Signal Reconstruction With The IFFT

The Inverse Fast Fourier Transform computes a time-domain signal from a complex frequency-domain representation.
The result of the IFFT can 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 $\Re\{X[-k]\} = \Re\{X[k]\}$ and $\Im\{X[-k]\} = -\Im\{X[k]\}$ .
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 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.
Frequency-domain processing and resynthesis of sounds generally involves the following sequence of steps, starting and ending with time-domain signals:
1. Apply the STFT to the sound, using a time-domain window with an appropriate overlap factor so that the windows sum to a constant (which avoids an amplitude modulation of the result). Transform each windowed segment with the FFT and save the array of complex data.
2. Perform frequency-domain processing on each array of FFT data (guaranteeing or assuming Hermetian symmetry is maintained).
3. Then use the IFFT to transform the processed frequency-domain data back to the time-domain.
4. Apply the same window to each IFFTed segment, overlap segments by the same amount as in step $\char93 1$ and add them together to produce the new time-domain signal.
The Matlab script stfttest.m demonstrates the STFT and subsequent overlap-add process applied to a test sound.