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1
SUMMARY OF IIR FILTER THEORY
Yogananda Isukapalli
Analog Filter Specifications
w’p w’
s w’
Analog Frequency
AmplItude
ds
0
1+dp1
1-dp
|)(| w¢jH
Pass band Stop band
,for 1)(1 ppp jH wwdwd ¢£¢+£¢£-
,for )( ¥£¢£¢£¢ wwdw ssjH
,)(log20
,)1(log20
10
10
dBdB
ss
pp
da
da
-=
--=
where ap and as are the peak pass band ripple and minimum stop band attenuation in dB respectively
2
3
Analog Filterx(t) y(t)
åå==
+=M
kk
k
k
N
kk
k
k dttyd
bdttxd
aty00
)()()(
å
å
=
=
-= M
i
ii
N
i
ii
sb
sasH
0
0
1)(
Transfer function
0 0t
1
0
1
1
1
10
0)(
1)(
:)()(
³
<
-
=
-=
=
t
tbe
ba
th
sbasH
ExamplesHLth
Impulse response
4
Analog filters are stable when their poles are in the left half of the s-plane, where as digital filters are stable when their poles are confined within the unit circle.
z- plane Im(s)
Re(s)
S- plane Im(z)
Re(z)
Stable Stable
Analog Filter : Magnitude
Function :
Magnitude Square Function :
)()(|)()( www w ¢<¢==¢ ¢= jHjHsHjH js
ww ¢=-=¢ jssHsHjH |)()()( 2
Stability
5
Analog Butterworth Filters
Define : Lowpass Butterworth Filters (Nth order)
N
N
p
jH
or
jH
22
2
2
)(11|)(|
)(1
1|)(|
ww
www
¢+=¢
¢¢
+=¢
N > 50
N =1N =2
0
0.5
1.0
|H(j )|
w¢pw¢
w¢
( normalized prototype, =1)pw¢
Maximally flat response
6
Poles of the Lowpass Prototype (Normalized = 1)
Butterworth Filter
Magnitude Square Function :
N
jssN
N
ssHsH
sHsH
jHp
)(11)()(
11)()(
11|)(|
2
)(2
22
22
1
-+=-
¢+=-
¢+=¢
¢=-=¢
=¢
www
ww
w
pw¢
7
Roots :
b. N is even :
1 + s2N = 0 s2N = -1
Roots :
12,.........2,1,0.1.1
)()22(
-===Nkees N
kjN
kj
k
pp
12,.........2,1,0.1
)22(
-==
+
Nkes N
kj
k
pp
Poles: Solve: 1 + (-s2)N = 0
a. N is odd 1 - s2N = 0
s2N = 1
8
Poles of a normalized ( = 1) Butterworth (LPF) filter lie uniformly spaced on a unit circle in the s-plane at the following locations:
pw¢
NkNNkj
NNkes NNkj
k
,...,2,1 2
)12(sin
2)12(
cos2/)12(
=
úûù
êëé -+
+úûù
êëé -+
== -+ ppp
The poles occur in complex conjugate pairs and lie on the left-hand side of the s-plane.
9
Example: N = 3
s0 = 1, s1 = 1.ejp/3, s2 = 1.ej2p/3
s3 = 1.ejp, s4 = 1.ej4p/3, s5 = 1.ej5p/3
Im(s)
Re(s)
|sk|=1
1221)( 23 +++
=sss
sH
÷÷ø
öççè
梢
÷÷÷
ø
ö
ççç
è
æ
-
-
³
p
s
A
A
p
s
N
wwlog2
110
110log10
10
Butterworth Filter order N
10
Analog Chebyshev Filters
)(11|)(|
)/(1|)(|
:
222
222
wew
wwew
¢+=¢
¢¢+=¢
NLP
pN
CjH
CKjH
Define
p
Where :
CN( ) is the Nth order chebyshev polynomial.
