GeezerVison

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).

For moderate sound levels and young listeners, Moore & Glasberg (1983) suggest that the bandwidth of human auditory filters can be approximated by the polynomial equation:^[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 100–6500Hz.^[1]

Seven years later, Glasberg & Moore (1990) published another, simpler approximation:^[2]

where f is in kHz and ERB(f) is also in kHz. The approximation is applicable at moderate sound levels and for values of f between 100 and 10000Hz.^[2]

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:

$\{\begin{array}{c}ERBS\left(0\right)=0\\ \frac{df}{dERBS\left(f\right)}=ERB\left(f\right)\end{array}$

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:

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

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

$ERBS\left(f\right)=11.17268\cdot \ln (47.065-\frac{676170.42\cdot f}{f+14678.49})$^[4]

$f=\frac{676170.4}{47.06538-e^{0.08950404\cdot ERBS\left(f\right)}}-14678.49$^[5]

where f is in Hz.

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

$ERBS\left(f\right)=21.4\cdot {\log }_{10}(1+4.37\cdot f)$^[6]

where f is in kHz.

• Critical bands
• Bark scale

• Hartmut Traunmüller (1997). "Auditory scales of frequency representation". Phonetics at Stockholm University. Archived from the original on 2011-04-27. Retrieved 2019-08-09.