Equivalent rectangular bandwidth

The equivalent rectangular bandwidth or ERB is a measure used in psychoacoustics.

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:

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