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Equivalent rectangular bandwidth

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The equivalent rectangular bandwidth or ERB is a measure used in <a href="Psychoacoustics" title="Psychoacoustics">psychoacoustics</a>, which gives an approximation to the bandwidths of the filters in <a href="Human_hearing" class="mw-redirect" title="Human hearing">human hearing</a>, using the unrealistic but convenient simplification of modeling the filters as rectangular <a href="Band-pass_filter" title="Band-pass filter">band-pass filters</a>, or band-stop filters, like in tailor-made notched music training (TMNMT).

<div class="mw-heading mw-heading2"><h2 id="Approximations">Approximations</h2></div>

For moderate sound levels and young listeners, <a href="#CITEREFMooreGlasberg1983">Moore & Glasberg (1983)</a> suggest that the bandwidth of human auditory filters can be approximated by the <a href="Polynomial" title="Polynomial">polynomial</a> equation:<a href="#cite_note-mooreglasberg-1">[1]</a>

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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\mathsf {ERB}} (\ F\ )=6.23\cdot F^{2}+93.39\cdot F+28.52}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {\mathsf {ERB}} (\ F\ )=6.23\cdot F^{2}+93.39\cdot F+28.52}</annotation>

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</math><img src="./fe4e216c98148a951cb625c6611f92f7a39bc9b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.626ex; height:3.176ex;" alt="{\displaystyle \operatorname {\mathsf {ERB}} (\ F\ )=6.23\cdot F^{2}+93.39\cdot F+28.52}" loading="lazy"> </td> <td></td> <td class="nowrap">Eq.1</td></tr></tbody></table>

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 <a href="Simultaneous_masking" class="mw-redirect" title="Simultaneous masking">simultaneous masking</a> experiments and is valid from 0.1–6500Hz.<a href="#cite_note-mooreglasberg-1">[1]</a>

Seven years later, <a href="#CITEREFGlasbergMoore1990">Glasberg & Moore (1990)</a> published another, simpler approximation:<a href="#cite_note-glasbergmoore-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle \,\operatorname {\mathsf {ERB}} (\ f\ )=24.7\ {\mathsf {Hz}}\ \cdot \log _{10}\left({\frac {4.37\cdot f}{\ 1000\ {\mathsf {Hz}}\ }}+1\right)\,}</annotation>

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</math><img src="./926dc6a02da20c3996e96f12d8102ac17732328d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.35ex; height:6.176ex;" alt="{\displaystyle \,\operatorname {\mathsf {ERB}} (\ f\ )=24.7\ {\mathsf {Hz}}\ \cdot \log _{10}\left({\frac {4.37\cdot f}{\ 1000\ {\mathsf {Hz}}\ }}+1\right)\,}" loading="lazy"> <a href="#cite_note-glasbergmoore-2">[2]</a></td> <td></td> <td class="nowrap">Eq.2</td></tr></tbody></table>where f is in Hz and ERB(f) is also in Hz. The approximation is applicable at moderate sound levels and for values of f between 100 and 10000Hz.<a href="#cite_note-glasbergmoore-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="ERB-rate_scale">ERB-rate scale</h2></div>

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.<a href="#cite_note-3">[3]</a> The scale can be constructed by solving the following <a href="Differential_equation" title="Differential equation">differential</a> system of equations:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\mathrm {ERBS} (0)=0\\{\frac {df}{d\mathrm {ERBS} (f)}}=\mathrm {ERB} (f)\\\end{cases}}}</annotation>

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</math><img src="./475e76ef8009ce5c49ddedba98cee5671ee3105c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.036ex; height:7.509ex;" alt="{\displaystyle {\begin{cases}\mathrm {ERBS} (0)=0\\{\frac {df}{d\mathrm {ERBS} (f)}}=\mathrm {ERB} (f)\\\end{cases}}}" loading="lazy"></dd></dl>

The solution for ERBS(f) is the integral of the reciprocal of ERB(f) with the <a href="Constant_of_integration" title="Constant of integration">constant of integration</a> set in such a way that ERBS(0) = 0.<a href="#cite_note-mooreglasberg-1">[1]</a>

Using the second order polynomial approximation (<a href="#math_Eq.1">Eq.1</a>) for ERB(f) yields:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ERBS} (f)=11.17\cdot \ln \left({\frac {f+0.312}{f+14.675}}\right)+43.0}</annotation>

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</math><img src="./48617992a054224c037f10034a413290fda96559.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.143ex; height:6.176ex;" alt="{\displaystyle \mathrm {ERBS} (f)=11.17\cdot \ln \left({\frac {f+0.312}{f+14.675}}\right)+43.0}" loading="lazy"> <a href="#cite_note-mooreglasberg-1">[1]</a></dd></dl>

where f is in kHz. The VOICEBOX speech processing toolbox for <a href="MATLAB" title="MATLAB">MATLAB</a> implements the conversion and its <a href="Inverse_function" title="Inverse function">inverse</a> as:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ERBS} (f)=11.17268\cdot \ln \left(1+{\frac {46.06538\cdot f}{f+14678.49}}\right)}</annotation>

