digital filter design - catholic university of...

1Copyright © 2001, S. K. Mitra

Digital Filter DesignDigital Filter Design

• Objective - Determination of a realizabletransfer function G(z) approximating agiven frequency response specification is animportant step in the development of adigital filter

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

• Digital filter design is the process ofderiving the transfer function G(z)


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

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

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

• In most practical applications, the problemof interest is the development of a realizableapproximation to a given magnituderesponse 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 andof infinite length, these filters are notrealizable

• In practice, the magnitude responsespecifications of a digital filter in thepassband and in the stopband are given withsome acceptable tolerances

• In addition, a transition band is specifiedbetween the passband and stopband


Digital Filter SpecificationsDigital Filter Specifications• For example, the magnitude response

of a digital lowpass filter may be given asindicated below

)( ωjeG


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 , werequire that with an error ,i.e.,

1)( ≅ωjeG

0)( ≅ωjeG sδ

pδ±pω≤ω≤0

π≤ω≤ωs

ppj

p eG ω≤ωδ+≤≤δ− ω ,1)(1

π≤ω≤ωδ≤ωss

jeG ,)(


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 ω,

and of a real-coefficient digitalfilter is an even function of ω

• As a result, filter specifications are givenonly for the frequency range

pω

sω

sδpδ

)( ωjeG)( ωjeG

π≤ω≤0


Digital Filter SpecificationsDigital Filter Specifications

• Specifications are often given in terms ofloss function A indB

• Peak passband ripple dB• Minimum stopband attenuation dB

)(log20)( 10ω−=ω jeG

)1(log20 10 pp δ−−=α

)(log20 10 ss δ−=α


Digital Filter SpecificationsDigital Filter Specifications• Magnitude specifications may alternately be

given in a normalized form as indicatedbelow


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 themagnitude in the passband

• - Maximum stopband magnitudeA1

21/1 ε+



• For the normalized specification, maximumvalue of the gain function or the minimumvalue of the loss function is 0 dB

• Maximum passband attenuation - dB• For , it can be shown that dB

( )210max 1log20 ε+=α

1<<δ p)21(log20 10max pδ−−≅α


Digital Filter SpecificationsDigital Filter Specifications• In practice, passband edge frequency

and stopband edge frequency arespecified in Hz

• For digital filter design, normalizedbandedge frequencies need to be computedfrom specifications in Hz using

TFF

FF p

T

p

T

pp π=

π=

Ω=ω 2

2

TFF

FF s

T

s

T

ss π=

π=

Ω=ω 22

sFpF



• Example - Let kHz, kHz, andkHz

• Then

7=pF 3=sF25=TF

π=××π=ω 56.01025

)107(23

3

p

π=××π=ω 24.01025

)103(23

3

s


• The transfer function H(z) meeting thefrequency response specifications should bea causal transfer function

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

• H(z) must be a stable transfer function andmust be of lowest order N for reducedcomputational complexity

Selection of Filter TypeSelection of Filter Type

1−z

NMzdzdzddzpzpzppzH N

N

MM ≤

++++++++= −−−

−−−,)( 2

21

10

22

110

L

L


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

transfer function is a polynomial inwith real coefficients:

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

• If a linear phase is desired, the filtercoefficients must satisfy the constraint:

∑==

−N

n

nznhzH0

][)(

][][ nNhnh −±=

1−z


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 withquantized coefficients

• Disadvantages in using an FIR filter - Orderof an FIR filter, in most cases, isconsiderably higher than the order of anequivalent IIR filter meeting the samespecifications, and FIR filter has thus highercomputational complexity


Digital Filter Design:Digital Filter Design:Basic ApproachesBasic Approaches

• Most common approach to IIR filter design -(1) Convert the digital filter specificationsinto an analog prototype lowpass filterspecifications

• (2) Determine the analog lowpass filtertransfer function

• (3) Transform into the desired digitaltransfer function

)(sHa

)(zG)(sHa



• This approach has been widely used for thefollowing reasons:(1) Analog approximation techniques arehighly advanced(2) They usually yield closed-formsolutions(3) Extensive tables are available foranalog filter design(4) Many applications require digitalsimulation of analog systems



• An analog transfer function to be denoted as

where the subscript “a” specificallyindicates the analog domain

• A digital transfer function derived fromshall be denoted as

)()()( sD

sPsHa

aa =

)()()( zD

zPzG =

)(sHa



• Basic idea behind the conversion ofinto is to apply a mapping from thes-domain to the z-domain so that essentialproperties of the analog frequency responseare 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

Ωj



• FIR filter design is based on a directapproximation of the specified magnituderesponse, with the often added requirementthat the phase be linear

• The design of an FIR filter of order N maybe accomplished by finding either thelength-(N+1) impulse response samplesor the (N+1) samples of its frequencyresponse

][nh

)( ωjeH



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


IIR Digital Filter Design: BilinearIIR Digital Filter Design: BilinearTransformation MethodTransformation Method

