Signals and Systems - Electrical Engineering

6.5 Analog Filtering 401

H = num/den; % frequency response

Y = X∗H; % convolution property

y = ifourier(Y, t); % inverse Fourier n

nExample 6.9

In this example we will compare the performance of Butterworth and Chebyshev low-pass filters

in the filtering of an analog signalx(t)=[−2 cos( 5 t)+cos( 10 t)+4 sin( 20 t)]u(t)using MATLAB.

We would like the two filters to have the same half-power frequency.

The magnitude specifications for the low-pass Butterworth filter are

αmax=0.1dB,p=5rad/sec (6.54)

αmin=15dB,s=10 rad/sec (6.55)

and a dc loss of 0 dB. Once this filter is designed, we would like the Chebyshev filter to have the

same half-power frequency. In order to obtain this, we need to change thepspecification for the

Chebyshev filter. To do that we use the formulas for the half-power frequency of this type of filter

to find the new value forp.

The Butterworth filter is designed by first determining the minimum orderNand the half-power

frequencyhpusing the functionbuttord, and then finding the filter coefficients by means of the

functionbutter. Likewise, for the design of the Chebyshev filter we use the functioncheb1ord

to find the minimum order and the cut-off frequency (the newpis obtained from the half-

power frequency). The filtering is implemented using the Fourier transform as before.

There are two significant differences between the designed Butterworth and Chebyshev filters.

Although both of them have the same half-power frequency, the transition band of the Cheby-

shev filter is narrower, [6.88 10], than that of the Butterworth filter, [5 10], indicating that the

Chebyshev is a better filter. The narrower transition band is compensated by a lower minimum

order of five for the Chebyshev compared to the six-order Butterworth. Figure 6.24 displays the

poles of the Butterworth and the Chebyshev filters, their magnitude responses, as well as the input

signalx(t)and the outputy(t)for the two filters (the two perform very similarly).

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% Example 6.9 -- Filtering with Butterworth and Chebyshev filters

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clear all;clf

syms t w

x = cos(10∗t)− 2 ∗cos(5∗t) + 4∗sin(20∗t); X = fourier(x);

wp = 5;ws = 10;alphamax = 0.1;alphamin = 15; % filter parameters

% butterworth filter

[N, whp] = buttord(wp, ws, alphamax, alphamin, ’s’)

[b, a] = butter(N, whp, ’s’)

% cheby1 filter

epsi = sqrt(10ˆ(alphamax/10)−1)