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).

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:

${\displaystyle \operatorname {\mathsf {ERB}} (\ F\ )=6.23\cdot F^{2}+93.39\cdot F+28.52}$

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 0.1–6500Hz.

Seven years later, Glasberg & Moore (1990) published another, simpler approximation:

${\displaystyle \,\operatorname {\mathsf {ERB}} (\ f\ )=24.7\ {\mathsf {Hz}}\ \cdot \left({\frac {4.37\cdot f}{\ 1000\ {\mathsf {Hz}}\ }}+1\right)\,}$

Eq.2

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

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. 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.

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

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

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

${\displaystyle \mathrm {ERBS} (f)=11.17268\cdot \ln \left(1+{\frac {46.06538\cdot f}{f+14678.49}}\right)}$

${\displaystyle f={\frac {676170.4}{47.06538-e^{0.08950404\cdot \mathrm {ERBS} (f)}}}-14678.49}$

where f is in Hz.

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

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

where f is in Hz.

• 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.
• Auditory Scales by Giampiero Salvi: shows comparison between Bark, Mel, and ERB scales