# 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

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

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