Equivalent rectangular bandwidth

Template:Short description 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, Template:Harvp suggest that the bandwidth of human auditory filters can be approximated by the polynomial equation:

where Template:Mvar is the center frequency of the filter, in kHz, and Template:Nobr 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–Template:Gaps.

Seven years later, Template:Harvp published another, simpler approximation:

Template:NumBlkwhere Template:Mvar is in Hz and Template:Nobr is also in Hz. The approximation is applicable at moderate sound levels and for values of Template:Mvar between 100 and Template:Gaps.

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

$\{\begin{array}{c}\mathrm{E}\mathrm{R}\mathrm{B}\mathrm{S}(0)=0\\ \frac{df}{d\mathrm{E}\mathrm{R}\mathrm{B}\mathrm{S}(f)}=\mathrm{E}\mathrm{R}\mathrm{B}(f)\end{array}$

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.

Using the second order polynomial approximation (Template:EquationNote) for ERB(f) yields:

$\mathrm{E}\mathrm{R}\mathrm{B}\mathrm{S}(f)=11.17\cdot \ln \left(\frac{f+0.312}{f+14.675}\right)+43.0$

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

$\mathrm{E}\mathrm{R}\mathrm{B}\mathrm{S}(f)=11.17268\cdot \ln (1+\frac{46.06538\cdot f}{f+14678.49})$

$f=\frac{676170.4}{47.06538-e^{0.08950404\cdot \mathrm{E}\mathrm{R}\mathrm{B}\mathrm{S}(f)}}-14678.49$

where f is in Hz.

Using the linear approximation (Template:EquationNote) for ERB(f) yields:

$\mathrm{E}\mathrm{R}\mathrm{B}\mathrm{S}(f)=21.4\cdot {\log }_{10}(1+0.00437\cdot f)$

where f is in Hz.

See also

• Critical bands
• Bark scale

External links

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