Matlab FFT - Result not understood - matlab

Question

I try to understand a result I get on with Matlab FFT and especially with the function fftshift
I first calculate the FFT of a simple signal (y = cos (2 * pi * f_signal * t) and the inverse transform to try to find the original signal.
I compare the results with and without the "zero padding" by adding 100 points higher at the starting signal before performing the FFT.
here is the code:
% Frequency input signal and time vector
f_signal=10;
nb_points = 100;
f_sampling=nb_points*f_signal;
step_time=1/f_sampling;
t=linspace(0,1/f_signal,nb_points);
%% Original signal with 100 points
y=cos(2*pi*f_signal*t);
%% Adding 100 more points for y
n_zero_padding=100;
y_f=padarray(y,[0 n_zero_padding],'post');
% Plot Input Signal
figure(1);
plot(y);
title('Input Signal');
% Perform Forward FFT
z=real(fft(y_f));
% Center FFT
z_centered=fftshift(z);
figure(2);
plot(z_centered);
title('FFT centrered with zero padding');
% Without centering
figure(3);
plot(z);
title('FFT with zero padding');
% Perform Inverse FFT
z_inv=ifft(z);
figure(4);
plot(z_inv);
title('Inverse FFT with zero padding');
% Inverse FFT with centered spectrum
z_inv_centered=ifft(z_centered);
figure(5);
plot(z_inv_centered);
title('Inverse FFT centered with zero padding');
I do not understand the result on Figure 5 (inverse transform of a shifted spectrum with fftshift)
here is the first spectrum obtained (not shifted) (Figure (3)):
here is the first shifted spectrum with fftshift (Figure (2)):
and the inverse FT of the shifted spectrum above (Figure (5)):
I do not understand the latter figure. If I take the theory, the fact of shifting the spectrum with fftshift leads to the obtaining of two Dirac centered around f_sampling / 2, more precisely on f_sampling / 2-f_signal and f_sampling / 2 + f_signal.
So I have the sum of these two Dirac and I do Inverse FFT of this sum: so I should get for Inverse FFT 2 cosine with frequencies very closed: it seems like I get on the figure (5 ) the product of two cosine signals but I do not see where is my misinterpretation ...
If someone could explain this result?
Thank you in advance

Answer 1

Because the spectrum of a real signal are symmetric about the origin, we prefer to see the spectrum like this, i.e., the zero frequency in the center. However, when computing the spectrum by FFT, what in the center are is the high frequency, thus, fftshift() is used to shift the zero spectrum to the center.
Since the fftshift() is used for visual matter, you can not perform ifft() on the shifted spectrum directly. On the other hand, the shifted spectrum is the modulated version of the original signal, and you obtain the modulated signal in Fig.5.