Signals and Systems - Electrical Engineering

6.5 Analog Filtering 399

where the last term is the definition of the Chebyshev polynomial for

′

hp>1. Thus, we get

hp=pcosh

[

1

N

cosh−^1

(

1

ε

)]

(6.52)

Factorization
The factorization of the magnitude-squared function is a lot more complicated for the Cheby-
shev filter than for the Butterworth filter. If we let the normalized variableS=s/pequalj′, the
magnitude-squared function can be written as

H(S)H(−S)=

1

1 +ε^2 C^2 N(S/j)

=

1

D(S)D(−S)

As before in the Butterworth case, the poles in the left-hands-plane givesH(S)= 1 /D(S), a stable
filter.

The poles of theH(S)can be found to be in an ellipse. They can be connected with the poles of the
corresponding order Butterworth filter by an algorithm due to Professor Ernst Guillemin. The poles
ofH(S)are given by the following equations fork=1,...,N, withNthe minimal order of the filter:

a=

1

N

sinh−^1

(

1

ε

)

σk=−sinh(a)cos(ψk) (real part)

′k=±cosh(a)sin(ψk) (imaginary part) (6.53)

where 0≤ψk< π/2 (refer to Equation 6.39) are the angles corresponding to the Butterworth filters
(measured with respect to the negative real axis of thes-plane).

Remarks

n The dc gain of the Chebyshev filter is not easy to determine as in the Butterworth filter, as it depends on
the order N. We can, however, set the desired dc value by choosing the appropriate value of a gain K so
thatHˆ(S)=K/D(S)satisfies the dc gain specification.
n The poles of the Chebyshev filter depend now on the ripple factorεand so there is no simple way to find
them as it was in the case of the Butterworth.
n The final step is to replace the normalized variable S=s/pin H(S)to get the desired filter H(s).

nExample 6.8

Consider the low-pass filtering of an analog signalx(t)=[−2 cos( 5 t)+cos( 10 t)+4 sin( 20 t)]u(t)

with MATLAB. The filter is a third-order low-pass Butterworth filter with a half-power frequency

hp=5 rad/sec—that is, we wish to attenuate the frequency components of the frequencies

10 and 20 rad/sec. Design the desired filter and show how to do the filtering.

The design of the filter is done using the MATLAB functionbutterwhere besides the specification of

the desired order,N=3, and half-power frequency,hp=5 rad/sec, we also need to indicate that