DSP-IIR数字滤波器设计算法_Chebyshev

fs 20kHz
f pass 5kHz, Apass 0.5dB, fstop 4kHz, Astop 10dB
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Nexact 3.37 N 4
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0.4405 1 z1 2
0.1618 1 z1 2
H
z
0.9441 1
G0
0 0 1
a01
0 0
1 1
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Example
fs 20kHz
f pass 4kHz, Apass 0.5dB, fstop 5kHz, Astop 10dB
dB
0 -2 -4 -6 -8 -10 -12
0
0 -2 -4 -6 -8 -10 -12
0
Given analog filter spicifications
Analog lowpass Chebyshev2 filter design
H s H0 s H1 sL HK s
Hi
s
1 1
s zi s si
1
1
s zi*
s si*
1, if N 2K
H0
s
0 0
s
,
if
N 2K 1
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Digital lowpass Chebyshev2 filter design
Hi
z
Hi
s
s
1 1
z z
1 1
1
Gi 1 ai1z 1
z 1 2 ai2 z 2
i 1, 2,L K
1
1
2 pass
, if
H0
z
H0
s
G s
1 1
z z
1 1
0
1 z 1
1 a01z 1
, if
N 2K N 2K 1
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Gi
1
20
1 201 cosi 02
1 i2
ai3
4c
1
01 cosi 02 i2 201 cosi 02 i2
, ai4
1 201 cosi 1 201 cosi
02 02
i2 i2
i pass sini ,i 1, 2,L K 0 pass sinh
G0
0 0 1
H s H0 s H1 sL HK s
H z H0 z H1 zL HK z
Hi
z
Hi
s
s
1 1
z z
1 1
1
Gi 1 ai1z 1
z 1 2 ai2 z 2
i 1, 2,L K
1
1
2 pass
, if
H0
z
H0
s
G s
1 1
z z
1 1
0
1 z 1
H0
s
0 0
s源自文库
,
if
N 2K 1
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Digital bandstop Chebyshev2 filter 设计公式
H s H0 s H1 sL HK s H z H0 z H1 zL HK z
Hi
z
Hi
s
s
1 2 cz 1
1 z 2
z
2
Gi 1 bi1z 1 bi2 z 2 bi3z 3 z 4 1 ai1z 1 ai2 z 2 ai3z 2 ai4z 4
CN
x
cos
cosh
N cos1 x ,if x 1 N cosh1 x ,if x 1
sinh(x)=sin(-jx)=(exp(x)-exp(-x))/2 cosh(x)=cos(-jx)=(exp(x)+exp(-x))/2
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Analog Lowpass Chebyshev Filter Design
1 a01z 1
, if
N 2K 1
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Gi
1
2
1 0
1 i2
cos i
2 0
i2
, bi1
21 1
i2 i2
ai1
2
1
2 0
i2
1
2
1 0
cos i
2 0
i2
ai 2
1 1
2
1 0
2
1 0
cos i cos i
2 0
02
i2
2 i
i pass sini ,i 1, 2,L K 0 pass sinh
模拟低通滤波器设计：Poles of Chebyshev1 H(s)
• Type 1
H 2
1
1
C 2
2
pass N
pass
1
C 2 2
pass N
s j pass
0
si pass sinh cosi j pass cosh sin i
si* pass sinh cosi j pass cosh sin i
0.0526 z 1
0.7095z 2
1
5843z 1
0.2314 z 2
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Zeros and Poles of Chebyshev2 H(s)
• Type 2
H 2
C
2 N
stop
C
2 N
stop
2 stop
CN2
jstop s
0
zi
jstop
sin i
2 0
i2
cos i
2 0
i2
ai1
1
2
2 0
2 i
20 cosi
1
2 0
i2
ai 2
1 20 cosi 1 20 cosi
2 0
2 0
2 i
i2
i pass sini ,i 1, 2,L K
0 pass sinh
G0
0 0 1
a01
0 0
1 1
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Example
1 a01z 1
, if
N 2K N 2K 1
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Gi
1
20
2 0
2 i
cos i
2 0
2 i
ai1
2
2 0
2 i
1
1 20 cos i
2 0
2 i
ai 2
1 20 cosi 1 20 cos i
2 0
2 0
2 i
2 i
i pass sini ,i 1, 2,L K
• Type 2: the magnitude response is characterized by the filter order N
H 2
CN2
stop
CN2
stop
2 stop
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根据性能指标，推导N=?
