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:

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

A newer approximation is:

$\frac{df}{dv} = 107.94 \cdot f + 24.7$

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