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, or band-stop filters, like in taylor-made notched music training (TMNMT).
Approximations
For moderate sound levels and young listeners, the bandwidth of human auditory filters can be approximated by the polynomial equation:
| [1] | (Eq.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 (linear) approximation:[2]
| [2] | (Eq.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 ERB-number scale, can be defined as a function ERBS(f) which returns the number of equivalent rectangular bandwidths below the given frequency f. The units of the ERB-number scale are Cams. The scale can be constructed by solving the following differential system of equations:
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:
where f is in kHz. The VOICEBOX speech processing toolbox for MATLAB implements the conversion and its inverse as:
where f is in Hz.
Using the linear approximation (Eq.2) for ERB(f) yields:
where f is in Hz.
See also
References
- ^ 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.
- ^ 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.
- ^ Brookes, Mike (22 December 2012). "frq2erb". VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013.
- ^ Brookes, Mike (22 December 2012). "erb2frq". VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013.
- ^ 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.