Signals and Systems - Electrical Engineering

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

is sampled using the Nyquist frequency and processed with a discrete filterH(z)that is obtained

from a second-order, high-pass analog filter

H(s)=

s^2

s^2 +

√

2 s+ 1

The discrete-time outputy[n] is then converted into analog. Apply MATLAB’sbilinearfunction to

obtain the discrete filter with half-power frequenciesωhp=π/2. Use MATLAB to plot the poles

and the zeros of the discrete filter in thez-plane and the corresponding magnitude and phase

responses. Use the functionplotto plot the sampled input and the filter output and consider these

approximations to the analog signals. Change the frequency scale of the discrete filter intofin

hertz and indicate what is the corresponding half-power frequency in hertz.

Solution

The coefficients of the numerator and denominator of the discrete filter are found fromH(s)using

the MATLAB functionbilinear. The inputFsin this function equalsKb/2 whereKbcorresponds to

the transformation of the discrete half-power frequencyωhpinto the normalized analog half-power

frequencyhp=1. The following script is used.

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

% Example 11.6

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

b = [1 0 0]; a = [1 sqrt(2) 1]; % coefficients of analog filter

whp = 0.5∗pi; % desired half-power frequency of discrete filter

Kb = 1/tan(whp/2); Fs = Kb/2; [num, den]=bilinear(b,a,Fs); % bilinear transformation

Ts = 1/500; % sampling period

n = 0:499; x1 = cos(2∗pi∗ 20 ∗n∗Ts)+cos(2∗pi∗ 250 ∗n∗Ts); % sampled signal

zplane(num, den) % poles/zeros of discrete filter

[H,w] = freqz(num,den); % frequency response of discrete filter

phi = unwrap(angle(H)); % unwrapped phase of discrete filter

y = filter(num,den,x1); % output of discrete filter with input x1

We find the transfer function of the discrete filter to be

H(z)=

0.2929( 1 −z−^1 )^2

1 +0.1715z−^2

The poles and the zeros ofH(z)can be found with the MATLAB functionrootsand plotted with

zplane. The frequency response is obtained usingfreqz. To have the frequency scale in hertz we

consider thatω=Ts, letting= 2 πf, then

f=

ω

2 πTs

=

(ω

π

)(fs

2

)