Page 431 - Analog and Digital Filter Design

428 Analog and Digital Filter Design

between their pole-locating formulae are expected. In fact, pole locations for the
Chebyshev response are given by:

(2 K - 1 ) ~ (2 K - 1)~
-sin .sinh n + jcos .cost4 v
2n 2n

1. 1
Where v = -.sin h-' -
n E
The filter order is given by n, and E depends on the passband ripple.

E = where R is the ripple in decibels (dB).
Correction is necessary if a 3dB cutoff point is required. The correction
factor is:

The 3 dB point occurs at a higher frequency than the natural cutoff point at the
ripple attenuation. To have a 3 dB cutoff point, the magnitude of each pole loca-
tion must be reduced by C3dB.

For example, suppose we have a third-order Chebyshev response with 0.15 dB
ripple in the passband.

-
R =0. 15, SO E = 4

E = = 0.1 875
1/~ 5.3344
=
1 1 1
v=-.sinli-'-=-.sinli-'5.3344
n E3
v = 2.37613 = 0.792
sinhv = 0.8774
COS^ = 1.3304

The real pole given by the equation for the Butterworth response is S = -1.0.
This moves towards the imaginary axis to give a Chebyshev pole location at
S = -0.8774, since the real part of the Butterworth pole location has to be
multiplied by sinh v.
The real part of the two imaginary poles for the Butterworth response is sin(d6)
or S = -0.5, so multiplying this by sinhv to give the Chebyshev response moves
it to S = -0.4387 in the horizontal direction. The imaginary part of these poles