Signals and Systems - Electrical Engineering

668 C H A P T E R 11: Introduction to the Design of Discrete Filters

n Like in the discrete Butterworth, for Chebyshev filters the dc gain (i.e., gain atω= 0 ) can be set to any

desired value by allowing a constant gain G in the numerator such that

|HN(ej^0 )|=|HN( 1 )|=G

|N( 1 )|

|D( 1 )|

=desired gain (11.40)

n MATLAB provides two functions to design Chebyshev filters. The functioncheby1is for designing the

filters covered in this section, whilecheby2is to design filters with a flat response in the passband and

with ripples in the stopband. The order of the filter is found usingcheb1ordandcheb2ord. The functions

cheby1andcheby2will give the filter coefficients.

nExample 11.9

Consider the design of two low-pass Chebyshev filters. The specifications for the first filter are:

α(ej^0 )=0 dB

ωp=0.47πrad αmax=2 dB

ωst=0.6πrad αmin=6 dB

For the second filter, letωp=0.48πrad and keep the other specifications. Determine the half-

power frequency of the two filters. Use MATLAB for the design.

Solution

We obtained in Example 11.7 a third-order Butterworth low-pass filter that satisfies the specifica-

tions of the first filter. According to the results in this example a second-order Chebyshev filter

satisfies the same specifications. It is always so that a Chebyshev filter satisfies the same specifica-

tions as a Butterworth with a lower order. For the second filter we narrow the transition band by

0.01πradians, and so the order increases by one. The following is the script for the design of the

two filters.

%%%%%%%%%%%%%%%%%%%%%%%%%

% Example 11.9---LP Chebyshev

%%%%%%%%%%%%%%%%%%%%%%%%%

alphamax = 2; alphamin = 9; % loss specs

figure(1)

for i = 1:2,

wp = 0.47 + (i-1)∗0.01; ws = 0.6; % normalized frequency specs

[N,wn] = cheb1ord(wp,ws,alphamax,alphamin)

[b,a] = cheby1(N,alphamax,wn);

wp = wp∗pi;

% magnitude and phase

[H,w] = freqz(b,a); w = w/pi; M = length(H);H = H/H(1);

% to verify specs

spec0 = zeros(1,M); spec1 = alphamax∗ones(1,M)∗(-1)ˆ(N+1);

spec2 = alphamin∗ones(1,M);

alpha = -20∗log10(abs(H));