Question

the subject is in digital signal processing

5. Consider a CT system with transfer function This system is called an integrutor because t by he d to the ingent t y)-x(r)dr. Discretize the above system using the bilinear transform. (a) What is the transfer function H'(:) of the resulting DT system b) If xin] is the input and yin] is the output of the resulting DT system, write the (c) Obtain an expression for the frequency response H'(o) of the DT system. Plot the difference equation relating the input and the output. magnitude and phase of the DT systern fr lol π using Matlab Compare them with the magnitude and phase of the CT integrator. Is the obtained DT integrator a good approximation of the CT integrator? Now, consider a CT system with transfer function GL(s) = s. This system is called a differentiator because the output y(t) is the derivative of the input x). Discretize it using the bilinear transform. (d) What is the transfer function G'() of the resulting DT system? (e) Obtain an expression for the frequency response G (a) of the discrete-time system. Plot the magnitude and phase of the discrete-time system for ω π using Matlab. Compare them with the magnitude and phase of the CT differentiator. Under what conditions can the discrete-time differentiator be considered a good approximation to the CT differentiator? (f) The transfer functions of CT integrator and differentiator are exact inverses of oue another. Is the same true for their DT approximations? Total for Question &: IS

Answer 1

Explanation:

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MATLAB:

clc;clear all;close all;
figure(1)
w=-pi:pi/500:pi
%Transfer function of CT Integrator
H=1./(j*w)
subplot(321)
plot(w/pi,abs(H),'r')
xlabel('w/pi')
ylabel('|H(w)|')
title('CT integrator -Frequency response')

subplot(322)
plot(w/pi,angle(H),'g')
xlabel('X pi')
ylabel('<H(w)>')
title('Phase response')

%Transfer fucntion on DT Integrator
HH=(1+exp(-j*w))./(2*(1-(exp(-j*w))))
subplot(323)
plot(w/pi,abs(HH),'r')
xlabel('w/pi')
ylabel('|H(exp(jw)|')
title('DT integrator -Frequency response')

subplot(324)
plot(w/pi,angle(HH),'g')
xlabel('X pi')
ylabel('<H(exp(jw)>')
title('Phase response')

%Transfer fucntion on CT differentiator
G=(j*w)
subplot(325)
plot(w/pi,abs(G),'b')
xlabel('w/pi')
ylabel('|H(w)|')
title('CT Differentiator -Frequency response')

subplot(326)
plot(w/pi,angle(G),'g')
xlabel('X pi')
ylabel('<H(w)>')
title('Phase response')

figure(2)
%Inverse of CT differentiator
G=1./(j*w)
subplot(211)
plot(w/pi,abs(G),'b')
xlabel('w/pi')
ylabel('|H(w)|')
title('Inverse of CT Differentiator')

subplot(212)
plot(w/pi,angle(G),'g')
xlabel('X pi')
ylabel('<H(w)>')
title('Phase response')

CT integrator -Frequency response Phase response 200 150 1 00 50 2 0.5 0.5 Хрі w/pi Phase response DT integrator -Frequency response 200 150 100 I 50 2 I -1 0.5 0.5 w/pi Хрі CT Differentiator -Frequency response Phase response 3.5 3 2.5 2 I 1.5 0.5 0.5 0.5 Хрі w/pi

Inverse of CT Differentiator 200 r 150 100 50 0.5 0.5 w/pi Phase response 2 -2 0.5 0.5 X pi

clc;clear all;close all;
%Transfer fucntion of DT differentiator
fs=0.5
Ts=1/fs;
w=-pi:pi/1000:pi
z=exp(j*w/fs)
GG=(2/Ts)*(z-1)./(z+1)

subplot(321)
plot(w/pi,abs(GG),'r')
xlabel('w/pi')
ylabel('|H(exp(jw)|')
title('DT Differentiator -Frequency response')

subplot(322)
plot(w/pi,angle(GG),'g')
xlabel('X pi')
ylabel('<H(exp(jw)>')
title('Phase response')

fs=2
Ts=1/fs;
z=exp(j*w/fs)
GG=(2/Ts)*(z-1)./(z+1)
subplot(323)
plot(w/pi,abs(GG),'r')
xlabel('w/pi')
ylabel('|H(exp(jw)|')
title('DT Differentiator -Frequency response')

subplot(324)
plot(w/pi,angle(GG),'g')
xlabel('X pi')
ylabel('<H(exp(jw)>')
title('Phase response')

fs=1000
Ts=1/fs;
z=exp(j*w/fs)
GG=(2/Ts)*(z-1)./(z+1)
subplot(325)
plot(w/pi,abs(GG),'r')
xlabel('w/pi')
ylabel('|H(exp(jw)|')
title('DT Differentiator -Frequency response')

subplot(326)
plot(w/pi,angle(GG),'g')
xlabel('X pi')
ylabel('<H(exp(jw)>')
title('Phase response')

Phase response DT Differentiator -Frequency response 2 2e+016 S 1.5e+016 1e+016 5e+015 I-1 0.5 0.5 Хрі w/pi Phase response DT Differentiator -Frequency response 2 I -1 0.5 0.5 X pi w/pi Phase response DT Differentiator -Frequency response 2 3.5 3 2.5 0.5 0.5 0.5 Хрі w/pi

Observation:

CT integrator and integrators provide same frequency response.

The inverse of CT differentiator is same as CT integrator and it is true for all sampling frequencies.

But the DT differentiator and DT differentiator responses are not same except at high sampling frequencies.