## The Bark Frequency Scale

Based on the results of many psychoacoustic experiments, the
*Bark scale* is defined so that the critical bands of human
hearing each have a width of one Bark. By representing spectral
energy (in dB) over the Bark scale, a closer correspondence is
obtained with spectral information processing in the ear
(§7.3).

The Bark scale ranges from 1 to 24 Barks, corresponding to the first 24 critical bands of hearing [304]. The published Bark band edges are given in Hertz as [0, 100, 200, 300, 400, 510, 630, 770, 920, 1080, 1270, 1480, 1720, 2000, 2320, 2700, 3150, 3700, 4400, 5300, 6400, 7700, 9500, 12000, 15500]. The published band centers in Hertz are [50, 150, 250, 350, 450, 570, 700, 840, 1000, 1170, 1370, 1600, 1850, 2150, 2500, 2900, 3400, 4000, 4800, 5800, 7000, 8500, 10500, 13500]. These center-frequencies and bandwidths are to be interpreted as samplings of a continuous variation in the frequency response of the ear to a sinusoid or narrow-band noise process. That is, critical-band-shaped masking patterns should be seen as forming around specific stimuli in the ear rather than being associated with a specific fixed filter bank in the ear.

Note that since the Bark scale is defined only up to 15.5 kHz, the highest sampling rate for which the Bark scale is defined up to the Nyquist limit, without requiring extrapolation, is 31 kHz. The 25th Bark band certainly extends above 19 kHz (the sum of the 24th Bark band edge and the 23rd critical bandwidth), so that a sampling rate of 40 kHz is implicitly supported by the data. We have extrapolated the Bark band-edges in our work, appending the values [20500, 27000] so that sampling rates up to 54 kHz are defined. While human hearing generally does not extend above 20 kHz, audio sampling rates as high as 48 kHz or higher are common in practice.

The Bark scale is defined above in terms of frequency in Hz versus
Bark number. For computing optimal bilinear transformations, it is
preferable to optimize the fit to the *inverse* of this
map, *i.e.*, Barks versus Hz, so that the mapping error will be measured
in Barks rather than Hz.

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The Bilinear Transform

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A Sum of Gaussian Random Variables is a Gaussian Random Variable