Conclusion and outlooks - Signal Processing in MRS with Biomedical Applications - page 389

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Conclusion and outlooks
This topic is on theory and practice of spectral analysis as it applies to quan
tification of a wide class biomedical time signals. The presented methodology
is general and can be applied to many other interdisciplinary fields which need
not have an overlap with biomedicine. Our principal method selected for this
challenging task of solving inverse synthesistype problems in data interpreta
tion is the fast Pade transform, FPT. This method, which can autonomously
pass from the time to the frequency domain with no recourse to Fourier inte
grals, represents a novel unification of the customary Pade approximant and
the causal Pade−z−transform. The FPT automatically and simultaneously
performs interpolation and extrapolation of the examined data. The idea of
synthesis of time signals in a search for an adequate explanation of the ob
served variation in studied phenomena consisting of composite effects, is to
find a subclass of simpler constituent elements related to the fundamental
structure of the examined system which produces a response to external per
turbations. Such decompositions of complicated into simpler effects is in the
heart of quantification of time signals through their parametrizations.
Finding a relatively small number of fundamental parameters, eliciting poles
and zeros, that could capture the main features of the investigated system as
sociated with a given time signal is of paramount theoretical and practical
importance. In this way, theoretical explanations of phenomena involving
time signals exhibit a great potential in simplifying the stated initial task and
coordinating its different parts by the decomposition analysis of observed com
posite phenomena. As such, the theory of time signals becomes an essential
complement of the corresponding measurements. This complementarity does
not stop with theoretical explanations and interpretations, but also provides
practical tools that enable interpolation where measured data have not been
recorded, and extrapolation to the ranges where predictions could be made
about the possible behavior of the system under study. Measurements in this
field yield time signals, but it is theory which provides frequency spectra and
decomposition of encoded data into their constituent fundamental elements.
Such inverse problems are di cult due to mathematical illconditioning and
the possible solutions are further hampered by inevitable noise.
The present book shows how this type of important problem, known as
spectral analysis, quantification or harmonic inversion, can be solved by the
FPT either with machine accuracy for theoretically generated/simulated time
signals or with the best possible precision for the corresponding experimen
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