How to write a sigma calculation

I'm trying to plot a Dtft,

the question is number 2 in the picture.

This is how I wrote

x=ones(1,9);

N=length(x);

n=-4:N-5;

q=@(omega)exp(-1i.*omega.*n);

wn=linspace(-2,2,9);

plot(wn,real(q(wn)), '-b', wn, imag(q(wn)), '-r');

grid

there is no problems, but the graph isn't pretty.

My thought is to change omega's length somehow or is there a better way?

Answer 1

[Abraham Boayue]

In part two of the problem, you will have to use the definition of the DTFT to compute X(omega). The resultant function is just a periodic version of X(jw) from part one of the problem. See the following code below.

%%Part 1
clear variables
close all
N= 200;
T = 2;
t  = -2:4/(N-1):2;
x = rectpuls(t,T);
f= -2:4/(N-1):2;
X = T*sin(pi*f*T)./(pi*f*T);
figure
subplot(121)
plot(t,x,'linewidth',2,'color','b')
grid;
a = title('x(t)');
set(a,'fontsize',14);
a = ylabel('x');
set(a,'Fontsize',14);
a = xlabel('t');
set(a,'Fontsize',14);
subplot(122)
plot(f,abs(X)/(max(X)),'linewidth',2,'color','m')
a = title('|X(jw)|');
set(a,'fontsize',14);
a = ylabel('X');
set(a,'Fontsize',14);
a = xlabel('f');
set(a,'Fontsize',14);
grid

%%Part 2
M = 1000;
N = 7;
n = 0:N-1;
xn = ones(1,N);
w = 8*pi;
omega = -w:2*w/(M-1):w; 
Xn= exp(-1i.*omega.*(N-1)./2).*(sin(omega*N/2)./sin(omega/2)); %define DTFT function
Mag = abs(Xn)/max(Xn);   %compute magnitude
Phase = angle(Xn);       %compute phase

figure
subplot(3,1,1)
stem(n,xn,'linewidth',2,'color','b'); 
a = title('x(n)');
set(a,'fontsize',14);
a = ylabel('xn');
set(a,'Fontsize',14);
a = xlabel('n');
set(a,'Fontsize',14);
grid

subplot(3,1,2)
plot(omega/pi,real(Mag),'linewidth',2,'color','k'); 
a = title('|X(\omega)|');
set(a,'fontsize',14);
a = ylabel('X');
set(a,'Fontsize',14);
a = xlabel('f');
set(a,'Fontsize',14);
grid

subplot(3,1,3)
plot(omega./pi,Phase,'linewidth',2,'color','k'); 
a = title('<X(\omega)');
set(a,'fontsize',14);
a = ylabel('X');
set(a,'Fontsize',14);
a = xlabel('f');
set(a,'Fontsize',14);
grid

[doyoun Kim]

Thx

Then is f meaning to divide pi?

[Abraham Boayue]

fs = 4;
fo = 10;
f = -fo:2*fo/(M-1):fo; % -10<f<10
omega = 2*pi*f/fs; 
fn = omega/2*pi; % You can plot the function vs the normalized frequency  f/fs               % if you wish

[doyoun Kim]

got it