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Equivalent rectangular bandwidth

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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.


  • Approximations 1
  • ERB-rate scale 2
  • See also 3
  • References 4
  • External links 5


For moderate sound levels and young listeners, the bandwidth of human auditory filters can be approximated by the polynomial equation:

\mathrm{ERB}(f) = 6.23 \cdot f^2 + 93.39 \cdot f + 28.52 [1]






where f is the center frequency of the filter in kHz and ERB(f) is the bandwidth of the filter in Hz. The approximation is based on the results of a number of published simultaneous masking experiments and is valid from 0.1 to 6.5 kHz.[1]

The above approximation was given in 1983 by Moore and Glasberg,[1] who in 1990 published another approximation:[2]

\mathrm{ERB}(f) = 24.7 \cdot (4.37 \cdot f + 1) [2]






where f is in kHz and ERB(f) is in Hz. The approximation is applicable at moderate sound levels and for values of f between 0.1 and 10 kHz.[2]

ERB-rate scale

The ERB-rate scale, or simply ERB scale, can be defined as a function ERBS(f) which returns the number of equivalent rectangular bandwidths below the given frequency f. It can be constructed by solving the following differential system of equations:

\begin{cases} \mathrm{ERBS}(0) = 0\\ \frac{df}{d\mathrm{ERBS}(f)} = \mathrm{ERB}(f)\\ \end{cases}

The solution for ERBS(f) is the integral of the reciprocal of ERB(f) with the constant of integration set in such a way that ERBS(0) = 0.[1]

Using the second order polynomial approximation (Eq.1) for ERB(f) yields:

\mathrm{ERBS}(f) = 11.17 \cdot \ln\left(\frac{f+0.312}{f+14.675}\right) + 43.0 [1]

where f is in kHz. The VOICEBOX speech processing toolbox for MATLAB implements the conversion and its inverse as:

\mathrm{ERBS}(f) = 11.17268 \cdot \ln\left(1 + \frac{46.06538 \cdot f}{f + 14678.49}\right) [3]
f = \frac{676170.4}{47.06538 - e^{0.08950404 \cdot \mathrm{ERBS}(f)}} - 14678.49 [4]

where f is in Hz.

Using the linear approximation (Eq.2) for ERB(f) yields:

\mathrm{ERBS}(f) = 21.4 \cdot log_{10}(1 + 0.00437 \cdot f) [5]

where f is in Hz.

See also


  1. ^ a b c d e B.C.J. Moore and B.R. Glasberg, "Suggested formulae for calculating auditory-filter bandwidths and excitation patterns" Journal of the Acoustical Society of America 74: 750-753, 1983.
  2. ^ a b c B.R. Glasberg and B.C.J. Moore, "Derivation of auditory filter shapes from notched-noise data", Hearing Research, Vol. 47, Issues 1-2, p. 103-138, 1990.
  3. ^ Brookes, Mike (22 December 2012). "frq2erb". VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013. 
  4. ^ Brookes, Mike (22 December 2012). "erb2frq". VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013. 
  5. ^ Smith, Julius O.; Abel, Jonathan S. (10 May 2007). "Equivalent Rectangular Bandwidth". Bark and ERB Bilinear Transforms. Center for Computer Research in Music and Acoustics (CCRMA), Stanford University, USA. Retrieved 20 January 2013. 

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