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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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Spectrum of a Memoryless Nonlinearities