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COMM 602: Digital Signal Processing
Lecture 10
Digital Filter Design
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Filter Types
:Remember
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Practical Filter specifications
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Practical Filter specifications
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Practical Filter specifications
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IIR Filter
M
M
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IIR Filter Design
Design means calculation of the coefficients of the difference
equation or the transfer function.
We use Analog filter design and then we convert the design into
digital domain because:
Analog filter design is a well developed field and highly advanced.
Analog filter design usually give closed form solutions.
Extensive tables are available for analog filter design.
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IIR Filter Design
Methods of IIR Filter Design:
- Impulse invariant- Step invariant
- Bilinear transformation.
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Impulse Invariant Method
Steps of Design:
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Example: Convert the following analog filter system function into a digital IIR
filter by means of the impulse invariance method:
1
s
s
s
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sHth 11 te 5.03
nTnTnt aeth 3325.0
where
Tea 5.0
1
00 1
33z3
az
zaznhnThZHn
nn
n
n
s
5.0
3
ssH
Example:
Convert the following analog filter system function into a digital
IIR filter by means of the impulse invariance method:
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Characteristics of Analog Filters Butterworth Filters:
All pole filter (no zeros), with no ripples in both the passband and
stop band. The transfer functions of Butterworth filter withis:
OrderFilter FunctionTransfer
11
s
12
12 ss
122
123 sss
1
2
3
16131.24142.36131.2
1234
ssss
4
filteranatheof
frequencyoffCutc
log
:
1c
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Example:
Using the impulse invariant method design a LP digital filter
satisfying the following requirements: cut off frequency 500 Hz,
Sampling rate=500 Hz, 2nd order Butterworth filter.Solution
2/100sin21002/sin2)(
)2/()2/()2/(2
2)2/(
2
1/2/
1)(
)2/100()2/(
22
22
2
2
22
2
2
teteth
s
sss
ss
sH
t
c
t
c
cc
cc
ccc
c
cc
c
cc
a
c
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002.0500
11
,2/100sin2100)( )2/100(
s
nT
fT
nTenTh
)(
cos21)sin()sin(
)()(
2
zHThen
zaTzaTznaTzand
zeXenXSinceaTnaT
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Step Invariant MethodSteps of Design:
nRealizatio(4)
)(
1
Find(3)
samplingbyFind)3(
)(responsestepFind(2)
filter.analogtheofFind(1)
1
nTaz
z
H(Z)
a(nT)s
sHa(t)
H(s)
stu
1)(
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Example:
TT
nTt
ez
z
ez
z
z
z
z
zzH
nTuenTatuetaThen
s
)/(
s
)/(
)s(s
ssHa(t)
)s(ss
H(s)
sH(s)
55
55
11-
11
5
1
15
11)(
)()5
1
5
1()()()
5
1
5
1()(
5
5151
5
1
5)s(s1)(
5
1Solution
filterdigitalit toconvert5
1:T.FanalogGiven the
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Bilinear Transformation
See Text book
For the derivation
Of this equation
2tan
2
T
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Bilinear Transformation
Derivation of:
(1)
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Frequency Warping Effect
2
tan2
T
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Properties of Bilinear Transformation
0)(tan2
01
01
01
:havewe(3)and(2)From
)3(1
tan1
tan)arg(
)2(1
1
:bydefinedareangleandradiusthewhere:as
representmaywe(1),equationinPutting.2Let
1
11
21
22
22
for
forr
forr
forr
zand
)(
)(zr
erzz
jsT
/
j
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BILINEAR ZTRANSFORMATION(Contd.)
S-Plane
LHS RHSJ-axis
Z-Plane
Unit Circle z =1z = -1
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Steps of Bilinear Transformation
)(H
H(z)
2tan
2
T
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Example:
ssH )(
)(H
3249.0)1.0tan(2
tan
2tan
2
T
For T=2
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H
ssH )(
H(z)
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Example:
5.0
3
s
sH
1
1
1
12
z
z
Ts
5.0
1
12
3
1
1
z
z
T
zH
11
1
1
15.01
12
13
zz
z
T
zzH
Convert the following analog filter system function into a digital
IIR filter by means of the bilinear transformation method:
Solution
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Example (Contd.)
1
1
1
1
15.01
12
13
zz
z
T
zzH
11
1
15.012
13
zzT
zzH
1
1
25.05.0
2
13
zTT
zzH
1
1
1
1
1
1
5.02
25.0
15.02
13
z
z
z
T
T
T
zzH
s
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Example (Contd.)
2
ny
1z
nx
1z
33.0
1
1
33.01
12
z
zzH