Editing Equivalent rectangular bandwidth (section)

== 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.