )

0 c c

1

HHP(e j )

11

c1 c1 c2 c2

HBP (e j )

1

c1 c1 c2 c2

HBS(e j )


Digital Filter SpecificationsDigital Filter Specifications As the impulse response corresponding to

each of these ideal filters is noncausal and of infinite length, these filters are not realizable

In practice, the magnitude response specifications of a digital filter in the passband and in the stopband are given with some acceptable tolerances

In addition, a transition band is specified between the passband and stopband


Digital Filter SpecificationsDigital Filter Specifications For example, the magnitude response

of a digital lowpass filter may be given as indicated below

)( jeG


Digital Filter SpecificationsDigital Filter Specifications As indicated in the figure, in the passband,

defined by , we require that with an error , i.e.,

In the stopband, defined by , we require that with an error , i.e.,

1)( jeG

0)( jeG s

pp0

spp

jp eG + ,1)(1

ssjeG ,)(


Digital Filter SpecificationsDigital Filter Specifications - passband edge frequency - stopband edge frequency - peak ripple value in the passband - peak ripple value in the stopband Since is a periodic function of ,

and of a real-coefficient digital filter is an even function of

As a result, filter specifications are given only for the frequency range

ps

sp

)( jeG)( jeG

08

Copyright 2005, S. K. Mitra

Digital Filter SpecificationsDigital Filter Specifications

Specifications are often given in terms of loss function A in dB

Peak passband rippledB

Minimum stopband attenuationdB

)(log20)( 10= jeG

)1(log20 10 pp =

)(log20 10 ss =


Digital Filter SpecificationsDigital Filter Specifications Magnitude specifications may alternately be

given in a normalized form as indicated below


Digital Filter SpecificationsDigital Filter Specifications Here, the maximum value of the magnitude

in the passband is assumed to be unity

- Maximum passband deviation, given by the minimum value of the magnitude in the passband

- Maximum stopband magnitudeA1

21/1 +



For the normalized specification, maximum value of the gain function or the minimum value of the loss function is 0 dB

Maximum passband attenuation -dB

For , it can be shown thatdB

( )210max 1log20 +=1



Example - Let kHz, kHz, and kHz

Then

7=pF 3=sF25=TF

== 56.01025

)107(23

3

p

== 24.01025

)103(23

3

s


The transfer function H(z) meeting the frequency response specifications should be a causal transfer function

For IIR digital filter design, the IIR transfer function is a real rational function of :

H(z) must be a stable transfer function and must be of lowest order N for reduced computational complexity

Selection of Filter TypeSelection of Filter Type

1z

NMzdzdzddzpzpzppzH N

N

MM ++++

++++=

,)( 22

110

22

110

LL


Selection of Filter TypeSelection of Filter Type For FIR digital filter design, the FIR

transfer function is a polynomial in with real coefficients:

For reduced computational complexity, degree N of H(z) must be as small as possible

If a linear phase is desired, the filter coefficients must satisfy the constraint:

==

N

n

nznhzH0

][)(

][][ nNhnh =

1z


Selection of Filter TypeSelection of Filter Type Advantages in using an FIR filter -

(1) Can be designed with exact linear phase,(2) Filter structure always stable with quantized coefficients

Disadvantages in using an FIR filter - Order of an FIR filter, in most cases, is considerably higher than the order of an equivalent IIR filter meeting the same specifications, and FIR filter has thus higher computational complexity


Digital Filter Design: Digital Filter Design: Basic ApproachesBasic Approaches

Most common approach to IIR filter design -(1) Convert the digital filter specifications into an analog prototype lowpass filter specifications

(2) Determine the analog lowpass filter transfer function

(3) Transform into the desired digital transfer function

)(sHa

)(zG)(sHa



This approach has been widely used for the following reasons:(1) Analog approximation techniques are highly advanced(2) They usually yield closed-form solutions(3) Extensive tables are available for analog filter design(4) Many applications require digital simulation of analog systems



