Homework 5

Textbook exercises [PDF]

M3.14. Below is the matlab script to verify exercise 3.55:

% M3.14
Xk = [-1+3j 3+4j 1-5j -2+2j 6+3j -2-3j];
Xk = [12 Xk 10 fliplr(conj(Xk))];
xn = ifft(Xk);

% (a)
xn(1)

ans =

    2.2857

16/7

ans =

    2.2857


% (b)
xn(8) % = x[7]

ans =

  -0.8571 - 0.0000i

-12/14
ans = -0.8571

% (c)
sum(xn)

ans =

  12.0000 + 0.0000i


% (d)
n = 0:13;
yn = xn.*exp(j*4*pi/7.*n);
sum(yn)

ans =

  -2.0000 - 2.0000i


% (e)
sum(xn.^2)

ans =

  35.5714 - 0.0000i

498/14

ans =

   35.5714

M3.16. Below are the pole-zero plots generated during this matlab session:

num =[4 15.6 6 2.4 -6.4];
den = [3 2.4 6.3 -11.4 6];
[r,p,k] = residuez(num,den)

r =

   0.0175 - 0.7087i
   0.0175 + 0.7087i
   1.1825 - 0.3934i
   1.1825 + 0.3934i


p =

  -1.0000 + 1.7321i
  -1.0000 - 1.7321i
   0.6000 + 0.3742i
   0.6000 - 0.3742i


k =

   -1.0667

% so G(z) = -1.0667 + (0.0175 - 0.7087i)/(1 - (-1.0000 + 1.7321i)z^-1) + ...

% make the pole-zero plot
zplane(num,den)  
% add in some ROC circles at radius of magnitude of poles..?
ww = abs(p(1))*exp(j*[0:100]*2*pi/100);
hold on
plot(real(ww),imag(ww))
% cool...
ww = abs(p(3))*exp(j*[0:100]*2*pi/100);
plot(real(ww),imag(ww),'r')
title('M3.16 (a) Pole-zero plot with ROC shown')

There are three regions of convergence: inside the red circle, in between the red and solid blue circles, and outside the blue circle. These correspond, respectively, to a left-sided, two-sided, and right-sided sequence.


% now part (b)
num = [2 6.4 4.9 -.1 -.6];
den = [5 15.5 31.7 22.52 4.8];
[r,p,k] = residuez(num,den)

r =

   0.1848 - 0.1472i
   0.1848 + 0.1472i
   0.0542          
   0.1012          


p =

  -1.0500 + 1.7022i
  -1.0500 - 1.7022i
  -0.6000          
  -0.4000          


k =

   -0.1250

zplane(num,den)
ww = abs(p(1))*exp(j*[0:100]*2*pi/100);
hold on
plot(real(ww),imag(ww),'r')
ww = abs(p(3))*exp(j*[0:100]*2*pi/100);
plot(real(ww),imag(ww),'g')
plot(real(ww),imag(ww),'r')
ww = abs(p(4))*exp(j*[0:100]*2*pi/100);
plot(real(ww),imag(ww),'r')
plot(real(ww),imag(ww),'g')
plot(real(ww),imag(ww),'b')
ww = abs(p(3))*exp(j*[0:100]*2*pi/100);
plot(real(ww),imag(ww),'g')
title('M3.16 (b) Pole-zero plot with ROC shown')

There are four regions of convergence: Inside the blue circle, between the blue and green circles, between the green and red circles, and outside the red circle. The inside region corresponds to a left-handed sequence, the two annular regions to two-sided sequences , and the outside region to a right-sided sequence. NOte that the DTFT will only exist for the larger annular region, in other words for a two-sided sequence.

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