Chebyshev Filters

Spring 2008

© Ammar Abu-Hudrouss -Islamic ١ University Gaza

Chebychev I filter

It has ripples in the passband and no in the stop band It has the following

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   

Ck1  2Ck Ck 1 

Digital Signal Processing ٢ Slide

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

Digital Signal Processing ٣ Slide

Filter roots

The poles of LP H (p ) are given by

pk  sinh2 sink  j cosh2 cosk  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

Digital Signal Processing ٤ Slide

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 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 Hp  0 p 2  0.626456p 1.142447p  0.62645

To determine H0 we know that H (p = 0) = 1 (N is odd)

H 0 1  H  0.715693 1.1424470.62645 0

And to have 5 dB gain we multiply H0 by 1.7783 0.7156931.7783 H p  p2  0.626456p 1.142447p  0.62645

Digital Signal Processing ٦ Slide

٣ Chebyshev filter type I

When we substitute by p =s/2500

19.8861012 Hs  s2 1566s  714106 s 1566

Digital Signal Processing ٧ Slide

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/ '

Digital Signal Processing ٨ Slide

٤ Chebyshev filter type II

It has ripples in the passband and ripple in the stop band It has the following transfer function

Digital Signal Processing ٩ Slide

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/ sinh2 sin k  j cosh2 cosk  k 1,2,3,, N

1 1 1  2k 1 2  sinh     N    k 2N

Digital Signal Processing ١٠ Slide

٥ Chebyshev filter type II Design

5. The zeros of H ( p ) is given by

zk  0k   j seck 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

Digital Signal Processing ١١ Slide

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)

Digital Signal Processing ١٢ Slide

٦ Chebyshev filter type II

1   sinh 199.995 1.05965847. 2 5

pk   0.155955926  j0.6108703175,- 0.524799485 j0.485389011, and - 0.7877702666 The zeros is calculated by

zk   j0k  j seck k 1,2,,m  n / 2

z1   j1.0515 z2   j1.7013

The transfer function H p2 1.05152 p 2 1.70132  Hp  0 p 2  0.3118311852p  0.3974722176 p 2 1.04959897p  0.5110169847p  0.787702666

Digital Signal Processing ١٣ Slide

Chebyshev filter type II H p 4  4.04p 2  3.2002 Hp  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

Digital Signal Processing ١٤ Slide

٧

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

Digital Signal Processing ١٥ Slide

٨