The Inverse Fast Fourier Transform computes a time-domain signal from its frequency-domain representation.
In general, the result of the IFFT will be a sequence of complex values.
If we want the IFFT result to be a real valued signal, then our frequency domain representation must be Hermetian symmetric.
A Hermetian symmetric spectrum is one in which its real part is even symmetric about the frequency bin and its imaginary part is odd symmetric about the frequency bin . This can also be expressed as
and
.
A real time-domain signal always has a Hermetian frequency-domain transform.
Thus, any manipulation or processing that is performed in the frequency-domain should maintain Hermetian symmetry. In practice, most processing is applied to only half of the transformed signal and symmetry is assumed before the inverse transform is calculated.
If time-domain windows were not overlapped and frequency-domain processing was performed, the resulting IFFT'ed blocks would most likely have discontinuities that would be heard as clicks or pops.
If spectral processing and subsequent resynthesis using the IFFT is planned, time-domain windows must be “overlap-added” during reconstruction to avoid clicks at IFFT boundaries.
The Matlab script stfttest.m
demonstrates the STFT and subsequent overlap-add process applied to a test sound.