Page 419 - Analog and Digital Filter Design

4 1 6 Analog and Digital Filter Design

(2R - 1)~
aR =sin[ 2,~ ] R = 1,2,3,. . . n

R = 1,2,3,. . .n
201
c, =-
1-6
R = 2,3,4,. . .n

Chebyshev Filter Response

Attenuation of Chebyshev filters is more difficult to calculate than for
Butterworth filters. The following expression is used.

A = I0.10g(l+E2Ctz2(Q)) dB
E = m, Ap is the passband ripple in decibels (e.g., 0.1 dB)
where
Cn(Q) is the Chebyshev polynomial and can be found from the
equation:

c,,,, (Q) = 2WC, (Q) - C,,-I (Q>
Co(Q) = 1 and C,(Q) = 0, hence a table can be built up:
C2(Q)=2w2 -1
c3 (Q) = 403 - 30
C,(Q) = 80," -8w' +1

The Chebyshev polynomial can be reproduced using this iterative process, but
there exists an alternative--an entirely equivalent solution:

Up to the ripple bandwidth, Cn(Q) = cos(uz.cos-~R).

Beyond the ripple bandwidth, Ctz(Q) = cos h(n.cosh-'Q).

The ratio of cutoff frequency to stopband edge is represented by the symbol Q.
When a 3dB cutoff frequency is required, R must be multiplied by a function
to give the correct results.

(31
Q(3dB) =Q.cosh -.cosh-l