Page 433 - Analog and Digital Filter Design

430 Analog and Digital Filter Design

(2K-1)n . (2K - 1)n
- sin . Sld1 I/ + J cos . COSl? u
2n 211
1. 1
Where n = - .smh-' - and K = 1,2, . . . , n
n E
The filter order is given by n, and E depends on the passband ripple.

E = -1 where R is the ripple in decibels (dB).
Pole locations for the Inverse Chebyshev response are based on the Chebyshev
response. There is no passband ripple, but the value for E can be found from the
stopband attenuation:

1
4- where A is the stopband attenuation (dB).

For example, if A = 20dB:

m-m-
E=-- 1 1 - 0.1005

Using this and the Chebyshev pole-locating formulae, the pole locations can be
found. These can be expressed in the form:

-0; k jm, where i is an integer from 1 to n/2.

These can now be transformed to give Inverse Chebyshev poles using the fol-
lowing equations:

This gives us the natural pole locations where the stopband equals lrads. A
more practical normalized response with a 3dB passband cutoff point can be
obtained by modifying these values. The 3 dB frequency is given by:

1
m3dB =
cosh c -cosh-t(C,,))
c,, = Jm

For example, if A = 20dB, C,, = 4% = 9.499 and n = 3.

m3dB = l/cosh(0.99605) = 0.65 rad/s.