Inverse FFT SynthesisWhen the number of partial overtones is large, an explicit sinusoidal oscillator bank requires a significant amount of computation, and it becomes more efficient to use the inverse FFT to synthesize large ensembles of sinusoids [35,239,143,142,139]. This method gives the added advantage of allowing non-sinusoidal components such as filtered noise to be added in the frequency domain [246,249].
Inverse-FFT (IFFT) synthesis was apparently introduced by Hal Chamberlin in his classic 1980 book ``Musical Applications of Microprocessors'' . His early method consisted of simply setting individual FFT bins to the desired amplitude and phase, so that an inverse FFT would efficiently synthesize a sum of fixed-amplitude, fixed-frequency sinusoids in the time domain. This idea was extended by Rodet and Depalle  to include shaped amplitudes in the time domain. Instead of writing isolated FFT bins, they wrote entire main lobes into the buffer, where the main lobes corresponded to the desired window shape in the time domain.G.7 (Side lobes of the window transform were neglected.) They chose the triangular window ( main-lobe shape), thereby implementing a linear cross-fade from one frame to the next in the time domain. A remaining drawback of IFFT synthesis was that inverse FFTs generate sinusoids at fixed frequencies, so that a rapid glissando may become ``stair-cased'' in the resynthesis, stepping once in frequency per output frame. An extension of IFFT synthesis to support linear frequency sweeps was devised by Goodwin and Kogon . The basic idea was to tabulate window main-lobes for a variety of sweep rates. (The phase variation across the main lobe determines the frequency variation over time, and the width of the main lobe determines its extent.) In this way, frequencies could be swept within an FFT frame instead of having to be constant with a cross-fade from one static frame to the next.
FFT Implementation of the Phase Vocoder