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![Page 1: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/1.jpg)
Discrete-Time IIR Filter Design from Continuous-Time Filters
Quote of the DayExperience is the name everyone gives to their
mistakes. Oscar Wilde
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall Inc.
![Page 2: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/2.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 2
Filter Design Techniques• Any discrete-time system that modifies certain frequencies• Frequency-selective filters pass only certain frequencies• Filter Design Steps
– Specification• Problem or application specific
– Approximation of specification with a discrete-time system• Our focus is to go from spec 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/Cxc(t) yr(t)C/D H(ej) nx ny
T/jHeH cj
![Page 3: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/3.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 3
Filter Specifications• Specifications
– Passband
– Stopband
• Parameters
• Specs in dB– Ideal passband gain =20log(1) = 0 dB– Max passband gain = 20log(1.01) = 0.086dB– Max stopband gain = 20log(0.001) = -60 dB
200020 01.1jH99.0 eff
30002 001.0jHeff
30002
20002
001.0
01.0
s
p
2
1
![Page 4: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/4.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 4
Butterworth Lowpass Filters• Passband is designed to be maximally flat• The magnitude-squared function is of the form
N2
c
2
cj/j1
1jH
N2c
2
cj/s1
1sH
1-0,1,...,2Nkfor ej1s 1Nk2N2/jcc
N2/1k
![Page 5: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/5.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 5
Chebyshev Filters• Equiripple in the passband and monotonic in the stopband• Or equiripple in the stopband and monotonic in the passband
xcosNcosxV /V1
1jH 1
Nc
2N
2
2
c
![Page 6: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/6.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 6
Filter Design by Impulse Invariance• Remember impulse invariance
– Mapping a continuous-time impulse response to discrete-time– Mapping a continuous-time frequency response to discrete-time
• If the continuous-time filter is bandlimited to
• If we start from discrete-time specifications Td cancels out– Start with discrete-time spec in terms of – Go to continuous-time = /T and design continuous-time filter– Use impulse invariance to map it back to discrete-time = T
• Works best for bandlimited filters due to possible aliasing
dcd nThTnh
k ddc
j kT2
jT
jHeH
T
jHeH
T/ 0jH
dc
j
dc
![Page 7: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/7.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 7
Impulse Invariance of System Functions • Develop impulse invariance relation between system functions• Partial fraction expansion of transfer function
• Corresponding impulse response
• Impulse response of discrete-time filter
• System function
• Pole s=sk in s-domain transform into pole at
N
1k k
kc ss
AsH
0t0
0teAth
N
1k
tsk
c
k
N
1k
nTskd
N
1k
nTskddcd nueATnueATnThTnh dkdk
N
1k1Ts
kd
ze1AT
zHdk
dkTse
![Page 8: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/8.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 8
Example• Impulse invariance applied to Butterworth
• Since sampling rate Td cancels out we can assume Td=1• Map spec to continuous time
• Butterworth filter is monotonic so spec will be satisfied if
• Determine N and c to satisfy these conditions
3.0 0.17783eH
2.00 1eH89125.0
j
j
3.0 0.17783jH
2.00 1jH89125.0
0.17783 3.0jH and 89125.0 2.0jH cc
N2
c
2
cj/j1
1jH
![Page 9: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/9.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 9
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 17783.013.0
1 and 89125.0
12.01
70474.0 and 68858.5N c
0,1,...,11kfor ej1s 11k212/jcc
12/1k
21
1
21
1
21
1
z257.0z9972.01z6303.08557.1
z3699.0z0691.11z1455.11428.2
z6949.0z2971.11z4466.02871.0
zH
4945.0s3585.1s4945.0s9945.0s4945.0s364.0s12093.0
sH 222
![Page 10: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/10.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 10
Example Cont’d
![Page 11: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/11.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 11
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 Td 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
d z1z1
T2
s
1
1
dc z1
z1T2
HzH
2/Tj2/T1
2/Tj2/T1s2/T1s2/T1
zdd
dd
d
d
js
![Page 12: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/12.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 12
Bilinear Transformation• On the unit circle the transform becomes
• To derive the relation between and
• Which yields
j
d
d e2/Tj12/Tj1
z
2tan
Tj2
2/cose22/sinje2
T2
je1e1
T2
sd
2/j
2/j
dj
j
d
2T
arctan2or 2
tanT2 d
d
![Page 13: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/13.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 13
Bilinear Transformation
![Page 14: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/14.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 14
Example• Bilinear transform applied to Butterworth
• Apply bilinear transformation to specifications
• We can assume Td=1 and apply the specifications to
• To get
3.0 0.17783eH
2.00 1eH89125.0
j
j
23.0
tanT2
0.17783jH
22.0
tanT2
0 1jH89125.0
d
d
N2
c
2
c/1
1jH
2N2
c
2N2
c 17783.0115.0tan2
1 and 89125.0
11.0tan21
![Page 15: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/15.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 15
Example Cont’d
• Solve N and c
• The resulting transfer function has the following poles
• Resulting in
• Applying the bilinear transform yields
6305.51.0tan15.0tanlog2
189125.0
11
17783.01
log
N
22
766.0c
0,1,...,11kfor ej1s 11k212/jcc
12/1k
5871.0s4802.1s5871.0s0836.1s5871.0s3996.0s20238.0
sH 222c
21
2121
61
z2155.0z9044.011
z3583.0z0106.11z7051.0z2686.11z10007378.0
zH
![Page 16: Discrete-Time IIR Filter Design from Continuous-Time Filters Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content.](https://reader036.fdocuments.us/reader036/viewer/2022082709/56649d755503460f94a56338/html5/thumbnails/16.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 16
Example Cont’d