MATLAB: How to plot a correct fft of cosine wave

Question

I'm starting with Matlab and our professor told us to do some exercises to get it going.

One of them it's to plot the Fourier Transform of a cosine (cos(2*pi*t)), but sampled with sample frequency of fs=12Hz.

So I did the following:

>> fs=12;

>> n=0:1/fs:11/fs;

>> x=cos(2*pi*n);

>> stem(n,x)

Which gives me the following:

Then I calculated the fft of x:

>> y=abs(fft(x));

>> k=0:fs:11*fs;

>> stem(k,y)

Which should give me the real part of the Fourier Transform of a cosine, but If I recall correctly, the FT of a cosine is two spikes, one at (wave frequency)/2*pi and another at -(wave frequency)/2*pi, but I got this:

Why I'm getting both spikes at positive side of the axis? One could argue the DFT is periodic, which is, but it should be periodic with 2*pi, no?

I think I'm really messing up DTFT and DFT of cosine, but I can't figure out what can I do to correct my plot.

Thanks in advance.

Answer 1

You forgot to use fftshift(). Remember, with fft, the origin is at element 1 and then "wraps around" so to see the origin in the middle like you'd expect, and positive frequencies to the right and negative frequencies to the left, you need to call fftshift().