Sketch the magnitude and phase plot of x(n)=sin(4​πn/21)+cos​(10πn/21)+​1 using DTFS representation.

function [Xk]=dfs(xn, N)
% Computes Discrete Fourier Series Coefficients
% ---------------------------------------------
% [Xk] = dfs(xn,N)
% Xk = DFS coeff. array over 0 <= k <= N-1
% xn = One period of periodic signal over 0 <= n <= N-1
% N = Fundamental period of xn
n=-N:0:N; % row vector for n
k=-N:0:N; % row vector for k
WN=exp(-1i.*2.*pi/N); % Wn factor
nk=n'*k; % creates a N by N matrix of nk values
WNnk=WN.^nk; % DFS matrix
Xk=xn*WNnk; % row vector for DFS coefficients
subplot(2,1,1);
stem(n,abs(Xk));
xlabel('n'); 
ylabel('Magnitude plot');
subplot(2,1,2);
stem(n,angle(Xk));
xlabel('n'); 
ylabel('Phase plot');

Command window:

N=10; 
xn=1+sin(4*pi/21*n)+cos(10*pi/21*n); 
Xk=dfs(xn,N)

[GandaBerunda]

Hi,

Could you please be more specific with the issue?

[Pratiksha Sahoo]

The figure isn't displaying, could you check the code please?

[KSSV]

The below lines give you empty arrays.

n=-N:0:N; % row vector for n
k=-N:0:N; % row vector for k

[Pratiksha Sahoo]

Could you let me know what to changes do I have to make instead in those lines?

[Steven Lord]

What were you hoping or expecting those lines of code to do? What specific vector were you hoping to create?

If (as an example) you were hoping to create the vector [-5 -4 -3 -2 -1 0 1 2 3 4 5] when N was 5, just use colon with the default increment of 1 (or you can explicitly specify the increment.)

N = 5;
k1 = -N:N

k1 = 1×11

-5 -4 -3 -2 -1 0 1 2 3 4 5

k2 = -N:1:N

k2 = 1×11

-5 -4 -3 -2 -1 0 1 2 3 4 5

Your version failed to do what you wanted because it's not possible to get from -5 to 5 by taking steps of length 0 as you asked MATLAB to do.

[Pratiksha Sahoo]

I made those changes but the output still ain't coming, vector ranges from -10 to 10.

Magnitude plot is of amplitude 1 at [0] and 0.5 each at [-5,-2,2,5]

For the phase plot, the amplitude is pi/2 at [-2] and -pi/2 at [2]

Answer 1

[Harsh Kumar]

Hi Pratiksha ,

It is my understanding that you are trying to implement the Discrete Time Fourier Series programmatically in MATLAB and unable to get the plots of output.

To resolve this issue, you have to define a global variable n to parse into your input function ‘xn’ and have to use non-zero steps in your variables ‘n’ and ‘k’ instead of 0 as currently they are pointing to a NULL Array.

Please refer to the below code snippet for better understanding.

N=10; 
n=-N:1:N; 
xn=1+sin(4*pi/21*n)+cos(10*pi/21*n);  
Xk=dfs(xn,N); 
function [Xk]=dfs(xn, N) 
% row vector for k 
k=-N:1:N;  
n=-N:1:N; 
% Wn factor 
WN=exp(-1i.*2.*pi/N); 
% creates a N by N matrix of nk values 
nk=n'*k;  
% DFS matrix 
WNnk=WN.^nk; 
% row vector for DFS coefficients  
Xk=xn*WNnk;  
subplot(2,1,1); 
stem(n,abs(Xk)); 
xlabel('n');  
ylabel('Magnitude plot'); 
subplot(2,1,2); 
stem(n,angle(Xk)); 
xlabel('n');  
ylabel('Phase plot'); 
end