Equivalent rectangular bandwidth

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 filters, or band-stop filters, like in tailor-made notched music training (TMNMT).

Approximations

For moderate sound levels and young listeners, Moore & Glasberg (1983) suggest that the bandwidth of human auditory filters can be approximated by the polynomial equation:^[1]

\operatorname {\mathsf {ERB}} (\ F\ )=6.23\cdot F^{2}+93.39\cdot F+28.52

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 0.1–6500Hz.^[1]

Seven years later, Glasberg & Moore (1990) published another, simpler approximation:^[2]

\,\operatorname {\mathsf {ERB}} (\ f\ )=24.7\ {\mathsf {Hz}}\ \cdot \left({\frac {4.37\cdot f}{\ 1000\ {\mathsf {Hz}}\ }}+1\right)\,

^[2]

Eq.2

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.^[2]

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.^[3] The scale can be constructed by solving the following differential system of equations:

{\begin{cases}\mathrm {ERBS} (0)=0\\{\frac {df}{d\mathrm {ERBS} (f)}}=\mathrm {ERB} (f)\\\end{cases}}

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.^[1]

Using the second order polynomial approximation (Eq.1) for ERB(f) yields:

\mathrm {ERBS} (f)=11.17\cdot \ln \left({\frac {f+0.312}{f+14.675}}\right)+43.0

^[1]

where f is in kHz. The VOICEBOX speech processing toolbox for MATLAB implements the conversion and its inverse as:

\mathrm {ERBS} (f)=11.17268\cdot \ln \left(1+{\frac {46.06538\cdot f}{f+14678.49}}\right)

^[4]

f={\frac {676170.4}{47.06538-e^{0.08950404\cdot \mathrm {ERBS} (f)}}}-14678.49

^[5]

where f is in Hz.

Using the linear approximation (Eq.2) for ERB(f) yields:

\mathrm {ERBS} (f)=21.4\cdot \log _{10}(1+0.00437\cdot f)

^[6]

where f is in Hz.

References

^ ^a ^b ^c ^d 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.
^ ^a ^b ^c Glasberg, B.R.; Moore, B.C.J. (1990). "Derivation of auditory filter shapes from notched-noise data". Hearing Research. 47 (1–2): 103–138.
^ Hartmann, William M. (2004). Signals, Sound, and Sensation. Springer Science & Business Media. p. 251. ISBN 9781563962837. 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.
^ Brookes, Mike (22 December 2012). "frq2erb". VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013.
^ Brookes, Mike (22 December 2012). "erb2frq". VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013.
^ Smith, Julius O.; Abel, Jonathan S. (10 May 2007). "Equivalent Rectangular Bandwidth". Bark and ERB Bilinear Transforms. Center for Computer Research in Music and Acoustics (CCRMA), Stanford University, USA. Retrieved 20 January 2013.

External links

Hartmut Traunmüller (1997). "Auditory scales of frequency representation". Phonetics at Stockholm University. Archived from the original on 2011-04-27. Retrieved 2019-08-09.
Auditory Scales by Giampiero Salvi: shows comparison between Bark, Mel, and ERB scales

[mooreglasberg-1] 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.

[glasbergmoore-2] Glasberg, B.R.; Moore, B.C.J. (1990). "Derivation of auditory filter shapes from notched-noise data". Hearing Research. 47 (1–2): 103–138.

[3] Hartmann, William M. (2004). Signals, Sound, and Sensation. Springer Science & Business Media. p. 251. ISBN 9781563962837. 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.

[4] Brookes, Mike (22 December 2012). "frq2erb". VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013.

[5] Brookes, Mike (22 December 2012). "erb2frq". VOICEBOX: Speech Processing Toolbox for MATLAB. Department of Electrical & Electronic Engineering, Imperial College, UK. Retrieved 20 January 2013.

[josabel99-6] Smith, Julius O.; Abel, Jonathan S. (10 May 2007). "Equivalent Rectangular Bandwidth". Bark and ERB Bilinear Transforms. Center for Computer Research in Music and Acoustics (CCRMA), Stanford University, USA. Retrieved 20 January 2013.

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