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Memoryless Nonlinearities

Memoryless or instantaneous nonlinearities form the simplest and most commonly implemented form of nonlinear element. Furthermore, many complex nonlinear systems can be broken down into a linear system containing a memoryless nonlinearity.

Given a sampled input signal $ x(n)$ , the output of any memoryless nonlinearity can be written as

$\displaystyle y(n) = f(x(n))
$

where $ f(\cdot)$ is ``some function'' which maps numbers to real numbers. We exclude the special case $ f(x)=\alpha x$ which defines a simple linear gain of $ \alpha$ .

The fact that a function may be used to describe the nonlinearity implies that each input value is mapped to a unique output value. If it is also true that each output value is mapped to a unique input value, then the function is said to be one-to-one, and the mapping is invertible. If the function is instead ``many-to-one,'' then the inverse is ambiguous, with more than one input value corresponding to the same output value.



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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