Fast Fourier Transform function

I tried to make a FFT of a sine wave but I didn't get what I expected.

t = 0:0.01:5;   % time interval
w = 4*pi;       % rad/s , so w = 2*pi*f, f=frequency
x = sin(w*t);
r = fft(x);     % fast fourier transform function
R = real(r);  
plot(R)         % plotted only the real part of r

What I don't understand is that the only frequency should be 2 Hz, since f=w/(2*pi) = (4*pi)/(2*pi) = 2, but I get a total different value.

Can anyone help me understand the result?

Thanks!

Answer 1

[Paul]

The code in the question is plotting R vs index, but it sounds like the goal is to plot R vs frequency (Hz). So we need to compute the frequency vector that corresponds to the samples of r.

t = 0:0.01:5;   % time interval
Ts = t(2);      % sampling period
w = 4*pi;       % rad/s , so w = 2*pi*f, f=frequency
x = sin(w*t);
r = fft(x);     % fast fourier transform function
R = real(r);

The corresponding frequency vector in Hz is:

fhz = (0:(numel(r)-1))/numel(r)/Ts;

Plot the real part as in the Question

figure;plot(fhz,R),xlabel("Hz")

We see two peaks at around 2Hz and 98 Hz, whchi corresponds to -2Hz.

Zoom in

figure;plot(fhz,R,'-o'),xlim([0 10])

We see a peak at 2 Hz, but small non-zero values around 2Hz. These values come about because the samples of x do not exactly an integer number of periods of the underlying sine wave. The easiest way to get the expetect result to make the number of samples of x equal to an integer multiple of P/Ts, where P is the period of the sine wave

P = 1/2;
P/Ts

ans = 50

But the actual number of samples is

N = numel(t)

N = 501

Because we really want 500 samples of data, jsut chop off the last point in the time vector

t(end) = [];   % time interval
x = sin(w*t);
r = fft(x);     % fast fourier transform function
R = real(r);
fhz = (0:(numel(r)-1))/numel(r)/t(2);

For the sine wave the real part of its FFT is zero, so also plot the imaginary part.

figure;
subplot(211);plot(fhz,R,'-o'),xlim([0 10])
subplot(212);plot(fhz,imag(r),'-o'),xlim([0 10])
xlabel('Hz')

As expected the real, part is basically zero and the imaginary part has a spike at 2Hz.

Answer 2

[Mitch Lautigar]

Couple things i'd like to comment on to try and help you. Please see bullets below.

• You're original graph is setup correctly, but i'd recommend the following change to allow for more samples and therefore better end results. "t = 0:0.01:5;" should probably be changed to "t = 0:0.001:5;"
• When you take the FFT, you need to set the axis up correctly. This is a common complexity in signals and systems, but looking at the MATLAB FFT command will help you set it up (https://www.mathworks.com/help/matlab/ref/fft.html).

Try these two tips, and if you are still stuck, let me know.