Question

Use MATLAB to find the 8 point DFT of x(n) = cos(2πmn/8) (m=3) for 0 ≤ n ≤ 7.

Plot both x(n) and its DFT and explain your results. The "dct" and "fft" functions in MATLAB may be useful. Please post MATLAB code.

Answer 1

The matlab code is attached below. It is well commented and self explanatory. Following it are the screen shots of code for readability and resultant plots.

% Code starts from here

clc % Clear the command window
close all % Close any open window
n = 0:1:7; % Create index vector for n [0,7]
m = 3; % defining variable m
x = cos(2*pi*m.*n./8); % defining the signal x(n)
figure % creating a new figure
plot(n,x,'r','Linewidth',2); % plotting x[n] against n
grid on; % Turning on the grid
xlabel('Index n-->'); % labelling X axis
ylabel('Amplitude-->'); % labelling Y axis

X = fft(x); % FFT of the signal x[n]
X_mag = abs(X); % Finding the magnitude spectrum of x
figure % creating a new figure
plot(n,X_mag,'r','Linewidth',2); % plotting X against n
grid on; % Turning on the grid
xlabel('Digital frequency bin n-->'); % labelling X axis
ylabel('Magnitude-->'); % labelling Y axis

% Code ends here

Outputs

We see that there are two peaks one at n = 3 which corresponds to m = 3 and its aliased image at n = 5.