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Auditory frequency-scale warping is closely related to the topic of
auditory filter banks which are non-uniform bandpass filter
banks designed to imitate the frequency resolution of human hearing
[30,32]. Classical auditory filter banks include
constant-Q filter banks such as the widely used third-octave filter
bank. More recently, constant-Q filter banks for audio have been
devised based on the wavelet transform, including the auditory
wavelet filter bank
[8]. Auditory filter banks have also been based more
directly on psychoacoustic measurements, leading to approximations of the
auditory filter frequency response in terms of a Gaussian function
[27], a ``rounded exponential'' [29],
and more recently the gammatone (or ``Patterson-Holdsworth'') filter
bank [30,32].
The gamma-chirp filter bank further adds a level-dependent asymmetric
correction to the basic gammatone channel frequency response, thereby
providing a yet more accurate approximation to the auditory frequency
response [10,9].
All auditory filter banks can be seen as defining some linear to
warped frequency mapping, since the filter-bank output signals are
non-uniformly distributed versus frequency. While this paper is
concerned primarily with approximating the Bark frequency scale using
a first-order conformal map, the same approach can be used to
approximate the warping defined by any pre-existing auditory filter
bank.
As another application of the results of this paper, an alternative
to the use of an auditory filter bank is a simpler uniform filter
bank, such as an FFT, applied to a signal having a warped spectrum,
where the warping is designed to approximate whatever auditory frequency
axis is deemed most appropriate. It so happens that the earliest related
work we are aware of was concerned with exactly this application, as we
take up in the next subsection.
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