Chebyshev Filters
Spring 2008
© Ammar Abu-Hudrouss -Islamic ١ University Gaza
Chebychev I filter
It has ripples in the passband and no ripple in the stop band It has the following transfer function
2 1 H () 2 2 1 CN
th CN(x) is N order Chebyshev polynomial defined as
1 cos( N cos x) x 1 C x N 1 cosh( N cosh x) x 1
The Chebyshev polynomial can be generated by recursive technique
C0 1 and C1
Ck1 2Ck Ck 1
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١ Chebyshev polynomial
Chebyshev polynomial characteristics
CN 1 for all 1
CN 1 1 forall N
All theroots of CN occurs in theinterval 1 x 1
The filter parameter is related to the ripple as shown in the figures
1 N odd 2 H 0 1 N even 1 2
2 1 H 1 1 2
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Filter roots
The poles of LP prototype filter H (p ) are given by
pk sinh2 sink j cosh2 cosk k 1,2,3,, N
1 1 1 2 sinh N 2k 1 k 2N And the order of the filter is given by
A /10 cosh 1 10As /10 1/10 p 1 N 1 cosh s
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٢ Chebyshev filter type I
Example: Design lowpass Chebyshev filter with maximum gain of 5 dB. Ap = 0.5-dB ripple in the passband, a bandwidth of 2500 rad/sec. A stopband frequency of 12.5 Krad/sec, and As = 30 dB or more for > s
p = 2500 rad/s , s = 12.5 Krad/s,
For the LP prototype filter As = 30 dB, Ap = 0.5 dB
p = p / p = 1 rad/s
s = s / p = 5 rad/s To determine the order of the filter cosh 1 103 1/100.05 1 N 2.2676 3 cosh 1(5) To calculate the ripple factor 10log(1 2 ) 0.5 0.34931 Digital Signal Processing ٥ Slide
Chebyshev filter type I
1 1 1 2 sinh 0.591378 2 0.34931
pk 0.313228 j1.02192 and 0.626456 The transfer function H Hp 0 p 2 0.626456p 1.142447p 0.62645
To determine H0 we know that H (p = 0) = 1 (N is odd)
H 0 1 H 0.715693 1.1424470.62645 0
And to have 5 dB gain we multiply H0 by 1.7783 0.7156931.7783 H p p2 0.626456p 1.142447p 0.62645
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٣ Chebyshev filter type I
When we substitute by p =s/2500
19.8861012 Hs s2 1566s 714106 s 1566
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Chebyshev filter type II
This class of filters is called inverse Chebyshev filters. Its transfer function is derived by applying the following 1) Frequency transformation of ’ = 1/ in H (j ) of lowpass normalized prototype filter to achieve the following highpass filter 2 1 1 H 2 2 j' 1 Cn 1/ '
2) When it subtracted from one we get transfer function of Chebyshev II filter 1 1 1 2 2 1 1 Cn 1/ ' 1 2 2 Cn 1/ '
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٤ Chebyshev filter type II
It has ripples in the passband and ripple in the stop band It has the following transfer function
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Chebyshev filter type II Design
1. The frequencies are normalised by s instead of p.
2. The ripple factor i is calculated by
0.1As i 1/ 10 1
2 . The order of the filter is given by
A /10 cosh 1 10 As /10 1/10 p 1 N 1 cosh 1/ p 3. The poles of LP prototype filter H (p) are given by
pk 1/ sinh2 sin k j cosh2 cosk k 1,2,3,, N
1 1 1 2k 1 2 sinh N k 2N
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٥ Chebyshev filter type II Design
5. The zeros of H ( p ) is given by
zk 0k j seck k 1,2,3,,m N / 2
The filter transfer function is calculated by
m H p 2 2 0 k 1 0k H p n p p k 1 k
6. Restore the magnitude scale by calculating H0. 7. Restore the frequency scale to by frequency transformation
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Chebyshev filter type II
Example: Design lowpass Chebyshev II filter that has 0-dB ripple in the passband, p = 1000 rad/sec, Ap = 0.5dB, A stopband frequency of 2000 rad/sec and As = 40 dB.
p = 1000 rad/s , s = 2000 krad/s, For the LP prototype filter
p = p / s = 0.5 rad/s
s = s / s = 1 rad/s 10log(1 2 ) 40 1/ 99.995 To determine the order of the filter cosh 1 104 1/100.05 1 N 4.82 5 cosh 1(2)
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٦ Chebyshev filter type II
1 sinh 199.995 1.05965847. 2 5
pk 0.155955926 j0.6108703175,- 0.524799485 j0.485389011, and - 0.7877702666 The zeros is calculated by
zk j0k j seck k 1,2,,m n / 2
z1 j1.0515 z2 j1.7013
The transfer function H p2 1.05152 p 2 1.70132 Hp 0 p 2 0.3118311852p 0.3974722176 p 2 1.04959897p 0.5110169847p 0.787702666
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Chebyshev filter type II H p 4 4.04p 2 3.2002 Hp 0 p5 2.1491328p4 2.30818905p3 1.54997p 2 0.65725515p 0.15999426
To determine H0 we know that H (p = 0) = 1 (N is odd) 3.2002H 1 0 H 0.049995 0.15999426 0
H (s) can be achieved by substituting by p =s/2000
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Elliptic filter is more efficient than Butterworth and Chebychev filters as it has the smallest order for a given set of specifications The drawback it suffers from high nonlinearity in the phase, especially near the band edges.
The filter order requires to achieve a given set of specifications in passband ripple 1 and 2
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