The Bark Frequency Scale

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

Noise (in signal processing) is a random signal. In the language of statistical signal processing, a noise signal is typically modeled as 'stochastic process', which is in turn defined as a sequence of random variables. — Click for https://ccrma.stanford.edu/~jos/sasp/What_Noise.html

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

The size of the passband, typically expressed in Hz. — Click for http://en.wikipedia.org/wiki/Bandpass

Hertz means 'cycles per second', and is abbreviated 'Hz'. It is the standard name for frequency units in cycles per second. Another common frequency unit is 'radians per second', which is two-pi times frequency in Hertz. — Click for http://en.wikipedia.org/wiki/Hertz

The Bark Frequency Scale

The Bark scale ranges from 1 to 24 Barks, corresponding to the first 24 critical bands of hearing [39]. 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 narrowband 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 allpass transformations, it is preferable to optimize the allpass 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.