Nonuniform Spectral Resampling

Recall sinc interpolation of a discrete-time signal [#!SmithAndGossett84!#]:

And recall asinc interpolation of a sampled spectrum (§2.5.2):

We see that resampling consists of an inner-product between the given
samples with a continuous *interpolation kernel* that is sampled
where needed to satisfy the needs of the inner product operation. In
the time domain, our interpolation kernel is a scaled sinc function,
while in the frequency domain, it is an asinc function. The
interpolation kernel can of course be horizontally *scaled* to
alter bandwidth [#!SmithAndGossett84!#], or more generally
*reshaped* to introduce a more general *windowing* in the
opposite domain:

- Width of interpolation kernel (main-lobe width)
1/width-in-other-domain
- Shape of interpolation kernel gain profile (window) in other domain

Getting back to non-uniform resampling of audio spectra, we have that
an auditory-filter frequency-response can be regarded as a
frequency-dependent *interpolation kernel* for nonuniformly
resampling the STFT frequency axis. In other words, an auditory
filter bank may be implemented as a non-uniform resampling of the
uniformly sampled frequency axis provided by an ordinary FFT, using
the auditory filter shape as the interpolation kernel.

When the auditory filters vary systematically with frequency, there
may be an equivalent *non-uniform frequency-warping* followed by
a *uniform sampling* of the (warped) frequency axis. Thus, an
alternative implementation of an auditory filter bank is to apply an
FFT (implementing a uniform filter bank) to a signal having a properly
*prewarped spectrum*, where the warping is designed to approximate
whatever auditory frequency axis is desired. This approach is
discussed further in Appendix E. (See also §2.5.2.)

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University