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</math><img src="./09825f2735c82c7908aa977f5091081645db619b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.984ex; height:6.176ex;" alt="{\displaystyle \mathrm {ERBS} (f)=11.17268\cdot \ln \left(1+{\frac {46.06538\cdot f}{f+14678.49}}\right)}" loading="lazy"> <a href="#cite_note-4">[4]</a></dd>

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<annotation encoding="application/x-tex">{\displaystyle f={\frac {676170.4}{47.06538-e^{0.08950404\cdot \mathrm {ERBS} (f)}}}-14678.49}</annotation>

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</math><img src="./ddc965647908a3ce044236de705b6d0da6075f7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.68ex; height:5.676ex;" alt="{\displaystyle f={\frac {676170.4}{47.06538-e^{0.08950404\cdot \mathrm {ERBS} (f)}}}-14678.49}" loading="lazy"> <a href="#cite_note-5">[5]</a></dd></dl>

where f is in Hz.

Using the linear approximation (<a href="#math_Eq.2">Eq.2</a>) for ERB(f) yields:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ERBS} (f)=21.4\cdot \log _{10}(1+0.00437\cdot f)}</annotation>

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</math><img src="./e3fd159a5163bf9224a5d4df9cd91cd935b40298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.471ex; height:2.843ex;" alt="{\displaystyle \mathrm {ERBS} (f)=21.4\cdot \log _{10}(1+0.00437\cdot f)}" loading="lazy"> <a href="#cite_note-josabel99-6">[6]</a></dd></dl>

where f is in Hz.

<ul><li><a href="Critical_bands" class="mw-redirect" title="Critical bands">Critical bands</a></li>

<li><a href="Bark_scale" title="Bark scale">Bark scale</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFMooreGlasberg1983" class="citation journal cs1">Moore, B.C.J.; Glasberg, B.R. (1983). "Suggested formulae for calculating auditory-filter bandwidths and excitation patterns". Journal of the Acoustical Society of America. 74: 750–753.</cite>

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<li id="cite_note-glasbergmoore-2">^ <a href="#cite_ref-glasbergmoore_2-0">a</a> <a href="#cite_ref-glasbergmoore_2-1">b</a> <a href="#cite_ref-glasbergmoore_2-2">c</a> <cite id="CITEREFGlasbergMoore1990" class="citation journal cs1">Glasberg, B.R.; Moore, B.C.J. (1990). "Derivation of auditory filter shapes from notched-noise data". Hearing Research. 47 (1–2): 103–138.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFHartmann2004" class="citation book cs1">Hartmann, William M. (2004). Signals, Sound, and Sensation. Springer Science & Business Media. p. 251. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781563962837</bdi>. <q>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.</q></cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFBrookes2012" class="citation web cs1">Brookes, Mike (22 December 2012). <a rel="nofollow" class="external text" href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/doc/voicebox/frq2erb.html">"frq2erb"</a>. VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBrookes2012" class="citation web cs1">Brookes, Mike (22 December 2012). <a rel="nofollow" class="external text" href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/doc/voicebox/erb2frq.html">"erb2frq"</a>. VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013.</cite>

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<li id="cite_note-josabel99-6"><a href="#cite_ref-josabel99_6-0">^</a> <cite id="CITEREFSmithAbel2007" class="citation web cs1">Smith, Julius O.; Abel, Jonathan S. (10 May 2007). <a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/bbt/Equivalent_Rectangular_Bandwidth.html">"Equivalent Rectangular Bandwidth"</a>. Bark and ERB Bilinear Transforms. Center for Computer Research in Music and Acoustics (CCRMA), Stanford University, USA. Retrieved 20 January 2013.</cite>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFHartmut_Traunmüller1997" class="citation web cs1">Hartmut Traunmüller (1997). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110427105916/http://www.ling.su.se/staff/hartmut/bark.htm">"Auditory scales of frequency representation"</a>. Phonetics at Stockholm University. Archived from <a rel="nofollow" class="external text" href="http://www.ling.su.se/staff/hartmut/bark.htm">the original</a> on 2011-04-27. Retrieved 2019-08-09.</cite></li>

<li><a rel="nofollow" class="external text" href="https://www.speech.kth.se/~giampi/auditoryscales/">Auditory Scales</a> by Giampiero Salvi: shows comparison between Bark, Mel, and ERB scales</li></ul></div><div><div class="zim-footer">

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