An analog transfer function to be denoted as

where the subscript a specifically indicates the analog domain

A digital transfer function derived from shall be denoted as

)()()( sDsPsH

a

aa =

)()()( zDzPzG =

)(sHa



Basic idea behind the conversion of into is to apply a mapping from the s-domain to the z-domain so that essential properties of the analog frequency response are preserved

Thus mapping function should be such that Imaginary ( ) axis in the s-plane be

mapped onto the unit circle of the z-plane A stable analog transfer function be mapped

into a stable digital transfer function

)(sHa)(zG

j



FIR filter design is based on a direct approximation of the specified magnitude response, with the often added requirement that the phase be linear

The design of an FIR filter of order N may be accomplished by finding either the length-(N+1) impulse response samplesor the (N+1) samples of its frequency response

{ }][nh)( jeH



Three commonly used approaches to FIR filter design -(1) Windowed Fourier series approach(2) Frequency sampling approach(3) Computer-based optimization methods


IIR Digital Filter Design: Bilinear IIR Digital Filter Design: Bilinear Transformation MethodTransformation Method

Bilinear transformation -

Above transformation maps a single point in the s-plane to a unique point in the z-plane and vice-versa

Relation between G(z) and is then given by

+=

1

1

112

zz

Ts

)(sHa

+== 11112)()(

z

zTs

a sHzG24


Bilinear TransformationBilinear Transformation Digital filter design consists of 3 steps:

(1) Develop the specifications of by applying the inverse bilinear transformation to specifications of G(z)(2) Design(3) Determine G(z) by applying bilinear transformation to

As a result, the parameter T has no effect on G(z) and T = 2 is chosen for convenience

)(sHa

)(sHa

)(sHa


Bilinear TransformationBilinear Transformation Inverse bilinear transformation for T = 2 is

For

Thus,

ssz

+= 11

oo js +=22

222

)1()1(

)1()1(

oo

oo

oo

oo zjjz +

++=++=

10 == zo10 zo 26


Bilinear TransformationBilinear Transformation

Mapping of s-plane into the z-plane


Bilinear TransformationBilinear Transformation For with T = 2 we have

or

)()(

11

2/2/2/

2/2/2/

+=+

= jjjjjj

j

j

eeeeee

eej

= jez

)2/tan(=)2/tan(

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

= jj


Bilinear TransformationBilinear Transformation Mapping is highly nonlinear Complete negative imaginary axis in the s-

plane from to is mapped into the lower half of the unit circle in the z-plane from to

Complete positive imaginary axis in the s-plane from to is mapped into the upper half of the unit circle in the z-plane from to

= 0=

0= =

1=z 1=z

1=z 1=z


Bilinear TransformationBilinear Transformation Nonlinear mapping introduces a distortion

in the frequency axis called frequency warping

Effect of warping shown below = tan(/2)


Bilinear TransformationBilinear Transformation Steps in the design of a digital filter -

(1) Prewarp to find their analog equivalents(2) Design the analog filter(3) Design the digital filter G(z) by applying bilinear transformation to

Transformation can be used only to design digital filters with prescribed magnitude response with piecewise constant values

Transformation does not preserve phase response of analog filter

),( sp ),( sp

)(sHa

)(sHa


IIR Digital Filter Design Using IIR Digital Filter Design Using Bilinear TransformationBilinear Transformation

