There are a number of reasons for the differences:
- You do not need to divide by No after either the forward or backward transform. Normalisations are applied by fft and ifft so that the inverse is correct. Removing these divisions removes the drastic change of scale (so you can omit the calls to ylim ).
- You must not call abs before taking the inverse. This merges the real and imaginary parts of Z_cos, losing information.
- You must not call abs after taking the inverse. This inverts the negative parts of the output curve, giving little cusps at the troughs of the wave.
- You must call ifftshift before ifft to correctly reverse the effect of fftshift . That is, you need x_2_recovered=ifft(ifftshift(Z_cos)).
- You must compare the result of the transform and its inverse with the correct original time series, so you need to plot z and not x_2 as the original if you want to see that that the fft has been inverted as expected.
After making these changes, I got identical-looking plots. You should find that the maximum absolute difference between z and x_2_recovered is very small, though they will not be identical due to arithmetic rounding.
