The Square Law

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When viewed as a Taylor series expansion such as Eq. $\,$ (T.2), the simplest nonlinearity is clearly the square law nonlinearity:

$\displaystyle f(x) = x + \alpha x^2$

where $\alpha$ is a parameter of the mapping.^T.2

Consider a simple signal processing system consisting only of the square-law nonlinearity:

$\displaystyle y(n) = x(n) + \alpha x^2(n)$

The Fourier transform of the output signal is easily found using the dual of the convolution theorem:^T.3

$\displaystyle Y(\omega) = X(\omega) + \alpha (X\ast X)(\omega)$

where `` $\ast$ '' denotes convolution. In general, the bandwidth of $X\ast X$ is double that of

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.

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