Analysis and application of the spectral warping transform to digital signal processing
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Date
2007
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Massey University
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Abstract
This thesis provides a thorough analysis of the theoretical foundations and
properties of the Spectral Warping Transform. The spectral warping transform
is defined as a time-domain-to-time-domain digital signal processing
transform that shifts the frequency components of a signal along the frequency
axis. The z -transform coefficients of a warped signal correspond to
z -domain ‘samples’ of the original signal that are unevenly spaced along the
unit circle (equivalently, frequency-domain coefficients of the warped signal
correspond to frequency-domain samples of the original signal that are unevenly
spaced along the frequency axis). The location of these unevenly
spaced frequency-domain samples is determined by a z -domain mapping
function. This function may be arbitrary, except that it must map the unit
circle to the unit circle.
It is shown that, in addition to the frequency location, the bandwidth,
duration and amplitude of each frequency component of a signal are affected
by spectral warping. Specifically, frequency components within bands that
are expanded in frequency have shortened durations and larger amplitudes
(conversely, components in compressed frequency bands become longer with
smaller amplitudes).
A property related to the expansion and compression of the duration of
frequency components is that if a signal is time delayed (its digital sequence
is prepended with zeroes) then each of the frequency components will have
a different delay after warping. This time-domain separation phenomenon
is useful for separating in time the frequency components of a signal. Such
separation is employed in the generation of spectrally flat chirp signals. Because
spectral warping will generally expand the duration of some frequency
components within a signal, the transform must produce more output samples
than there are (non-zero) input samples in order to avoid time-domain
aliasing. A discussion of the necessary output signal length is presented.
Particular attention is given to spectral warping using all-pass mapping
function, which can be realised as a cascade of all-pass filters. There exists
an efficient hardware implementation for this all-pass SW realisation [1, 2].
A proof-of-concept application-specific integrated circuit that performs the
core operations required by this algorithm was developed.
Another focus of the presented research is spectral warping using a piecewise-
linear mapping function. This type of spectral warping has the advantage
that the changes in frequency, duration and amplitude between the
non-warped and warped signals are constant factors over fixed frequency
bands.
A matrix formulation of the spectral warping transformation is developed.
It presents the spectral warping transform as a single matrix multiplication.
The transform matrix is the product of the three matrices that represent
three conceptual steps. The first step is to apply a discrete Fourier transform
to the time-domain signal, providing the frequency-domain representation.
Step two is an interpolation to produce the signal content at the desired
new frequency samples. This interpolation effectively provides the frequency
warping. The final step is an inverse DFT to transform the signal back into
the time domain. A special case of the spectral warping transform matrix
has the same result as a linear (finite-impulse-response) filter, showing that
spectral warping is a generalisation of linear filtering. The conditions for the
invertibility of the spectral warping transformation are derived.
Several possible realisation of the SW transform are discussed. These
include two realisation using parallel finite-impulse-response filter banks and
a realisation that uses a cascade of infinite-impulse-response filters.
Finally, examples of applications for the spectral warping transform are
given. These include: non-uniform spectral analysis (and signal generation),
approximate spectral analysis in the time domain, and filter design.
This thesis concludes that the SW transform is a useful tool for the manipulation
of the frequency content of digital signals, and is particularly
useful when the frequency content of a signal (or the frequency response of
a system) over a limited band is of interest. It is also claimed that the SW
transform may have valuable applications for embedded mixed-signal testing.
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Digital system processing