Gammatone filter

Sample gammatone

Gammatone filter is a linear filter used extensively in auditory neuroscience, psychoacoustics, and speech processing to model the human auditory system. It is designed to simulate the frequency selectivity of the auditory system, closely mimicking the response of the basilar membrane within the cochlea to sound. The gammatone filter's impulse response is characterized by a sinusoidal carrier modulated by a gamma distribution envelope, which gives the filter its name.

Overview

The concept of the gammatone filter was introduced to provide a more accurate representation of the human ear's filtering properties compared to earlier models, such as the band-pass filter. The gammatone filter's parameters can be adjusted to closely match the critical bands of hearing, which are specific frequency ranges over which the auditory system perceives sound as a single entity.

Mathematical Formulation

The impulse response of a gammatone filter is given by:

\[ g(t) = a \cdot t^{n-1} \cdot e^{-2\pi b t} \cdot \cos(2\pi f_c t + \phi) \]

where:

• \(a\) is the amplitude,
• \(t\) is time,
• \(n\) is the filter order, typically 4,
• \(b\) is the bandwidth parameter,
• \(f_c\) is the center frequency of the filter,
• \(\phi\) is the phase of the carrier.

The parameters \(b\) and \(n\) are crucial in determining the shape of the filter's response, with \(n\) affecting the slope of the filter skirts and \(b\) controlling the bandwidth.

Applications

Gammatone filters are widely used in various fields related to hearing and speech processing. In auditory neuroscience, they help in understanding how the human auditory system processes different frequencies. In psychoacoustics, gammatone filters are used to investigate how humans perceive sound, including the perception of pitch and timbre. In speech processing and automatic speech recognition, these filters are employed to extract features from speech signals that are robust to noise and other distortions.

Implementation

Implementing a gammatone filter bank involves creating a series of filters with center frequencies spaced along the cochlea's length, mimicking the tonotopic organization of the human auditory system. Each filter in the bank captures the energy of the sound signal in its respective frequency band, allowing for detailed analysis of the signal's spectral content.

See Also

• Auditory system
• Cochlea
• Psychoacoustics
• Speech processing

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