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.

Approximations

The center-frequency-dependent bandwidth of human auditory filters can be approximated by the polynomial equation:

$\mathrm {ERB} =6.23\cdot f^{2}+93.39\cdot f+28.52$

where f is the center frequency of the filter in kHz and ERB 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.

A newer approximation is:

$\mathrm {ERB} =107.94\cdot f+24.7$

ERB-rate scale

A scale relating frequency to ERB, the ERB-rate scale (sometimes called the ERB scale) can be constructed by solving the differential equation:

${\frac {df}{dv}}=6.23\cdot f^{2}+93.39\cdot f+28.52$

Constraining the scale to be zero at frequency zero results in:

$v=11.17\ln \left({\frac {f+0.312}{f+14.675}}\right)+43.0$

where v is frequency expressed in ERB-rate units and f is frequency in kHz. In this warped frequency scale, the filter bandwidth is constant.

The VOICEBOX speech processing toolbox for MATLAB implements the conversion and its inverse as:

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

$f={\frac {676170.4}{47.06538-e^{0.08950404\cdot v}}}-14678.49$

where f is frequency in Hz.

See also

• Critical bands
• Bark scale

External links

• http://ccrma.stanford.edu/~jos/bbt/Equivalent_Rectangular_Bandwidth.html
• http://www2.ling.su.se/staff/hartmut/bark.htm