89.12.191.249: Removed erroneous log10 in formula (refer to equation 3 in [Glasberg & Moore])

2025-08-18T11:22:16Z

Removed erroneous log10 in formula (refer to equation 3 in [Glasberg & Moore])

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{{short description|Measure used in psychoacoustics}}
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, or band-stop filters, like in tailor-made notched music training (TMNMT).

== Approximations ==
For moderate sound levels and young listeners, {{harvp|Moore|Glasberg|1983}} suggest that the bandwidth of human auditory filters can be approximated by the [[polynomial]] equation:<ref name=mooreglasberg>{{cite journal |first1=B.C.J. |last1=Moore |first2=B.R. |last2=Glasberg |year=1983 |title=Suggested formulae for calculating auditory-filter bandwidths and excitation patterns |journal=Journal of the Acoustical Society of America |volume=74 |pages=750-753 }}</ref>

{{NumBlk|:|<math>
\operatorname\mathsf{ERB}(\ F\ ) = 6.23 \cdot F^2 + 93.39 \cdot F + 28.52
</math> |{{EquationRef|1|Eq.1}}}}

where {{mvar|F}} is the center frequency of the filter, in kHz, and {{nobr|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–{{gaps|6|500|Hz}}.<ref name=mooreglasberg/>

Seven years later, {{harvp|Glasberg|Moore|1990}} published another, simpler approximation:<ref name=glasbergmoore>{{cite journal |first1=B.R. |last1=Glasberg |first2=B.C.J. |last2=Moore |year=1990 |title=Derivation of auditory filter shapes from notched-noise data |journal=Hearing Research |volume=47 |issue=1-2 |pages=103-138 }}</ref>

{{NumBlk|:|<math>\, \operatorname\mathsf{ERB}(\ f\ ) = 24.7\ \mathsf{Hz}\ \cdot \left( \frac{ 4.37 \cdot f }{\ 1000\ \mathsf{Hz}\ } + 1 \right)\, </math> <ref name=glasbergmoore/>|{{EquationRef|2|Eq.2}}}}where {{mvar|f}} is in Hz and {{nobr|ERB({{mvar|f}})}} is also in Hz. The approximation is applicable at moderate sound levels and for values of {{mvar|f}} between 100 and {{gaps|10|000|Hz}}.<ref name=glasbergmoore/>

== 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 known ERBs, or as Cams, following a suggestion by Hartmann.<ref>{{cite book |last1=Hartmann |first1=William M. |title=Signals, Sound, and Sensation |date=2004 |publisher=Springer Science & Business Media | page = 251 | isbn = 9781563962837 | quote = Unfortunately, the Cambridge unit has given the name 'ERB' in the literature, which stands for 'Equivalent rectangular bandwidths', and therefore does not distinguish it from any other measure of the critical band since the time of Fletcher. We call the Cambridge unit a 'Cam' instead. }}</ref> The scale 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 ==
* {{cite web|url=http://www.ling.su.se/staff/hartmut/bark.htm|title=Auditory scales of frequency representation|author=Hartmut Traunmüller|date=1997|website=Phonetics at Stockholm University|access-date=2019-08-09|archive-date=2011-04-27|archive-url=https://web.archive.org/web/20110427105916/http://www.ling.su.se/staff/hartmut/bark.htm|url-status=dead}}
* [https://www.speech.kth.se/~giampi/auditoryscales/ Auditory Scales] by Giampiero Salvi: shows comparison between Bark, Mel, and ERB scales

[[Category:Acoustics]]
[[Category:Hearing]]
[[Category:Signal processing]]

Equivalent rectangular bandwidth - Revision history

89.12.191.249: Removed erroneous log10 in formula (refer to equation 3 in [Glasberg & Moore])