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 usin ...

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 impulse (). More generally, an impulse response is the reac ...

that is the product of a

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

and

sinusoid A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is '' simple harmonic motion''; as rotation, it correspond ...

al tone. It is a widely used model of auditory filters in the

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

. A gammatone response was originally proposed in 1972 as a description of revcor functions 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. It is defined such that one radian is the angle subtended at ...

) 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 In psychoacoustics, a pure tone is a sound with a sinusoidal waveform; that is, a sine wave of constant frequency, phase-shift, and amplitude. By extension, in signal processing a single-frequency tone or pure tone is a purely sinusoidal signal ...

) 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 percept ...

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. Lindeberg and Friberg define a new family of generalized gammatone filters.

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