= cos(Ncos-1 ) 0£ £1
= cosh(Ncosh-1 ) > 1
e is the ripple parameter 0 < e < 1 sets the ripple amplitude in the ripple passband 0£ e £1
( normalized prototype, pw¢ =1)
w¢
CN( )w¢
CN( )w¢
w¢w¢
w¢w¢
11
The ripple amplitude in dB is given by :
÷øö
çèæ+
-= 210 11
log10e
g dB
( )210 1log10 e+=
Fig: Chebyshev prototype low pass response
12
w¢ w¢ w¢w¢CN+1( ) = 2 CN( ) - CN-1( ) Also,
÷÷ø
öççè
梢
÷÷÷
ø
ö
ççç
è
æ
-
-
³-
-
p
s
A
A
p
s
N
ww1
10
101
cosh2
110
110coshChebyshev Filter order N
13Fig: Prototype Chebyshev denominator polynomials
,...,2,1 ,2
)12(
);1
(sinh1
where
),sin()cosh()cos()sinh(
1
NkNNk
N
js
k
kkk
=-+
=
=
+=
-
pb
ea
baba
Poles of a normalized ( = 1) Chebyshev LPFpw¢
14
Mapping analog filters to digital domain:
• Mapping differentials:
Backward: Tnynyny
dttdy
nTt
a ]1[][]][[)( )1( --=Ñ=
=
Forward:T
nynynydttdy
nTt
a ][]1[]][[)( )1( -+=Ñ=
=
)1(Ñ Backward or forward difference operator
Also, ]]][[[]][[ )1()1()( nyny kk -ÑÑ=Ñ
with ][(.))0( ny=Ñ
15
å
å
=
== N
k
kk
M
k
kk
a
sc
sdsH
0
0)(å
å
=
-=
-
-
-
=N
k
kk
M
k
kk
Tzc
Tzd
zH
0
10
1
]1[
]1[)(
Analog Filters Digital Filters
Mapping Functions
Tzs11 --
=Tzs 1-
=
Backward Difference Mapping
Forward Difference Mapping
• Both the mapping functions may produce satisfactory results only for extremely small sampling times and are generally not recommended.
16
Impulse Invariance Mapping:
åå=
-= -
=«-
=N
kTpk
N
k k
ka ze
czHpscsH
k1
11 1
)( )(
• Evaluation – Pole Mapping:
(a) The impulse invariant mapping maps poles from the s-plane’s j axis to the z-plane’s unit circle.
(b) All s-plane poles with negative real parts map to z-plane poles inside the unit circle. In other words, stable analog poles are mapped to stable digital filter poles.
(c) All poles on the right half of the s-plane, i.e. with positive real parts, map to digital poles outside the unit circle.
The impulse invariant mapping, thus, preserves the stability ofthe filter.
w¢
17
• Deficiency of the impulse-invariance mapping:If s-plane poles have the same real parts and imaginary parts
that differ by some integer multiples of 2p/T, then there are an infinite number of s-plane poles that map to the same location in the z-plane. This will result in aliased poles in the z-plane.
• Aliasing in digital frequency response:
....)2(1)(1)( +±¢+¢=¢
TjjH
TjH
TeH aa
Tj pwww
Digital frequency response reproduces the analog frequency response every 2p/T.To avoid aliasing for lower sampling rates, the pole mapping is restricted to the primary strip
TTpwp
£¢£-Thus each horizontal strip of the s-plane of width 2p/T is mappedonto the entire z-plane.
18
s-plane
z-plane
Re(z)
Im(z)|z|=1
Im(s)
Re(s)Tp
Tp
-
To prevent significant aliasing:
TjHa /for 0)( pww >¢»¢
Band limit the analog filter.
19
• Example of the impulse-invariance mapping:
Third order Butterworth low-pass filter’s transfer function:
1) s 1)(s(s1 H(s) 2 +++
=
866.0-0.5p ,866.0-0.5p ,1p:are poles threeThe
)866.05.0(s0.577e
)866.05.0(s0.577e
1)(s1
:as tscoefficien thegives Algebra)866.05.0(s
C)866.05.0(s
C1)(s
C
j0.866)0.5j0.866)(s- 0.5 1)(s(s1 H(s)
:isexpansion fraction partial The
321
j2.62j2.62-
321
jj
jj
jj
-=+=-=
+++
-++
+=
+++
-++
+=
++++=
20.
