# 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 (TMNMT).

## Approximations

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

${\displaystyle \mathrm {ERB} (f)=6.23\cdot f^{2}+93.39\cdot f+28.52}$ [1]
(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]

${\displaystyle \mathrm {ERB} (f)=24.7\cdot (4.37\cdot f+1)}$ [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 ERB-number scale, can be defined as a function ERBS(f) 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.[3] The scale can be constructed by solving the following differential system of equations:

${\displaystyle {\begin{cases}\mathrm {ERBS} (0)=0\\{\frac {df}{d\mathrm {ERBS} (f)}}=\mathrm {ERB} (f)\\\end{cases}}}$

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:

${\displaystyle \mathrm {ERBS} (f)=11.17\cdot \ln \left({\frac {f+0.312}{f+14.675}}\right)+43.0}$ [1]

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

${\displaystyle \mathrm {ERBS} (f)=11.17268\cdot \ln \left(1+{\frac {46.06538\cdot f}{f+14678.49}}\right)}$ [4]
${\displaystyle f={\frac {676170.4}{47.06538-e^{0.08950404\cdot \mathrm {ERBS} (f)}}}-14678.49}$ [5]

where f is in Hz.

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

${\displaystyle \mathrm {ERBS} (f)=21.4\cdot \log _{10}(1+0.00437\cdot f)}$ [6]

where f is in Hz.