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 , the output of any memoryless nonlinearity can be written as

where is ``some function'' which maps numbers to real numbers. We exclude the special case which defines a simple

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.

- Clipping Nonlinearity
- Arctangent Nonlinearity
- Cubic Soft Clipper
- Series Expansions
- Arctangent Series Expansion
- Spectrum of a Memoryless Nonlinearity
- Square Law Series Expansion
- Power Law Spectrum
- Arctangent Spectrum
- Cubic Soft-Clipper Spectrum
- Stability of Nonlinear Feedback Loops
- Practical Advice

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University