)866.0cos(2
))866.0cos(2(
)866.06/5cos(154.1
)866.06/5cos(154.1)866.0cos(2
H(z)
:as expressed and simplifiedfurther becan This
)e-(1)866.06/5cos(154.1z-
)e-(zz H(z)
get weg,Simplifyin)e-(1
1 )e-(1
1 )e-(1
1 H(z)
:as written becan function transfer theTherefore,
23
5.02
5.01
5.11
5.05.00
322
13
12
0
1j0.866)T(-0.5
5.02
T-
1j0.866)T(-0.51j0.866)T(-0.51T-
T
TT
TT
TT
TTT
T
eaTeeaTeea
TeebTeeTeb
whereazazaz
zbzb
zTe
zzz
-
--
--
--
---
-+
-
---+-
-=
+=
+-=
++=
+++-=
++++
=
+-+=
++=
p
p
p
21
11
2)(
-+
=zzTzH
ssH 1)( =
112
or 11
21
+-
=-+
=-
zz
Ts
zzTs
Bilinear Mapping:
• Evaluation – Pole Mapping:Stable analog filters are mapped from s-plane to z-plane as stable digital filters. Aliasing problem is eliminated.• Analog and Digital frequencies relationship for prewarping:
frequency Digital frequency Analog
®®¢
ww)2/tan( Tk ww =¢
22
Summary of Bilinear Transformation method1
23
Example 1:Low pass filter: Obtain the transfer function H(z) of the digital low pass filter to approximate the following transfer function:
121)(
2 ++=
sssH
Use Bilinear Transformation method and assume a 3 dB cut off frequency of 150Hz and a sampling frequency of 1.28kHz.
Soln. The critical frequency, and Fs =1/T = 1.28kHz, giving a prewarped critical frequency of
rad/s, 1502 ´= pw p
0.3857T/2)tan( ==¢ pp ww
24
Use of BZT and Classical Analog Filters
Analog Filters Review
N
p
jH2
2
)(1
1|)(|
www
¢¢
+=¢
• Low pass Butterworth filter of order N
Poles of a normalized ( = 1) Butterworth LPFpw¢
NkNNkj
NNkes NNkj
k
,...,2,1 2
)12(sin
2)12(
cos2/)12(
=
úûù
êëé -+
+úûù
êëé -+
== -+ ppp
÷÷ø
öççè
梢
÷÷÷
ø
ö
ççç
è
æ
-
-
³
p
s
A
A
p
s
N
wwlog2
110
110log10
10
Frequency response
Filter order N
(8.25)
(8.26)
(8.24)
25
Analog Filters Review contd
2. Low pass Chebyshev Type 1 Filter of order N
)/(1 |)(| 22
2
pNCKjH
wwew
¢¢+=¢Frequency
response
Poles of a normalized ( = 1) Chebyshev LPFpw¢
,...,2,1 ,2
)12(
);1
(sinh1
where
),sin()cosh()cos()sinh(
1
NkNNk
N
js
k
kkk
=-+
=
=
+=
-
pb
ea
baba
Filter order N
÷÷ø
öççè
梢
÷÷÷
ø
ö
ççç
è
æ
-
-
³-
-
p
s
A
A
p
s
N
ww1
10
101
cosh2
110
110cosh
(8.27)
(8.29)
(8.28)
26
Fig: Sketches of frequency response of some classical analog filters
(a) Butterworth response (b) Chebyshev type I © Chebyshev type II
(d) Elliptic
Analog Filters Review contd
3. Low pass Elliptic Filter of order N
)(1 |)(| 22
2
wew
¢+=¢
NGKjHFrequency
response(8.30)
27
Analog Frequency Transformations
1. Desired Low pass Low pass prototypeThe low pass to low pass transformation is:
p
ssw¢
=
Denote frequencies for
p
lpp
p
lpp jjww
www
w¢
=¢
= i.e.
lpw PrototypepwLPF and
From 8.21 (a)
(8.31)
28
2. Desired High pass Low pass prototypeThe low pass to high pass transformation is:
ss pw¢= From 8.21 (b)
Denote frequencies for hpw Prototype pwHPF and
hp
pp
hp
pp jjww
www
w¢
-=¢
-= i.e. (8.32)
29
30
Design Example
31
32
References
1. “Digital Signal Processing – A Practical Approach” -Emmanuel C. Ifeachor and Barrie W. Jervis Second Edition
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