Pamantasan ng Lungsod ng MaynilaUniversity of the City of Manila

Intramuros, Manila

College of Engineering and TechnologyComputer Engineering Department

CPE 415.1

Digital Signal Processing (Laboratory)

ACTIVITY 8

Francisco, Marion Angelo V. Sinfuego, Ian C. Yamson, Eirry Rose Anne R.

BS CpE 4 - 1

Engr. Juarizo

August 26, 2015

Analysis of Z – Transforms

Q3.46. Using Program P3 1 evaluate the following z- transform on the unit circle:

G=2+5 z−1+9 z−2+5 z−2+3 z−4

5+45 z−1+2 z−2+z−3+z−4Eq .(3.32)

Answer:

function z ()k =(2500);num =[2 5 9 5 3];den = [5 45 2 1 1];w = 0:pi/(k-1):pi;h = freqz(num, den, w);subplot(2,2,1)plot(w/pi,real(h));gridtitle('Real part')xlabel('\omega/\pi'); ylabel('Amplitude')subplot(2,2,2)plot(w/pi,imag(h));gridtitle('Imaginary part')xlabel('\omega/\pi'); ylabel('Amplitude')subplot(2,2,3)plot(w/pi,abs(h));gridtitle('Magnitude Spectrum')xlabel('\omega/\pi'); ylabel('Magnitude')subplot(2,2,4)plot(w/pi,angle(h));gridtitle('Phase Spectrum')xlabel('\omega/\pi'); ylabel('Phase, radians')end

Q3.47. Write a MATLAB program to compute and display the poles and zeros, to compute and display the factored form, and to generate the pole-zero plot of a z–transform that is a ratio of two polynomials in z -1. Using this program, analyze the z-transform G(z) of Eq. (3.32)

Answer:

function zzz ()num=[0.3 2.5 -0.2+j*0.4 -0.2-j*0.4 ];den=[-0.5 -0.75 0.6+j*0.7 0.6-j*0.7];k=(3.9);[z,p,k]=tf2zp(num,den);disp('Zeros are at');disp(z);disp('Poles are at');disp(p);disp('Gain Constant');disp(k);radius=abs(p);disp('Radius of the poles');disp(radius);zplane(num,den)[num den] = zp2tf(z,p,k)end

Q3.48. From the pole-zero plot generated in Question Q3.47, determine the number of regions of convergence (ROC) of G(z). Show explicitly all possible ROCs. Can you tell from the pole-zero plot whether or not the DTFT exists.

Answer:

All possible ROCs of this z-transform are :R1 : | z | < 0.2718 (left-sided, not stable) R2 : 0.2718 < | z | < 0.2866 (two-sided, not stable) R3 : 0.2866 < | z | < 8.9576 (two-sided, stable) R4 : | z | > 8.9576 (right-sided, not stable)

From the acquired pole and zero plot it is hard to determine if the DTFT exists, especially if you are only given the graph. To determine the if the DTFT exists you must also know the ROC, and from the ROC we acquired, we can say that the DTFT exists in R3 because it is stable, for the other ROCs it hard to determine cause it is not stable.

Q3.49. Write a MATLAB program to compute and display the rational form of a z-transform from its zeros, poles, and gain constant. Using this program, determine the rational form of a z-transform whose zeros are at ξ1= 0.3, ξ2 = 0.2, ξ3 = -0.2 + j0.4, and ξ4 = -0.2 - j0.4; the poles are λ1 = 0.5, λ2 = -0.75, λ3 = 0.6 + j0.7, and λ4 = 0.6 – j0.7; and the gain constant k is 3.9.

Answer: