Page 101 - Analog and Digital Filter Design

98 Analog and Digital Filter Design

Chebyshev Pole locations
The Chebyshev response has ripple in its passband. This is because the trans-
fer function has poles that lie on an ellipse, rather than on a circle like the
Butterworth response. The positions of the poles are related to Butterworth
pole locations by hyperbolic trigonometric functions: sinh(x) and cosh(x).
In general terms, poles move away from the real axis by a constant multiplying
factor. They also move towards the imaginary axis by a different constant multi-
plying factor. This is shown in the S-plane diagram, in Figure 3.1 1.

Figure 3.1 1
Chebyshev Pole Locations I

Pole locations for the normalized Chebyshev response with a 3 dB cutoff point
are given in Tables 3.5 to 3.9. The passband ripple values used to produce these
tables are 0.01 dB, 0.1 dB, 0.25dB, OSdB, and 1 dB; these are the most popular
values. You may notice that in all these tables, a first-order response pole is
always real and positioned at -1.0. This should not be a great surprise since this
is the same for all responses.
To keep the purists happy, pole locations for the normalized Chebyshev response
with a “natural” cutoff frequency are given in Tables 3.10 to 3.14. If the
natural cutoff frequency is at w = 1, the 3dB attenuation frequency is at
(t 3
w = cosh -.cash-' - where E = and R is the passband ripple in dB
and n is the filter order. The 3dB attenuation frequency is always greater than
one, provided that the passband ripple is less than 3dB.