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 linear gain of .

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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Subsections

• Clipping Nonlinearity
• Arctangent Nonlinearity
• Cubic Soft Clipper
• Series Expansions
  • Arctangent Example
  
  
• Spectrum of a Memoryless Nonlinearities
  • The Square Law
  • Power Laws
  • Spectrum of the Arctangent Nonlinearity
  
  
• Cubic Soft-Clipper
• Practical Advice

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by co