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Analysis and application of the spectral warping transform to digital signal processing
Allen, Warwick Peter Malcolm
Thesis
Massey UniversityThis thesis provides a thorough analysis of the theoretical foundations and properties of the Spectral Warping Transform. The spectral warping transform is defined as a timedomaintotimedomain 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, frequencydomain coefficients of the warped signal correspond to frequencydomain samples of the original signal that are unevenly spaced along the frequency axis). The location of these unevenly spaced frequencydomain 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 timedomain 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 (nonzero) input samples in order to avoid timedomain aliasing. A discussion of the necessary output signal length is presented. Particular attention is given to spectral warping using allpass mapping function, which can be realised as a cascade of allpass filters. There exists an efficient hardware implementation for this allpass SW realisation [1, 2]. A proofofconcept applicationspecific 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 nonwarped 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 timedomain signal, providing the frequencydomain 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 (finiteimpulseresponse) 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 finiteimpulseresponse filter banks and a realisation that uses a cascade of infiniteimpulseresponse filters. Finally, examples of applications for the spectral warping transform are given. These include: nonuniform 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 mixedsignal testing.
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