Nonlinear Modifications

In Fourier terms, the simplest nonlinearity is a square law. Consider an FFT processor that squares each frame spectrum:

In the time domain, each frame is convolved with itself:

Since is time limited to samples, we can avoid time domain aliasing by requiring .

More generally, we can consider

This can be thought of as cascaded convolutions of with itself. The resulting signal will be at most samples long. We can avoid time domain aliasing in this case by requiring

We can express a general class of nonlinearities as a polynomial in the spectrum:

In this case, we require to avoid time aliasing.

For related information, look into Volterra series expansions [21]. The interated-convolution expansion above can be regarded as a special case of a Volterra series expansion.

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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.