• Type 1
H 2
1
1
C 2
2
pass N
pass
H
stop
2
1
1
C 2
2
pass N
stop pass
1
1
2 stop
ln e Nexat
ln w
e2 1 w2 1
e stop , w stop
pass
pass
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根据性能指标，推导N=?
• Type 2
H 2
C
2 N
Analog Lowpass Chebyshev Filter Design
• Type 1: the magnitude response is characterized
by the filter order N
H 2
1
1
C 2
2
pass N
pass
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Chebyshev polynomial 多项式
, if
N 2K 1
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Gi
1
201
1 i2
cosi 02
i2 , bi1
4ci2 1 i2
bi3, bi2
2
2c2 1 i2 1 1 i2
ai1
4c 01 cosi 02 i2 1 201 cosi 02 i2
, ai2
2 2c2 1 02 i2
H s H0 s H1 sL HK s
Bilinear Transformation
H z H0 z H1 zL HK z
Hi z Hi s s f z , i 1, 2,L K
H0 z H0 s s f z
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Digital lowpass Chebyshev1 filter 设计公式
1
2
3
4
5
6
7
8
9
10
f in kHz
Desired digital filter spicifications
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
w in pi
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dB
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Nexact 3.37 N 4
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0.7612 1 0.1580z1 z2 0.5125 11.4890z1 z2
H s H0 s H1 sL HK s
H z H0 z H1 zL HK z
Hi
z
Hi
s
s
1 1
z 1 z 1
Gi 1 bi1z 1 z 2 1 ai1z 1 ai2 z 2
i 1, 2,L K
1, if
N 2K
H0
z
H0
s
s
1 1
z z
1 1
G0 1 z 1
CN2
jstop s
2 stop
0
si
sinh
stop
cosi j cosh sini
, si*
i 1, 2,L K N 2K, N 2K 1
s0
stop
sinh
,N
2K
1
1 ln N
stop
2 stop
1
i
N 1 2i , i 1, 2,L
2N
K
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, zi*
stop
sin i
i
1, 2,L
K
N 2K, N 2K 1
i
N 1 2i , i 1, 2,L
2N
K
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Zeros and Poles of Chebyshev2 H(s)
• Type 2
H 2
C
2 N
stop
C
2 N
stop
2 stop
Nexact 3.4 N 4
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Analog bandstop Chebyshev2 filter design
H s H0 s H1 sL HK s
Hi
s
1 1
s zi s si
1
1
s zi*
s si*
1,if N 2K 1
0
Given analog filter spicifications
1
2
3
4
5
6
7
8
9
10
f in kHz
Desired digital filter spicifications
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
w in pi
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dB
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Nexact 3.37 N 4
H z 1 0.0615z1 0.7043z2 1 0.5653z1 0.2228z2
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Digital bandpass Chebyshev2 filter 设计公式
H s H0 s H1 sL HK s H z H0 z H1 zL HK z
Hi
z
Hi
s
s
1 2 cz 1
1 z 2
z
2
Gi 1 bi1z 1 bi2 z 2 bi3z 3 z 4 1 ai1z 1 ai2 z 2 ai3z 2 ai4z 4
i 1, 2,L K
1, if
N 2K
H0
z
H0
s
s
1 2 cz 1
1 z 2
z
2
G0 1 z 2 1 a01z 1 a02 z 2
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0.3091 1 z1 2
0.1043 1 z1 2
H
z
0.9441 1
0.483z1
0.7194z2
1
0.9004z1
0.3177z2
Digital highpass Chebyshev1 filter design
H s H0 s H1 sL HK s
H z H0 z H1 zL HK z
0 pass sinh
G0
0 0 1
a01
0 0
1 1
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Example
fs 20kHz
f pass 4kHz, Apass 0.5dB, fstop 5kHz, Astop 10dB
dB
0 -2 -4 -6 -8 -10 -12
0
0 -2 -4 -6 -8 -10 -12
i 1, 2,L K
N 2K, N 2K 1
s0 pass sinh, N 2K 1
1 N
ln
1
pass
1
2 pass
1
i
N 1 2i , i 1, 2,L
2N
K
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Each analog SOS will be transformed into a SOS of the digital filter by bilinear transformation
stop
C
2 N
stop
2 stop
2
H pass
C
2 N
stop pass
C
2 N
stop pass
2 stop
1
1
2 pass
ln e Nexat
ln w
e2 1 w2 1
e stop , w stop
pass
pass
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a01
2c 0 1 , a02
1 0 1 0
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Example
fs 20kHz
f pa 2kHz, f pb 4kHz, Apass 0.5dB,
fsa 1.5kHz, fsb 4.5kHz, Astop 10dB
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