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Effect of Pole and Zero Locations on the Frequency Response Of an LTI Discrete Time System In this experiment you will investigate the effect of pole and zero placement in the z-plane on the frequency response of an LTI DTS IIR digital filter. MATLAB will be used for this. The transfer function H z( ) of the LTI DTS that we will operate is given by 1 2 ( ) ( ) ( ) ( ) ( ) j j z z z z H z z re z re         where the two zeros of H z( ) are: 1z and 2 z , and the two poles of H z( ) are: j re  and j re   , and where r is the magnitude of the poles and  and  are the angles of the poles. The transfer function can also be written as 1 1 1 2 1 2 1 2 1 2 1 1 1 2 2 (1 ) (1 ) 1 ( ) ( ) (1 ) (1 ) 1 2 cos( ) j j z z z z z z z z z z H z re z re z r z r z                        By inspection of H z( ) , the difference equation relating the output y n( ) to the input x n( ) is given by 2 1 2 1 2 y n x n z z x n z z x n r y n r y n ( ) ( ) ( ) ( 1) ( 2) 2 cos( ) ( 1) ( 2)            Depending on the pole and zero locations in the z-plane, this algorithm can do different kinds of filtering. Let’s investigate the possibilities. 1) Set the sampling frequency to 8 sf KHz  , and get the input x n( ) from sampling x t t t t ( ) cos(400 ) cos(3600 ) cos(7000 )       (a) What are the frequencies in Hz of the sinusoids in xt( ) ? Set the poles and zeros as follows: 1 2 z z r      1, 1, 0.95, and    / 20 . (b) In the z-plane and with respect to the unit circle, plot the poles and zeros of the transfer function. (c) Calculate and plot the magnitude frequency response over the frequency range, 0 s   f f . (d) What kind of a filter is this? (e) Calculate the coefficients of the difference equation, and give the difference equation. (f) Apply the input to the difference equation, and run the algorithm long enough to reach steady-state behavior, and plot the input and output. (g) In view of the frequency response, discuss the performance of the filter. 2) Repeat part (1) for r  0.65. How did the pole locations change? Discuss what happened to the frequency response? 3) Repeat part (1) for r  0.99. How did the pole locations change? Discuss what happened to the frequency response? 4) Repeat part (1) for: 1 2 z z r      1, 1, 0.95, and    / 2.1 . 5) Repeat part (1) for: 1 2 z z r      1, 1, 0.95, and    /1.1 . 6) Using the coefficients of parts (1), (2), and (3), obtain and plot the unit pulse response. How does the unit pulse response change? 7) Repeat part (1) for: 1 2 z z r      1, 1, 1.05, and    / 2.1 . Explain what happened? 8) Overall, give a discussion about the relationship between pole magnitude and angle and the LTI DTS frequency response. How do the zero locations affect the frequency response? 9) Set the zeros to: 1 2 z z   0, 0, and set the pole magnitude to: r  1.0. Three values of  will be tried, which are: 1 2       / 20, /10, and 3    / 5. Set the sampling frequency to 20 . sf KHz  For each pole angle, give a pole zero plot. Describe how the poles are being moved. (a) For 1 , calculate the coefficients and give the difference equation. For a unit pulse input, run the algorithm just long enough to reach steady state conditions. Plot the unit pulse response. What is the frequency of the steady-state response? b) Repeat part (a) for  2 . c) Repeat part (a) for 3 . d) This algorithm can be used as a sine wave generator. What value of  must be used to generate a 2.5 KHz sine wave?

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clc;clear;

fs=8*10^3;
t=0:1/fs:0.03;

%input signal
x = cos(400*pi*t) + cos(3600*pi*t) + cos(7000*pi*t);

plot(t,x);
title('input');
xlabel('time(s)');
ylabel('x(n)');

B = [1 2 1];
A = [1 -1*0.95*2*cos(pi/20) 0.95^2];

H = tf(B,A);
pzmap(H)
grid on


[hw,fw] = zerophase(B,A);

z = roots(B);
p = roots(A);

plot3(cos(fw),sin(fw),hw)
hold on
plot3(cos(fw),sin(fw),zeros(size(fw)),'--')
plot3(real(z),imag(z),zeros(size(z)),'o')
plot3(real(p),imag(p),zeros(size(p)),'x')
hold off
xlabel("Real")
ylabel("Imaginary")
view(35,40)
grid...

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