• Bilinear transformation -

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

• Relation between G(z) and is thengiven by

+−= −

−

1

1

112

zz

Ts

)(sHa

−+

−−==

11

112)()(z

zTsa sHzG


Bilinear TransformationBilinear Transformation• Digital filter design consists of 3 steps:

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

• As a result, the parameter T has no effect onG(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 Ω+σ=

22

222

)1()1(

)1()1(

oo

oo

oo

oo zjjz

Ω+σ−Ω+σ+=⇒

Ω−σ−Ω+σ+=

10 =→=σ zo

10 <→<σ zo

10 >→>σ zo


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/

ω−ωω−

ω−ωω−

ω−

ω−

+−=

+−=Ω jjj

jjj

j

j

eeeeee

eej

ω= jez

)2/tan(ω=Ω)2/tan(2

)2/cos(2)2/sin(2 ω=

ωω= jj


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

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

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

−∞=Ω 0=Ω

0=Ω ∞=Ω

1−=z 1=z

1=z 1−=z


Bilinear TransformationBilinear Transformation• Nonlinear mapping introduces a distortion

in the frequency axis called frequencywarping

• Effect of warping shown below


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

(1) Prewarp to find their analogequivalents(2) Design the analog filter(3) Design the digital filter G(z) by applyingbilinear transformation to

• Transformation can be used only to designdigital filters with prescribed magnituderesponse with piecewise constant values

• Transformation does not preserve phaseresponse of analog filter

),( sp ωω),( sp ΩΩ

)(sHa

)(sHa


IIR Digital Filter Design UsingIIR Digital Filter Design UsingBilinear TransformationBilinear Transformation

• Example - Consider

• Applying bilinear transformation to the abovewe get the transfer function of a first-orderdigital lowpass Butterworth filter

c

ca s

sHΩ+

Ω=)(

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

1

11

11 −−

−

−+

−−= +Ω+−+Ω==

zzzsHzGc

c

zzsa



• Rearranging terms we get

where

1

1

11

21)( −

−

−+⋅−=

zzzGα

α

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

11

c

c

c

cω+ω−

=Ω+Ω−

=α



• Example - Consider the second-order analognotch transfer function

for which

• is called the notch frequency• If then

is the 3-dB notch bandwidth

22

22)(

o

oa sBs

ssHΩ++

Ω+=

0)( =Ωoa jH1)()0( =∞= jHjH aa

oΩ2/1)()( 12 =Ω=Ω jHjH aa

12 Ω−Ω=B



• Then

where

1

1

11)()(

−

−

+−=

=zzsa sHzG

22122

22122

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

−−

−−

−Ω++Ω−−+Ω+Ω++Ω−−Ω+=

zBzBzz

ooo

ooo

21

21

)1(2121

21

−−

−−

++−+−⋅+=

zzzzααβ

βα

oo

o ω=Ω+Ω−=β cos

11

2

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

11

2

2

w

w

o

oBB

BB

+−=

+Ω+−Ω+=α



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

• Thus

• From the above values we get

π=π=ω 3.0)400/60(2oπ=π= 03.0)400/6(2wB

90993.0=α

587785.0=β



• Thus

• The gain and phase responses are shown below

21

21

90993.01226287.11954965.01226287.1954965.0)( −−

−−

+−+−=

zzzzzG

0 50 100 150 200-40

-30

-20

-10

0

Frequency, Hz

Gai

n, d

B

0 50 100 150 200-2

-1

0

1

2

Frequency, Hz

Pha

se, r

adia

ns


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

• Example - Design a lowpass Butterworthdigital filter with , ,

dB, and dB• Thus

• If this implies

π=ω 25.0p π=ω 55.0s15≥αs5.0≤α p

5.0)(log20 25.010 −≥πjeG

15)(log20 55.010 −≤πjeG

1)( 0 =jeG1220185.02 =ε 622777.312 =A


IIRIIR Lowpass Lowpass Digital Filter Design Digital Filter DesignUsing Bilinear TransformationUsing Bilinear Transformation

• Prewarping we get

• The inverse transition ratio is

• The inverse discrimination ratio is

4142136.0)2/25.0tan()2/tan( =π=ω=Ω pp

1708496.1)2/55.0tan()2/tan( =π=ω=Ω ss

8266809.21 =ΩΩ=

p

sk

841979.1511 2

1=

ε−= A

k



• Thus

• We choose N = 3• To determine we use

6586997.2)/1(log)/1(log

10

110 ==kkN

cΩ

222

11

)/(11)(

ε+=

ΩΩ+=Ω N

cppa jH



• We then get

• 3rd-order lowpass Butterworth transferfunction for is

• Denormalizing to get wearrive at

588148.0)(419915.1 =Ω=Ω pc

1=Ωc

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

=sss

sHan

=

588148.0)( sHsH ana

588148.0=Ωc



• Applying bilinear transformation towe get the desired digital transfer function

