Equivalent rectangular bandwidth

The equivalent rectangular bandwidth or ERB is a measure used in psychoacoustics, which gives an approximation to the bandwidths of the filters in human hearing, using the unrealistic but convenient simplification of modeling the filters as rectangular band-pass filters, or band-stop filters, like in tailor-made notched music training.

Approximations

For moderate sound levels and young listeners, suggest that the bandwidth of human auditory filters can be approximated by the polynomial equation:
where is the center frequency of the filter, in kHz, and is the bandwidth of the filter in Hz. The approximation is based on the results of a number of published simultaneous masking experiments and is valid from 0.1–.
Seven years later, published another, simpler approximation:
where is in Hz and is also in Hz. The approximation is applicable at moderate sound levels and for values of between 100 and.

ERB-rate scale

The ERB-rate scale, or ERB-number scale, can be defined as a function ERBS which returns the number of equivalent rectangular bandwidths below the given frequency f. The units of the ERB-number scale are known ERBs, or as Cams, following a suggestion by Hartmann. The scale can be constructed by solving the following differential system of equations:
The solution for ERBS is the integral of the reciprocal of ERB with the constant of integration set in such a way that ERBS = 0.
Using the second order polynomial approximation for ERB yields:
where f is in kHz. The VOICEBOX speech processing toolbox for MATLAB implements the conversion and its inverse as:
where f is in Hz.
Using the linear approximation for ERB yields:
where f is in Hz.