Example - Consider

Applying bilinear transformation to the above we get the transfer function of a first-order digital lowpass Butterworth filter

c

ca ssH +

=)(

)1()1()1()()( 11

1

11

11

+= ++

+==zz

zsHzGc

c

zzsa



Rearranging terms we get

where

1

1

11

21)(

+=

zzzG

)2/tan(1)2/tan(1

11

c

c

c

c+=+

=



Example - Consider the second-order analog notch transfer function

for which

is called the notch frequency If then

is the 3-dB notch bandwidth

22

22)(

o

oa sBs

ssH +++=0)( =oa jH

1)()0( == jHjH aao

2/1)()( 12 == jHjH aa12 =B



Then

where

1

1

11)()(

+==zzsa

sHzG

22122

22122

)1()1(2)1()1()1(2)1(

+++++++=

zBzBzz

ooo

ooo

21

21

)1(2121

21

++++=

zzzz

oo

o =+= cos

11

2

2)2/tan(1)2/tan(1

11

2

2

w

w

o

oBB

BB

+=++

+=



Example - Design a 2nd-order digital notch filter operating at a sampling rate of 400 Hzwith a notch frequency at 60 Hz, 3-dB notch bandwidth of 6 Hz

Thus

From the above values we get

== 3.0)400/60(2o== 03.0)400/6(2wB

90993.0=587785.0=



Thus

The gain and phase responses are shown below

21

21

90993.01226287.11954965.01226287.1954965.0)(

+

+=zz

zzzG

0 50 100 150 200-40

-30

-20

-10

0

Frequency, Hz

Gain

, dB

0 50 100 150 200-2

-1

0

1

2

Frequency, Hz

Phas

e, ra

dian

s


IIR IIR LowpassLowpass Digital Filter Design Digital Filter Design Using Bilinear TransformationUsing Bilinear Transformation

Example - Design a lowpass Butterworth digital filter with , ,

dB, and dB Thus

If this implies

= 25.0p = 55.0s15s5.0 p

5.0)(log20 25.010 jeG15)(log20 55.010 jeG

1)( 0 =jeG1220185.02 = 622777.312 =A



Prewarping we get

The inverse transition ratio is

The inverse discrimination ratio is

4142136.0)2/25.0tan()2/tan( === pp1708496.1)2/55.0tan()2/tan( === ss

8266809.21 ==

p

sk

841979.15112

1=

= Ak



Thus

We choose N = 3 To determine we use

6586997.2)/1(log)/1(log

10

110 ==kkN

c

222

11

)/(11)( +=+= Ncppa

jH



We then get

3rd-order lowpass Butterworth transfer function for is

Denormalizing to get we arrive at

588148.0)(419915.1 == pc

1=c)1)(1(

1)( 2 +++= ssssHan

=588148.0

)( sHsH ana

588148.0=c



Applying bilinear transformation to we get the desired digital transfer function

Magnitude and gain responses of G(z) shown below:

)(sHa

1

1

11)()(

+==zzsa sHzG

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

/

Mag

nitu

de

0 0.2 0.4 0.6 0.8 1-40

-30

-20

-10

0

/

Gain

, dB


Design of IIR Design of IIR HighpassHighpass, , BandpassBandpass, , and and BandstopBandstop Digital FiltersDigital Filters

First Approach -(1) Prewarp digital frequency specifications of desired digital filter to arrive at frequency specifications of analog filter of same type(2) Convert frequency specifications of into that of prototype analog lowpass filter

(3) Design analog lowpass filter

)(zGD)(sHD

)(sHD

)(sHLP)(sHLP



(4) Convert into using inverse frequency transformation used in Step 2(5) Design desired digital filter by applying bilinear transformation to

)(sHLP )(sHD

)(zGD)(sHD



Second Approach -(1) Prewarp digital frequency specifications of desired digital filter to arrive at frequency specifications of analog filter of same type(2) Convert frequency specifications of into that of prototype analog lowpass filter

)(zGD)(sHD

)(sHLP

)(sHD



(3) Design analog lowpass filter(4) Convert into an IIR digital transfer function using bilinear transformation(5) Transform into the desired digital transfer function

