Page 434 - Analog and Digital Filter Design

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Design Equations

To make wjdB = 1 radis, the pole locations must be scaied by 110.65 or 1.53846.
The following formula incorporates this scaling factor.

(
-G! .COS17 -.cosh-'C,7 W, .cash -.cosh-'C,,
Pole, = 1: 1 kj 1:
a,? +w,2 6,- +w,-

Tables 3.15, 3.17, and 3.19, in Chapter 3, show the pole locations for the Inverse
Chebyshev response having a normalized 3 dB passband cutoff and 20 dB, 30
dB, and 40dB stopband attenuation.

To find the filter order required, use the following equation:

cos 128 (c,z 1
12 = . where C2 is the ratio of stopband to passband.
cosh-'(n) '

For example, if the 3dB point is at lOHz and the stopband begins at IjHz,
L2 = 1.5. If 20dB of stopband attenuation is required, Cn = 9.95. This gives
IZ = 2.988/0.9624 = 3.1; the filter order must be four or more.

Inverse Chebyshev Zeroes
The zero frequency locations for any order of Inverse Chebyshev filter were
given by equations in Chapter 2 and are repeated below. Zero locations are given
as pn since Z, = aK + and the real part aK = 0. Applying the equations pro-
duces both positive and negative frequencies, but only the positive frequencies
are used. The proof for finding the equations is given in Huelsman'.

k = 1,2,. . . , IZ

Inverse Chebyshev zero locations found using these equations should be used
with pole locations for the natural (normalized to stopband) response. The
Inverse Chebyshev response can be normalized to have a 3 dB passband atten-
uation. The zero locations for this response can be found by modifying these
values. Previously, I showed that the poles moved away from the origin by a
frequency-scaling factor: