A gammatone filter is a

linear filter Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...

described by an

impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...

that is the product of a

gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...

and

sinusoid A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...

al tone. It is a widely used model of

auditory filters In audiology and psychoacoustics the concept of critical bands, introduced by Harvey Fletcher in 1933 and refined in 1940, describes the frequency bandwidth of the "auditory filter" created by the cochlea, the sense organ of hearing within the inne ...

in the

auditory system The auditory system is the sensory system for the sense of hearing. It includes both the sensory organs (the ears) and the auditory parts of the sensory system. System overview The outer ear funnels sound vibrations to the eardrum, increasin ...

. A gammatone response was originally proposed in 1972 as a description of

revcor function The reverse correlation technique is a data driven study method used primarily in psychological and neurophysiological research. This method earned its name from its origins in neurophysiology, where cross-correlations between white noise stimuli an ...

s measured in the cochlear nucleus of cats. The gammatone impulse response is given by :

g(t) = at^ e^ \cos(2\pi ft + \phi), \,

where

f

(in Hz) is the center frequency,

\phi

(in

radians The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

) 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 time-domain impulse response is a sinusoid (a

pure tone Pure may refer to: Computing * A pure function * A pure virtual function * PureSystems, a family of computer systems introduced by IBM in 2012 * Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool * Pure-FTPd, F ...

) with an amplitude envelope which is a scaled

function. Gammatone filterbank cepstral coefficients (GFCCs) are auditory features that have been used first in the speech domain, and later in the field of underwater target recognition. A bank of gammatone filters is used as an improvement on the triangular filters conventionally used in

mel scale The mel scale (after the word ''melody'') is a perceptual scale of pitches judged by listeners to be equal in distance from one another. The reference point between this scale and normal frequency measurement is defined by assigning a perce ...

filterbanks and MFCC features. Different ways of motivating the gammatone filter for auditory processing have been presented by Johannesma, Patterson et al., Hewitt and Meddis, and Lindeberg and Friberg.

Variations

Variations and improvements of the gammatone model of auditory filtering include the complex gammatone filter, 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.

References

External links

Slaney's implementation in Mathematica

Cooke's implementation in C

A real-time implementation in C++ (for PureData)

Ellis's "Gammatone-like spectrogram" in Matlab
Auditory system Linear filters {{neuroscience-stub