We illustrate the first approach

)(sHLP)(sHLP

)(zGLP

)(zGLP)(zGD


IIR IIR HighpassHighpass Digital Filter DesignDigital Filter Design Design of a Type 1 Chebyshev IIR digital

highpass filter Specifications: Hz, Hz,

dB, dB, kHz Normalized angular bandedge frequencies

700=pF 500=sF2=TF1= p 32=s

=== 7.02000

70022

T

pp F

F

=== 5.02000

50022T

ss F

F


IIR IIR HighpassHighpass Digital Filter DesignDigital Filter Design Prewarping these frequencies we get

For the prototype analog lowpass filter choose

Using we get Analog lowpass filter specifications: ,

, dB, dB

= pp

1= p962105.1=s

9626105.1)2/tan( == pp0.1)2/tan( == ss

1= p926105.1=s 1= p 32=s


IIR IIR HighpassHighpass Digital Filter DesignDigital Filter Design MATLAB code fragments used for the design

[N, Wn] = cheb1ord(1, 1.9626105, 1, 32, s)[B, A] = cheby1(N, 1, Wn, s);[BT, AT] = lp2hp(B, A, 1.9626105);[num, den] = bilinear(BT, AT, 0.5);

0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20

-10

0

/

Gain

, dB


IIR IIR BandpassBandpass Digital Filter DesignDigital Filter Design Design of a Butterworth IIR digital bandpass

filter Specifications: , ,

, , dB, dB Prewarping we get

= 45.01p = 65.02p= 75.02s= 3.01s 1= p 40=s

8540807.0)2/tan( 11 == pp6318517.1)2/tan( 22 == pp

41421356.2)2/tan( 22 == ss5095254.0)2/tan( 11 == ss


IIR IIR BandpassBandpass Digital Filter DesignDigital Filter Design

Width of passband

We therefore modify so that and exhibit geometric symmetry with

respect to We set For the prototype analog lowpass filter we

choose

777771.0 12 == ppwB393733.1 21

2 == ppo2

21 23010325.1 oss =1 s

2 s1 s

2 o5773031.0 1 =s

1= p


IIR IIR BandpassBandpass Digital Filter DesignDigital Filter Design

Using we get

Specifications of prototype analog Butterworth lowpass filter:

, , dB,dB

w

op B

= 22

3617627.2777771.05773031.0

3332788.0393733.1 ==s

1= p 3617627.2=s 1= p40=s


IIR IIR BandpassBandpass Digital Filter DesignDigital Filter Design MATLAB code fragments used for the design

[N, Wn] = buttord(1, 2.3617627, 1, 40, s)[B, A] = butter(N, Wn, s);[BT, AT] = lp2bp(B, A, 1.1805647, 0.777771);[num, den] = bilinear(BT, AT, 0.5);

0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20

-10

0

/

Gain

, dB


IIR IIR BandstopBandstop Digital Filter DesignDigital Filter Design Design of an elliptic IIR digital bandstop filter Specifications: , ,

, , dB, dB Prewarping we get

Width of stopband

= 45.01s = 65.02s= 75.02p= 3.01p 1= p 40=s

,8540806.0 1 =s ,6318517.1 2 =s,5095254.0 1 = p 4142136.2 2 = p

777771.0 12 == sswB393733.1 12

2 == sso2

12 230103.1 opp = 54Copyright 2005, S. K. Mitra

IIR IIR BandstopBandstop Digital Filter DesignDigital Filter Design We therefore modify so that and

exhibit geometric symmetry with respect to

We set For the prototype analog lowpass filter we

choose Using we get

1 p 1 p 2 p2 o

577303.0 1 = p1=s

22

=o

ws

B

0.42341263332787.0393733.1

777771.05095254.0 == p

10


IIR IIR BandstopBandstop Digital Filter DesignDigital Filter Design MATLAB code fragments used for the design

[N, Wn] = ellipord(0.4234126, 1, 1, 40, s);[B, A] = ellip(N, 1, 40, Wn, s);[BT, AT] = lp2bs(B, A, 1.1805647, 0.777771);[num, den] = bilinear(BT, AT, 0.5);

0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20

-10

0

/

Gai

n, dB

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