Magnetism (3, Review). Like poles repel Unlike poles attract.
P21 Summary IIR Filters4 Analog filters are stable when their poles are in the left half of the...
Transcript of P21 Summary IIR Filters4 Analog filters are stable when their poles are in the left half of the...
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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
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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
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Analog Butterworth Filters
Define : Lowpass Butterworth Filters (Nth order)
N
N
p
jH
or
jH
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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
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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¢
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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
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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.
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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
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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¢
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The ripple amplitude in dB is given by :
÷øö
çèæ+
-= 210 11
log10e
g dB
( )210 1log10 e+=
Fig: Chebyshev prototype low pass response
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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¢
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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
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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.
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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¢
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• 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.
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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.
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• 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,
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5.02
5.01
5.11
5.05.00
322
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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
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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 =¢
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Summary of Bilinear Transformation method1
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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
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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)
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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)
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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)
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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)
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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)
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Design Example
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References
1. “Digital Signal Processing – A Practical Approach” -Emmanuel C. Ifeachor and Barrie W. Jervis Second Edition