Arctangent Nonlinearity

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Arctangent Nonlinearity

A simple example of an invertible (one-to-one) memoryless nonlinearity is the arctangent mapping:

$\displaystyle f(x) = \frac{2}{\pi}\arctan(\alpha x), \quad x\in[-1,1]$

where normally $\alpha\gg 1$ . This function is graphed for $\alpha=10$ in Fig.T.1. (Recall that $\arctan(x)$ is defined as the angle whose tangent is

. Only angles between $-\pi/2$ and $\pi/2$ are needed to cover all real values of

**Figure T.1:** Arctangent nonlinearity.
$\includegraphics[width=3in]{eps/atanex}$

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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 courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Physical Audio Signal Processing
    Nonlinear Elements
       Memoryless Nonlinearities
          Arctangent Nonlinearity