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 modelling 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 $$