The equivalent rectangular bandwidth or ERB is a measure used in
psychoacoustics
Psychoacoustics is the branch of psychophysics involving the scientific study of sound perception and audiology—how humans perceive various sounds. More specifically, it is the branch of science studying the psychological responses associated ...
, 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 filter
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects ( attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two- ...
s, or band-stop filters, like in tailor-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
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
equation:
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.
[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.]
The above approximation was given in 1983 by Moore and Glasberg,
[ who in 1990 published another (linear) approximation:][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.]
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.[
]
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
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connect ...
set in such a way that ERBS(0) = 0.[
Using the second order polynomial approximation () for ERB(''f'') yields:
: ][
where ''f'' is in kHz. The VOICEBOX speech processing toolbox for ]MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
implements the conversion and its inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when ad ...
as:
:
:
See also
* Critical bands
* Bark scale
References
{{Reflist
External links
* https://web.archive.org/web/20110427105916/http://www.ling.su.se/staff/hartmut/bark.htm
Acoustics
Hearing
Signal processing