Equivalent Rectangular Bandwidth (ERB) Filter Bank

After the development of third-octave filter banks, psychoacousticians performed further studies to obtain more accurate estimates of the auditory filter bandwidths. Most recently, they arrived at a formula they use to refer to Equivalent Rectangular Bandwidth (ERB). While a formula to convert frequency values into ERB-based frequencies is provided in psychoacoustics laboratory assignment, the bandwidth of an ERB filter centered at a given frequency $f_c$ is

$$BW_{ERB} = 24.7 \left(0.00437 f_c + 1 \right).$$ (7)

It is important to note that the formula above converts a frequency (in Hz) to a bandwidth (also in Hz). To convert a frequency in Hz to a frequency in units of ERB-bands, the formula from psychoacoustics laboratory assignment should be used, namely

$$ERB_{rate} = 21.4 \log \left( 0.00437 f_c + 1 \right)$$ (8)