# 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:

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

$\mathrm{ERB}(f) = 24.7 \cdot (4.37 \cdot f + 1)$ [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 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:

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

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:

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

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

$\mathrm{ERBS}(f) = 11.17268 \cdot \ln\left(1 + \frac{46.06538 \cdot f}{f + 14678.49}\right)$ [3]
$f = \frac{676170.4}{47.06538 - e^{0.08950404 \cdot \mathrm{ERBS}(f)}} - 14678.49$ [4]

where f is in Hz.

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

$\mathrm{ERBS}(f) = 21.4 \cdot log_{10}(1 + 0.00437 \cdot f)$ [5]

where f is in Hz.