2. IIR as a class of LTI Filters
Discrete-time Filter is any discrete-time system that modifies certain
frequencies.
Frequency-selective filters pass only certain frequencies
The difference equation of IIR filters:
Transfer function:
To give an Infinite Impulse Response (IIR), a filter must be recursive, that is,
incorporate feedback N ≠ 0, M ≠ 0.
2
3. 3
Filter Design Techniques
Filter Design Steps:
Specification
• Problem or application specific
Approximation of specification with a discrete-time system
• Our focus is to go from specifications to discrete-time system
Implementation
• Realization of discrete-time systems depends on target technology
We already studied the use of discrete-time systems to implement a
continuous-time system.
If our specifications are given in continuous time we can use
D/C𝑥 𝑐 𝑡 𝑦𝑟(𝑡)C/D H(ej)
nx ny
/TjHeH c
j
5. Design of IIR Filters
A digital filter, , with infinite impulse response (IIR), can be designed
by first transforming it into a prototype analog filter and then design
this analog filter using a standard procedure.
Once the analog filter is properly designed, it is then mapped back to the
discrete-time domain to obtain a digital filter that meets the specifications.
5
)( jw
eH
)( jHc
6. Design of IIR Filters
The commonly used analog filters are
1. Butterworth filters: no ripples at all,
2. Chebyshev filters: ripples in the passband OR in the stopband, and
3. Elliptical filters: ripples in BOTH the pass and stop bands.
A disadvantage of IIR filters is that they usually have nonlinear phase.
Some minor signal distortion is a result.
There are two main techniques used to design IIR filters:
1. The Impulse Invariant method,
2. Matched z-transform method, and
3. The Bilinear transformation method.
6
7. Butterworth Low Pass Filters
Passband is designed to be maximally flat
The magnitude square response of a Butterworth filter of order N is
where is the 3-dB frequency of the filter.
The larger N is, the closer the Butterworth filter is to an ideal low pass filter.
7
N2
c
2
c
j/j1
1
jH
N2
c
2
c
j/s1
1
sH
c
8. 8
Chebyshev Filters
Equiripple in the pass-band and monotonic in the stop-band
The magnitude square response of a N-th order Chebyshev filter with a
ripple parameter of ε is
Where is the N-th order Chebyshev polynomial.
xNxVwhere
V
jH N
cN
c
1
22
2
coscos
/1
1
xVN
9. 9
Filter Design by Impulse Invariance
In this design method we want the digital filter impulse response to look “similar”
to that of a frequency-selective analog filter. Hence we sample ℎ 𝑎(𝑡) at some
sampling interval T to obtain h(n); that is,
Mapping a continuous-time impulse response to discrete-time
Mapping a continuous-time frequency response to discrete-time
The analog and digital frequencies are related by 𝜔 = Ω𝑇 𝑜𝑟 𝑒 𝑗𝜔 = 𝑒 𝑗Ω𝑇 𝑜𝑟 𝑧 = 𝑒 𝑠𝑇
If the continuous-time filter is band-limited to
nThnh c
k
c
j
k
T
j
T
jH
T
eH
21
1
d
c
j
T
jH
T
eH
dc TjH /for0
10. Design Procedure
Given the digital lowpass filter specifications 𝜔 𝑝 , 𝜔𝑠 , 𝑅 𝑝 , and 𝐴 𝑠 , we want to
determine 𝐻(𝑧) by first designing an equivalent analog filter and then mapping
it into the desired digital filter. The steps required for this procedure are:
1. Choose T and determine the analog frequencies
2. Design an analog filter 𝐻 𝑎(𝑠) using the specifications 𝜔 𝑝 , 𝜔𝑠 , 𝑅 𝑝 , and 𝐴 𝑠. This can
be done using any one of the three (Butterworth, Chebyshev, or elliptic) prototypes.
3. Using partial fraction expansion, expand 𝐻 𝑎(𝑠) into
4. Now transform analog poles {𝑝 𝑘} into digital poles {𝑒 𝑝 𝑘 𝑇
} to obtain the digital
filter:
10
11. 11
Impulse Invariant method: Steps
Sample the impulse response (quickly enough to avoid aliasing
problem)
Compute z-transform of resulting sequence
First order:
Second Order:
aT
ez
z
zH
as
sH
)(
1
)(
aTaT
aT
ebTezz
bTezz
zH
bas
as
sH 22
2
22
)cos(2
)cos(
)(
)(
)(
)(
aTaT
aT
ebTezz
bTez
zH
bas
b
sH 2222
)cos(2
)sin(
)(
)(
)(
12. 12
Summary of the Impulse Invariant Method
Advantage:
preserves the order and stability of the analogue filter.
The frequencies Ω and ω are linearly related.
