Inverse Fourier of a Discrete and Finite Sampling Window

Question

I am trying to find the inverse Fourier of a discrete sampling window with N_f = 11 elements. I am expecting to see a periodic Sinc-like signal in the time-domain:

However, this is the output I get:

Why the number of observed oscillations in my output is more than the expected output?

N_f = 11; % Number of samples in the finite sampling window in Frequency domain
delta_f = 1;
f = [-N_f/2+1:delta_f:N_f/2];
wf = ones(1, N_f)/N_f; % W(f):finite sampling window in Frequency Domain
N_t =128;
wt = ifftshift(ifft(wf, N_t))*N_t; % w(t):Inverse Fourier of the window

@CrisLuengo , sorry if I was not clear. I meant **the number** of observed oscillations between two consecutive Sinc. For example, in the expected w(t) on the first figure, there are 4 peaks between the maximums of two Sincs. However in my output, there are 9 peaks. — SaraG, Jan 19 '18 at 21:03
Have you written out the formula for the sinc function and counted the zero-crossings? It could be that the zeropadding is making more of them visible. — AnonSubmitter85, Jan 19 '18 at 21:29

score 2 · Accepted Answer · answered Jan 20 '18 at 05:45

I think one of the issues with your code is related to the location of the origin. You create an array wf with 11 ones (and normalized), then call ifft(wf,128). This pads the array with zeros to a size of 128, but the 11 ones are on the left side. You can see this by doing

fft(ifft(wf, N_t))

Thus, your window is shifted by 5 samples to the right, covering frequency bins 0 through 11, rather than -5 through 5 (or identically, 0 through 5 and 124 through 128).

The code below creates a signal with 128 samples, and fills the 11 frequency bins around the middle with 1/11. By calling ifftshift we rotate the signal such that the middle element is moved to the leftmost bin. Now bins 0 through 5 and 124 through 128 are non-zero. I then call ifft, and fftshift to move the 0 frequency back to the middle of the signal. Finally, I plot three repetitions of this signal.

N_f = 11; % Number of samples in the finite sampling window in Frequency domain
N_t = 128;
wf = zeros(1,N_t);
wf( N_t/2 - floor(N_f/2) + 1 : N_t/2 + floor(N_f/2) + 1 ) = 1 / N_f;
wt = fftshift(ifft(ifftshift(wf))) * N_t;
figure; plot(repmat(wt,1,3))

As you can see, the result is as you expected it. Note that the wt result of ifft is actually real-valued. The result of your inverse transform wasn't real-valued, you had to ignore the imaginary component to produce your plot. That's a sign that the input signal wasn't symmetric!

We can change N_f to be twice as large, yielding a result similar to yours, but with a purely real output:

N_f = 21;
N_t = 128;
wf = zeros(1,N_t);
wf( N_t/2 - floor(N_f/2) + 1 : N_t/2 + floor(N_f/2) + 1 ) = 1 / N_f;
wt = fftshift(ifft(ifftshift(wf))) * N_t;
figure; plot(repmat(wt,1,3))

Inverse Fourier of a Discrete and Finite Sampling Window

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