Issue using fft() function

I sampled the waveform x (t) = 10*cos(2*pi*1000t) + 6*cos(2*pi*2000t) + 2*cos(2*pi*4000t) with a sampling rate of 12000 Hz. And I want to plot the DFT of x (t) with N=64 points using fft() function. But the fft graph is not as expected. It is shifting. How can I solve this? How can I make it to plot correctly without any shifting?

I have attached my code below.

f = 1000;
fs = 12*f;
T = 0.01;
t = 0:1/fs:T;
x = 10*cos(2*pi*f*t) + 6*cos(2*pi*2*f*t) + 2*cos(2*pi*4*f*t);

plot(t,x);
title("x(t)");
xlabel("Time (in s)");
ylabel("Amplituide");

N = 64;
y = fft(x,N)/N;
freq = (-N/2:N/2 - 1)*fs/N;
stem(freq,abs(y));
xlabel("Normalized Frequency (in Hz)");
ylabel("|X(f)|/N");

Answer 1

[RANGA BHARATH]

Hi @Ganesh Jada. Here is the solution and code for your question.

Question: How to shift the DFT to the center of the spectrum?

Solution:

The fft() function automatically shifts the spectrum away from the center. So, there is a function called fftshift() which rearranges the outputs of fft(), fft2() and fftn() by moving the zero-frequency component to the center of the array. It is useful for visualizing a Fourier transform with the zero-frequency component in the middle of the spectrum.

So, you can use the below code which uses fftshift() function and plots that using stem() function.

Code:

f = 1000;

fs = 12*f;

T = 0.01;

t = 0:1/fs:T;

x = 10*cos(2*pi*f*t) + 6*cos(2*pi*2*f*t) + 2*cos(2*pi*4*f*t);

plot(t,x);

title("x(t)");

xlabel("Time (in s)");

ylabel("Amplituide");

N = 64;

y = fft(x,N)/N;

freq = (-N/2:N/2 - 1)*fs/N;

%% Here is the modification

y_shift = fftshift(y);

stem(freq,abs(y_shift));

xlabel("Normalized Frequency (in Hz)");

ylabel("|X(f)|/N");

Links to Documentation:

• Here is the link to the documentation of fftshift() function: https://www.mathworks.com/help/matlab/ref/fftshift.html#d124e458005

[Ganesh Jada]

Thank you for your response I am grateful to you