:-)
Introduction
Anatomy
Physiology
The auditory filter
Loudness
Pitch
Auditory Scene Analysis 1
Auditory Scene Analysis 2
Ecological Psychology - General
Ecological Psychology of Hearing
Sound Localization
Bibliography
CDs
Sound Examples

Dik J. Hermes
nospam-d.j.hermes@tue.nl-nospam

Eindhoven University of Technology
Department of Industrial Engineering & Innovation Sciences
Human-Technology Interaction
P.O. Box 513
NL 5600 MB Eindhoven
The Netherlands

The auditory filter


One of the most important functions of the peripheral auditory system is its function as a frequency analyzer.  The crucial role in this respect is played by the basilar membrane, on which every location is characterized by a best or characteristic frequency.

Physiologically the auditory filter can, e.g., be measured by:
  • neural tuning curves
    Slide: Tuning curves (Durrant & Lovrinic, 3rd ed., p. 217)
  • neural impulse responses
    Slide: Impulse response basilar membrane (Moore, p. 27)
  • Slide: Neural impulse responses (Yost, p. 126)
  • Slide: Simulated impulse responses (Moore, p. 193)

Psycho-acoustically, this has lead to the concept of critical band, which represents the frequency resolution of the auditory system.  Psycho-acoustically the critical bandwidth can, e.g., be measured by:

  • masking
    Auditory demonstrations. Tracks 2-6. "Critical bands by masking"
  • loudness comparison
    Auditory demonstrations. Track 7. "Critical bands by loudness comparison" 

The Bark scale

This has lead to the definition of the Bark scale, a frequency scale on which equal distances correspond with perceptually equal distances.   Above about 500 Hz this scale is more or less equal to a logarithmic frequency axis. Below 500 Hz the Bark scale becomes more and more linear.

The ERB-rate scale

Measuring the critical bands below 500 Hz appeared to be quite difficult, due to the fact that, at low frequencies,  the sensitivity and the efficiency of the auditory system rapidly diminishes, while headphone technology caused leakage of acoustic energy at low frequencies.  More accurate measurements of the auditory-filter bandwidth have now lead to the ERB-rate scale.  These measurements have used notched-noise maskers to measure the auditory filter bandwidth.  In general, on this ERB-rate scale the auditory-filter bandwidth, expressed in equivalent rectangular bandwidth (ERB), is smaller than on the Bark scale, a difference which becomes larger for lower frequencies. (From the towns of the research groups where they were developed, Hartmann, 1997, has doped the ERB-rate scale the Cambridge scale, while he doped the Bark scale, the Munich scale.)  If the unit on the ERB-rate scale is E, the number of ERBs, and f is frequency, the conversion formula's are:

  • f = 229 · (10E/21.4 - 1),
  • E = 21.4 · log10(0.00437·f+1).

All this has lead to the notion of an auditory filter bank. In this view, the peripheral hearing system consists of a large number of band-pass filters.  The frequency characteristics of these filters are constant on an ERB-rate scale.  Mathematically they can be expressed as so-called rounded exponentials. When we define g = |(f - fc )/ fc|,

| H(f) |2 = (1+pg) · e -pg.

In the time domain, the impulse response is modelled as a so-called gamma tone:

  • h(t) = c · t (g -1)  · e-t /t · cos(2pfct + j).

The notion of critical bandwidth or auditory-filter bandwidth arises in many different contexts in hearing research.

How to construct perceptual frequency scales

The ERB-rate scale z

We now want to construct a frequency scale z in such a way that equal intervals on that scale represent the same number of auditory filter bandwidths (for further details see Hartmann, 1997).  Let E(f) be the ERB at f. Point of departure is that the change in z, in number of ERBs, is such that

dz = (Dz/ Df) df.

In the equation 1/(Dz/ Df) = Df / Dz represents the change in f per unit z which, at that f, is exactly one ERB. Or, dz = (1/E(f)) df. Integration then yields:

which gives us the equation for calculating the ERB-rate scale from measured ERBs. The unit for this scale E is number of ERBs.

The Bark scale

The Bark scale is defined as a table (Zwicker, 1961).  The italics are the lower and higher bound of the critical bands with integer values.

Bark   frequency   Bark   frequency   Bark   frequency
0   20   8   920   16   3150
    50       1000       3400
1   100   9   1080   17   3700
    150       1170       4000
2   200   10   1270   18   4400
    250       1370       4800
3   300   11   1480   19   5300
    350       1600       5800
4   400   12   1720   20   6400
    450       1850       7000
5   510   13   2000   21   7700
    570       2150       8500
6   630   14   2320   22   9500
    700       2500       10500
7   770   15   2700   23   12000
    840       2900       13500
                24   15500

Both Bark en ERB-rate scale are presented in the figure below. The formula used for the Bark scale is from Traunmüller (1990), JASA 88, 97-1000.

 

Literature

Hartmann W.M.
Auditory filters.
In: Signals, sound, and sensation, Chapter 10.
Woodbury, NY: American Institute of Physics, 1997.

Moore B.C.J.
An introduction to the psychology of hearing, 3rd edition.
San Diego, CA: Academic Press, 1997.