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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 filter]]s.
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== Approximations ==
 
For moderate sound levels and young listeners, the bandwidth of human auditory filters can be approximated by the [[polynomial]] equation:
 
{{NumBlk|:|<math>
\mathrm{ERB}(f) = 6.23 \cdot f^2 + 93.39 \cdot f + 28.52
</math> <ref name=mooreglasberg/>|{{EquationRef|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&nbsp;kHz.<ref name=mooreglasberg>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.</ref>
 
The above approximation was given in 1983 by Moore and Glasberg,<ref name=mooreglasberg/> who in 1990 published another approximation:<ref name=glasbergmoore>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.</ref>
 
{{NumBlk|:|<math>
\mathrm{ERB}(f) = 24.7 \cdot (4.37 \cdot f + 1)
</math> <ref name=glasbergmoore/>|{{EquationRef|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&nbsp;kHz.<ref name=glasbergmoore/>
 
== 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 equation|differential]] system of equations:
 
:<math>
\begin{cases}
\mathrm{ERBS}(0) = 0\\
\frac{df}{d\mathrm{ERBS}(f)} = \mathrm{ERB}(f)\\
\end{cases}
</math>
 
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.<ref name=mooreglasberg/>
 
Using the second order polynomial approximation ({{EquationNote|Eq.1}}) for ERB(''f'') yields:
 
:<math>
\mathrm{ERBS}(f) = 11.17 \cdot \ln\left(\frac{f+0.312}{f+14.675}\right) + 43.0
</math> <ref name=mooreglasberg/>
 
where ''f'' is in kHz. The VOICEBOX speech processing toolbox for [[MATLAB]] implements the conversion and its [[Inverse function|inverse]] as:
 
:<math>
\mathrm{ERBS}(f) = 11.17268 \cdot \ln\left(1 + \frac{46.06538 \cdot f}{f + 14678.49}\right)
</math> <ref>{{cite web |url=http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/doc/voicebox/frq2erb.html |title=frq2erb |last1=Brookes |first1=Mike |date=22 December 2012  |work=VOICEBOX: Speech Processing Toolbox for MATLAB |publisher=Department of Electrical & Electronic Engineering, Imperial College, UK |accessdate=20 January 2013}}</ref>
:<math>
f = \frac{676170.4}{47.06538 - e^{0.08950404 \cdot \mathrm{ERBS}(f)}} - 14678.49
</math> <ref>{{cite web |url=http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/doc/voicebox/erb2frq.html |title=erb2frq |last1=Brookes |first1=Mike |date=22 December 2012  |work=VOICEBOX: Speech Processing Toolbox for MATLAB |publisher=Department of Electrical & Electronic Engineering, Imperial College, UK |accessdate=20 January 2013}}</ref>
 
where ''f'' is in Hz.
 
Using the linear approximation ({{EquationNote|Eq.2}}) for ERB(''f'') yields:
 
:<math>
\mathrm{ERBS}(f) = 21.4 \cdot log_{10}(1 + 0.00437 \cdot f)
</math> <ref name=josabel99>{{cite web |url=https://ccrma.stanford.edu/~jos/bbt/Equivalent_Rectangular_Bandwidth.html |title=Equivalent Rectangular Bandwidth |last1=Smith |first1=Julius O. |last2=Abel |first2=Jonathan S. |date=10 May 2007 |work=Bark and ERB Bilinear Transforms |publisher=Center for Computer Research in Music and Acoustics (CCRMA), Stanford University, USA |accessdate=20 January 2013}}</ref>
 
where ''f'' is in Hz.
 
==See also==
 
* [[Critical bands]]
* [[Bark scale]]
 
==References==
 
{{Reflist}}
 
== External links ==
* http://www2.ling.su.se/staff/hartmut/bark.htm
 
[[Category:Acoustics]]
[[Category:Hearing]]
[[Category:Signal processing]]

Revision as of 18:36, 4 February 2014

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