Disadvantages:
There is distortion of the shape of frequency response due to aliasing.
13. Example(1)
If the analog filter transfer function is
Find the digital filter
Combine the second and third terms, we obtain
13
14. 14
Example(2)
Impulse invariance applied to Butterworth
Since sampling rate T cancels out we can assume T=1
Map spec to continuous time
Butterworth filter is monotonic so specifications will be satisfied if
Determine N and c to satisfy these conditions
3.00.17783eH
2.001eH89125.0
j
j
3.00.17783jH
2.001jH89125.0
0.177833.0jHand89125.02.0jH cc
N2
c
2
c
j/j1
1
jH
15. 15
Example Cont’d
Satisfy both constrains
Solve these equations to get
N must be an integer so we round it up to meet the spec
Poles of transfer function
The transfer function
Mapping to z-domain
2N2
c
2N2
c 17783.0
13.0
1and
89125.0
12.0
1
70474.0and68858.5N c
0,1,...,11kforej1s 11k212/j
cc
12/1
k
21
1
21
1
21
1
257.09972.01
6303.08557.1
3699.00691.11
1455.11428.2
6949.02971.11
4466.02871.0
zz
z
zz
z
zz
z
zH
4945.0s3585.1s4945.0s9945.0s4945.0s364.0s
12093.0
sH 222
17. Matched z-transform method
Matched z-Transform: very simple method to convert analog filter into digital
filters.
Poles and zeros are transformed according to
Where Td is the sampling period
Poles using this method are similar to impulse invariant method.
Zeros are located at a new position.
This method suffers from aliasing problem.
17
18. 18
Example of Impulse Invariant vs Matched z transform methods
Consider the following analog filter into a digital IIR filter
Impulse invariant method
Matched z-transform
Some poles but different zeros
19. 19
Filter Design by Bilinear Transformation
Get around the aliasing problem of impulse invariance
Map the entire s-plane onto the unit-circle in the z-plane
Nonlinear transformation
Frequency response subject to warping
Bilinear transformation
Transformed system function
Again T cancels out so we can ignore it
We can solve the transformation for z as
Maps the left-half s-plane into the inside of the unit-circle in z
Stable in one domain would stay in the other
1
1
1
12
z
z
T
s
1
1
1
12
z
z
T
HzH c
2/2/1
2/2/1
2/1
2/1
TjT
TjT
sT
sT
z
js
21. 21
Bilinear Transformation
On the unit circle the transform becomes
To derive the relation between and
Which yields
j
d
d
e
2/Tj1
2/Tj1
z
2
tan
2
2/cos2
2/sin22
1
12
2/
2/
T
j
e
je
T
j
e
e
T
s j
j
j
j
d
2
arctan2or
2
tan
2 T
T
23. Design Procedure
Given the digital lowpass filter specifications 𝜔 𝑝 , 𝜔𝑠 , 𝑅 𝑝 , and 𝐴 𝑠 , we want to
determine 𝐻(𝑧) by first designing an equivalent analog filter and then mapping
it into the desired digital filter. The steps required for this procedure are:
1. Choose a value for T. This is arbitrary, and we may set T = 1.
2. Prewarp the cutoff frequencies ω p and ω s ; that is, calculate Ω 𝑝 and Ω 𝑠 using.
3. Design an analog filter 𝐻 𝑎(𝑠) using the specifications 𝜔 𝑝 , 𝜔𝑠 , 𝑅 𝑝 , and 𝐴 𝑠. This can
be done using any one of the three (Butterworth, Chebyshev, or elliptic) prototypes.
4. finally, apply the bilinear transform to get the digital IIR filter.
23
24. Example: Application of Bilinear Transform
Design a first order low-pass digital filter with -3dB frequency of 1kHz
and a sampling frequency of 8kHz using a the first order analogue low-
pass filter
which has a gain of 1 (0dB) at zero frequency, and a gain of -3dB ( =
√0.5 ) at Ωc rad/sec (the "cutoff frequency ").
Solution:
First calculate the normalized digital cutoff frequency:
Calculate the equivalent pre-warped analogue filter cutoff frequency
(rad/sec)
24
25. Example: Application of Bilinear Transform
Thus, the analogue filter has the system function
Apply Bilinear transform
As a direct form implementation
25
26. Example: Magnitude Frequency Response
Note that the digital filter response at zero frequency equals 1, as for the
analogue filter, and the digital filter response at 𝜔 = 𝜋 equals 0, as for the
analogue filter at Ω = ∞. The –3dB frequency is 𝜔𝑐 = 𝜋/4, as intended.
26
27. Example: Pole-zero diagram for digital design
Note that:
The filter is stable, as expected
The design process has added an extra zero compared to the prototype
• This is typical of filters designed by the bilinear transform.
27