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:

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.B.C.J. Moore and B.R. Glasberg, "Suggested formulae for calculating auditory-filter bandwidths and excitation patterns" Journal of the Acoustical Society of America 74: 750-753, 1983.

The above approximation was given in 1983 by Moore and Glasberg, who in 1990 published another (linear) approximation:B.R. Glasberg and B.C.J. Moore, "Derivation of auditory filter shapes from notched-noise data", Hearing Research, Vol. 47, Issues 1-2, p. 103-138, 1990.

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.

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 Cams. The scale can be constructed by solving the following differential system of equations:

:$\begin \mathrm(0) = 0\\ \frac = \mathrm(f)\\ \end$

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 () for ERB(''f'') yields:

:$\mathrm(f) = 11.17 \cdot \ln\left(\frac\right) + 43.0$

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

:$\mathrm(f) = 11.17268 \cdot \ln\left(1 + \frac\right)$
:$f = \frac - 14678.49$

where ''f'' is in Hz.

Using the linear approximation () for ERB(''f'') yields:

:$\mathrm(f) = 21.4 \cdot \log_(1 + 0.00437 \cdot f)$

where ''f'' is in Hz.

See also

* Critical bands
* Bark scale

References

{{Reflist

External links

* https://web.archive.org/web/20110427105916/http://www.ling.su.se/staff/hartmut/bark.htm

Acoustics
Hearing
Signal processing