Signals and Systems - Electrical Engineering

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

When designing the Butterworth filter we only need to find the minimal orderNgiven that the

half-power frequency is specified. We find thatN=15 satisfies the specifications. Using this value

with the given discrete half-power frequency the functionbuttergives the coefficients of the filter.

See Figure 11.17 for the results.

The design of the Chebyshev filter with orderN=15 and half-power frequencyωhp=0.4πcannot

be done directly with the functioncheby1, as we do not have the passband frequencyωp. Using

the equation to compute the half-power frequency, we solve forωpafter we give a value toαmax

(arbitrarily chosen to be 0.001 dB), which allows us to compute the ripple factorε. See the script

part corresponding to the Chebyshev design. The functioncheby1with inputsN,αmax, andωp

gives the coefficients of the designed filter. Using the coefficients we plot the poles and the zeros,

the magnitude and the phase responses, and the loss function, as shown in Figure 11.17. According

to the loss function plots the Chebyshev filter displays a sharper response in the transition band

than the Butterworth filter, as expected.

% Butterworth/Chebyshev filters for analog signal

wh = 0.4∗pi;ws = 0.5∗pi; alphamin = 40;Fs = 20000;

% Butterworth

N = log10((10ˆ(0.1∗alphamin)-1))/(2∗log10(tan(ws/2)/tan(wh/2))); N = ceil(N)

% [b,a] = butter(N,wh/pi); % to get Butterworth filter get rid of ‘ % ’

% Chebyshev

alphamax = 0.001;

epsi = sqrt(10ˆ(0.1∗alphamax)-1);

% computation of wp for Chebyshev design

wp = 2∗atan(tan(0.5∗wh)/(cosh(acosh(sqrt(10ˆ(0.1∗3.01)-1)/epsi)/N))); wp = wp/pi;

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

% magnitude and phase

[H,w] = freqz(b,a);w = w/pi;M = length(H);f = w∗Fs/2;

alpha0 = -20;H = H∗10;

% to verify specs

spec2 = alpha0 + alphamin∗ones(1,M);

hpf = alpha0 + 3.01∗ones(1,M);

alpha = -20∗log10(abs(H));

Ha = unwrap(angle(H)); n

11.4.4 Rational Frequency Transformations

As indicated before, the conventional approach to filter design is to obtain first a prototype low-pass

filter and then to transform it into different types of filters by means of frequency transformations.

The magnitude specifications remain unchanged.

When using analog filters to design IIR discrete filters the frequency transformation could be done in

two ways:

n Transform a prototype low-pass analog filter into a desired analog filter, which in turn is converted

into the desired discrete filter using the bilinear or other transformation.