Page 432 - Analog and Digital Filter Design

42
Design Equations

for the Butterworth response is at jcos(d6) or S = kj0.866, so multiplying this
by cosh v to give the Chebyshev response moves it to S = fjl ,1522 in the verti-
cal direction.

The pole locations for a third-order Chebyshev response with 0.15 dB ripple in
the passband are:

-0.8774
-0.4387 f jl. 1522

The Chebyshev pole locations produce a normalized frequency response
with attenuation equal to the ripple (0.15dR) at w = 1. The 3dB point will
have a frequency greater than w = 1. The magnitude of the pole locations given
for the Chebyshev response must now be reduced to correct for the 3 dB cutoff
point; they must each be divided by C3dB. Dividing by a constant factor (that
is greater than onej makes the pole positions move towards the origin of the
S-plane.

C,,, = cosh0.78614 = 1.32525

In other words, the 0.15 dB point occurs at w = 1, and the 3 dB point occurs at
o = 1.32525. So dividing the pole locations by CJdB gives:

-0.662 1

-0.33 10 -t. j0.8694.

All three poles are now within the unit circle, and the 3dB point occurs at
w= 1.

Inverse Chebyshev Pole and Zero Locations
As suggested by their name, Inverse Chebyshev filters are derived from
Chebyshev filters. The pole positions are the inverse of those given for
Chebyshev filters. The frequency response of Chebyshev filters was described in
Chapter 2. There are ripples in the passband with a smoothly decaying response
in the stopband. Inverting the pole positions produces a filter with a smooth
passband. The zeroes produce ripple in the stopband.

Pole locations for the Chebyshev response have been described earlier in the
previous subsection and are given by: