Fractionally Iterated Convolution

The above fractional-delay examples were special cases of fractionally iterated convolution. Given the transfer function , corresponding to impulse response , the iterated convolution corresponds to the squared transfer function . More generally, the th-order iterated convolution corresponds to the th power of the transfer function:

Fractionally iterated convolution occurs when is not an integer. While this would seem to have no meaning in the time domain, it is straightforward to implement in the frequency domain. As in the previous examples of designing fractional delay filters, results are well defined when we work with the polar-form frequency response

where denotes the unwrapped phase response.

Again as we saw in the fractional delay examples, a transition band is often desired about the point at half the sampling rate ( ), such that the desired response is tapered smoothly to zero at that point. Such a transition band reduces time aliasing caused by working on a finite grid in the frequency domain.

More generally, the use of phase unwrapping, spectral oversampling, and a transition band about can make a wide class of nonlinear spectral modifications work as expected in the time domain. In the context of short-time Fourier analysis, modification, and resynthesis, these principles may be applied to each spectral frame.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.