• Magnitude and gain responses of G(z) shownbelow:

)(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

ω/π

Mag

nitu

de

0 0.2 0.4 0.6 0.8 1-40

-30

-20

-10

0

ω/π

Gai

n, d

B


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

• First Approach -(1) Prewarp digital frequency specificationsof desired digital filter to arrive atfrequency specifications of analog filterof same type(2) Convert frequency specifications ofinto that of prototype analog lowpass filter

(3) Design analog lowpass filter

)(zGD)(sHD

)(sHD

)(sHLP

)(sHLP


Design of IIRDesign of IIR Highpass Highpass,, Bandpass Bandpass,,andand Bandstop Bandstop Digital Filters Digital Filters

(4) Convert into usinginverse frequency transformation used inStep 2(5) Design desired digital filter byapplying bilinear transformation to

)(sHLP )(sHD

)(zGD)(sHD



• Second Approach -(1) Prewarp digital frequency specificationsof desired digital filter to arrive atfrequency specifications of analog filterof same type(2) Convert frequency specifications ofinto that of prototype analog lowpass filter

)(zGD)(sHD

)(sHLP

)(sHD



(3) Design analog lowpass filter(4) Convert into an IIR digitaltransfer function using bilineartransformation(5) Transform into the desireddigital transfer function

• We illustrate the first approach

)(sHLP

)(sHLP)(zGLP

)(zGLP)(zGD


IIRIIR Highpass Highpass Digital Filter Design Digital 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=α p 32=αs

π=×π=π

=ω 7.02000

70022

T

pp F

F

π=×π

=π

=ω 5.02000

50022T

ss F

F


IIRIIR Highpass Highpass Digital Filter Design Digital Filter Design• Prewarping these frequencies we get

• For the prototype analog lowpass filter choose

• Using we get• Analog lowpass filter specifications: ,

, dB, dB

ΩΩΩ

−=Ω ˆˆ pp

1=Ω p

962105.1=Ωs

9626105.1)2/tan(ˆ =ω=Ω pp

0.1)2/tan(ˆ =ω=Ω ss

1=Ω p

926105.1=Ωs 1=α p 32=αs


IIRIIR Highpass Highpass Digital Filter Design Digital 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

ω/π

Gai

n, d

B


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

filter• Specifications: , ,

, , dB, dB• Prewarping we get

π=ω 45.01p π=ω 65.02p

π=ω 75.02sπ=ω 3.01s 1=α p 40=αs

8540807.0)2/tan(ˆ 11 =ω=Ω pp

6318517.1)2/tan(ˆ 22 =ω=Ω pp

41421356.2)2/tan(ˆ 22 =ω=Ω ss

5095254.0)2/tan(ˆ 11 =ω=Ω ss


IIR IIR Bandpass Bandpass 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

777771.0ˆˆ 12 =Ω−Ω= ppwB393733.1ˆˆˆ 21

2 =ΩΩ=Ω ppo2

21 ˆ23010325.1ˆˆ oss Ω≠=ΩΩ1ˆ sΩ

2ˆ sΩ1ˆ sΩ

2ˆ oΩ5773031.0ˆ 1 =Ωs

1=Ω p


IIRIIR Bandpass Bandpass Digital Filter Design Digital Filter Design

• Using we get

• Specifications of prototype analogButterworth lowpass filter:

, , dB, dB

w

op BΩ

Ω−ΩΩ−=Ω ˆ

ˆˆ 22

3617627.2777771.05773031.0

3332788.0393733.1 =×−=Ωs

1=Ω p 3617627.2=Ωs 1=α p40=αs


IIRIIR Bandpass Bandpass Digital Filter Design Digital 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

ω/π

Gai

n, d

B


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

, , dB, dB• Prewarping we get

• Width of stopband

π=ω 45.01s π=ω 65.02sπ=ω 75.02pπ=ω 3.01p 1=α p 40=αs

,8540806.0ˆ 1 =Ωs ,6318517.1ˆ 2 =Ωs

,5095254.0ˆ 1 =Ω p 4142136.2ˆ 2 =Ω p777771.0ˆˆ 12 =Ω−Ω= sswB

393733.1ˆˆˆ 122 =ΩΩ=Ω sso

212 ˆ230103.1ˆˆ opp Ω≠=ΩΩ


IIRIIR Bandstop Bandstop Digital Filter Design Digital 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

1ˆ pΩ 1ˆ pΩ 2ˆ pΩ

2ˆ oΩ

577303.0ˆ 1 =Ω p

1=Ωs

22 ˆˆˆ

Ω−ΩΩΩ=Ωo

ws

B

0.42341263332787.0393733.1

777771.05095254.0 =−×=Ω p


IIRIIR Bandstop Bandstop Digital Filter Design Digital 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

ω/π

Gai

n, d

B

digital filter design - catholic university of...

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