# Gammatone filter

Figure 1: A gammatone impulse response.

A gammatone filter is a linear filter described by an impulse response that is the product of a gamma distribution and sinusoidal tone. It is a widely used model of auditory filters in the auditory system.

The gammatone impulse response is given by

$g(t) = at^{n-1} e^{-2\pi bt} \cos(2\pi ft + \phi), \,$

where $f$ (in Hz) is the center frequency, $\phi$ (in radians) is the phase of the carrier, $a$ is the amplitude, $n$ is the filter's order, $b$ (in Hz) is the filter's bandwidth, and $t$ (in seconds) is time.

This is a sinusoid (a pure tone) with an amplitude envelope which is a scaled gamma distribution function.[1]

## Variations

Variations and improvements of the gammatone model of auditory filtering include the gammachirp filter, the all-pole and one-zero gammatone filters, the two-sided gammatone filter, and filter cascade models, and various level-dependent and dynamically nonlinear versions of these.[2]

## References

1. ^ Slaney, Malcolm (1993). "An Efficient Implementation of the Patterson-Holdsworth Auditory Filter Bank". Apple Computer Technical Report #35.
2. ^ Richard F. Lyon, Andreas G. Katsiamis, Emmanuel M. Drakakis (2010). "History and Future of Auditory Filter Models". Proc. ISCAS. IEEE.