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.

Researchers have derived a differential equation giving the ERB value $v$ in Hz of a human auditory filter with a center frequency of $f$ kHz

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

Solving this equation gives roughly the following relation between the ERB value $v$ and the frequency $f$ in Hz:

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

Or the converse:

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

See also

• Critical bands
• Bark scale

Reference

http://ccrma.stanford.edu/~jos/bbt/Equivalent_Rectangular_Bandwidth.html