# 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.

## Approximations

For moderate sound levels and young listeners, the bandwidth of human auditory filters can be approximated by the polynomial equation:

(Eq.1)

where f is the center frequency of the filter in kHz and ERB(f) 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 to 6.5 kHz.[1]

The above approximation was given in 1983 by Moore and Glasberg,[1] who in 1990 published another (linear) approximation:[2]

(Eq.2)

where f is in kHz and ERB(f) is in Hz. The approximation is applicable at moderate sound levels and for values of f between 0.1 and 10 kHz.[2]

## ERB-rate scale

The ERB-rate scale, or simply ERB scale, can be defined as a function ERBS(f) which returns the number of equivalent rectangular bandwidths below the given frequency f. It can be constructed by solving the following differential system of equations:

The solution for ERBS(f) is the integral of the reciprocal of ERB(f) with the constant of integration set in such a way that ERBS(0) = 0.[1]

Using the second order polynomial approximation (Eq.1) for ERB(f) yields:

[1]

where f is in kHz. The VOICEBOX speech processing toolbox for MATLAB implements the conversion and its inverse as:

[3]
[4]

where f is in Hz.

Using the linear approximation (Eq.2) for ERB(f) yields:

[5]

where f is in Hz.