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:<ref name=mooreglasberg>Template:Cite journal</ref>

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.<ref name=mooreglasberg/>

Seven years later, Template:Harvp published another, simpler approximation:<ref name=glasbergmoore>Template:Cite journal</ref>

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.<ref name=glasbergmoore/>

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.<ref>Template:Cite book</ref> The scale can be constructed by solving the following differential system of equations:

<math>

\begin{cases} \mathrm{ERBS}(0) = 0\\ \frac{df}{d\mathrm{ERBS}(f)} = \mathrm{ERB}(f)\\ \end{cases} </math>

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.<ref name=mooreglasberg/>

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

<math>

\mathrm{ERBS}(f) = 11.17 \cdot \ln\left(\frac{f+0.312}{f+14.675}\right) + 43.0 </math> <ref name=mooreglasberg/>

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

<math>

\mathrm{ERBS}(f) = 11.17268 \cdot \ln\left(1 + \frac{46.06538 \cdot f}{f + 14678.49}\right) </math> <ref>Template:Cite web</ref>

<math>

f = \frac{676170.4}{47.06538 - e^{0.08950404 \cdot \mathrm{ERBS}(f)}} - 14678.49 </math> <ref>Template:Cite web</ref>

where f is in Hz.

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

<math>

\mathrm{ERBS}(f) = 21.4 \cdot \log_{10}(1 + 0.00437 \cdot f) </math> <ref name=josabel99>Template:Cite web</ref>

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