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The Square Law

When viewed as a Taylor series expansion such as Eq. (R.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.R.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:R.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 $ X$.


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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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