EE 451 - Homework 1

Solutions

1.
Problem 2.7
(a)
$\pi N/6 = 2 \pi k \Rightarrow N = 12 k$. The smallest k for which N is an integer is 1, so the period is N = 12
(b)
$3 \pi N/4 = 2 \pi k \Rightarrow N = 8 k / 3$. The smallest k for which N is an integer is 3, so the perios is N = 8.
(c)
This is not periodic. $\sin(\pi n/5)$ is periodic with period 10. When you divide this by $\pi n$, it is no longer periodic - it gets smaller as n gets bigger, so is not periodic. 

     n = -30:30;
     wo = pi/5;
     x = sin(wo*n)./(pi *n);
     x(n==0) = wo/pi;          % L'Hospital's rule
     stem(n,x);
     grid
     title('x[n] = [sin(\pi n/5)]/(\pi n)');
     ylabel('x[n]');

\epsfig{file=fig1_c.eps,height=4in}

(d)
$\pi N / \sqrt{2} = 2 \pi k \Rightarrow N = 2 \sqrt{2} k$. There is no k which makes N an integer, so this is not periodic.
2.
Problem 2.29
(a) 
     n = -20:20;
     x = zeros(size(n));
     x((n>=-1)&(n<=3)) = 1;
     x(n==4) = 1/2;
     subplot(321)
     stem(n,x)
     grid
     axis([-10 10 0 1.5]);
     title('x[n]');
     
     subplot(322)
     na = n+2;
     stem(na,x);
     grid
     axis([-10 10 0 1.5]);
     title('x[n-2]');

     subplot(323)
     nb = 4-n;
     stem(nb,x);
     grid
     axis([-10 10 0 1.5]);
     title('x[4-n]');

     subplot(324)
     nc = -10:10;               % x goes from -20:20, so y will go from -10:10
     for k=-10:10               % (e.g., y[10] = x[20])
        y(nc==k) = x(n==(2*k));
     end
     stem(nc,y);
     grid
     axis([-10 10 0 1.5]);
     title('x[2n]');

     subplot(325)
     y = x.*((2-n)>=0);
     stem(n,y);
     grid
     axis([-10 10 0 1.5]);
     title('x[n] u[2-n]');

     subplot(326)
     ne = n+1;
     y = x.*(ne == 3);
     stem(ne,y);
     grid
     axis([-10 10 0 1.5]);
     title('x[n-1] \delta[n-3]');
\epsfig{file=fig2.eps,height=4in}
3. 
%(a)
      subplot(311)
      n=0:40;
      x = (0.9).^n .* cos(0.1*pi*n+pi/5);
      stem(n,x);
      grid
      ylabel('x[n]');
      title('x[n] = 0.9^n cos(0.1 \pi n + \pi/6)');

%(b)
      subplot(312)
      n=0:40;
      x = (0.9).^n .* cos(4.1*pi*n+pi/5);
      stem(n,x);
      grid
      xlabel('n');
      ylabel('x[n]');
      title('x[n] = 0.9^n cos(4.1 \pi n + \pi/6)');
      
%(c)
      subplot(313)
      n=-10:10;
      x=(0.75).^n .* (((n+6)>=0) - ((n-5)>=0)) + 5*(n==2);
      stem(n,x);
      grid
      xlabel('n');
      ylabel('x[n]');
      title('x[n] = 0.75^n (u[n+6] - u[n-5]) + 5 delta[n-2]');
\epsfig{file=fig3.eps,height=4in}
4.
Problem 4
(a)
The system is not time invariant. Putting in a signal of 1 at time 0 will result in an output of 0. Putting in a signal of 1 at time 4 will result in an output of 4, so the same signal at different times gives different outputs.
(b) 
       subplot(321)
       n = -20:20;
       x = (n>=0) - ((n-4)>=0);
       stem(n,x);
       grid
       axis([-10 10 -4 4])
       title('x[n] = u[n] - u[n-4]');

(c) 
       subplot(322)
       y = n.*x;
       stem(n,y);
       grid
       axis([-10 10 -4 4])
       title('y[n] = n x[n]');

(d) 
       subplot(323)
       stem(n+2,y)
       grid
       axis([-10 10 -4 4])
       title('y[n-2]');

(e) 
       subplot(324)
       n1 = n+2;
       x1 = x;
       stem(n1,x1);
       grid
       axis([-10 10 -4 4])
       grid
       title('x1[n] = x[n-2]');

(f) 
       subplot(325)
       y1 = n1.*x1;
       stem(n1,y1);
       grid
       axis([-10 10 -4 4])
       grid
       title('y1[n]');
\epsfig{file=fig4.eps,height=4in}

(g)
$y1[n] \ne y[n-2]$.

5.
Problem 5
(a)
The system is time invariant.
(b) 
       subplot(321)
       n = -30:30;
       x = (n>=0) - ((n-4)>=0);
       stem(n,x);
       grid
       axis([-10 10 0 1.5])
       title('x[n] = u[n] - u[n-4]');

(c) 
       subplot(322)
       ny=-15:15;
       for k=-15:15
           y(ny==k) = (x((n+1)==k) + x(n==k) + x((n-1)==k))/3;
       end
       stem(ny,y);
       grid
       axis([-10 10 0 1.5])
       title('y[n] = (x[n+1] + x[n] + x[n-1])/3');

(d) 
       subplot(323)
       stem(ny+2,y)
       grid
       axis([-10 10 0 1.5])
       title('y[n-2]');

(e) 
       subplot(324)
       n1 = n+2;
       x1 = x;
       stem(n1,x1);
       grid
       axis([-10 10 0 1.5])
       title('x1[n] = x[n-2]');

(f) 
       subplot(325)
       ny1 = -15:15;
       for k=-15:15
           y1(ny1==k) = (x((n1+1)==k) + x(n1==k) + x((n1-1)==k))/3;
       end
       stem(ny1,y1);
       grid
       axis([-10 10 0 1.5])
       title('y1[n]');
\epsfig{file=fig5.eps,height=4in}

(g)
y1[n] = y[n-2].


Bill Rison
1999-09-08