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Square Law Series Expansion

When viewed as a Taylor series expansion such as Eq.$ \,$ (6.18), the simplest nonlinearity is clearly the square law nonlinearity:

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

where $ \alpha$ is a parameter of the mapping.7.18

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:7.19

$\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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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