Conclusion and outlooks - Signal Processing in MRS with Biomedical Applications - page 394

task, since they are meromorphic functions whose poles are the zeros of the
denominator polynomial. Moreover, the zeros of the numerator polynomial
are the zeros of this response function in the form of the polynomial quotient.
The locations of these poles in complex planes give metabolite chemical shifts
and lifetimes in MRS from example (ii). Further, the Cauchy residues of the
rational polynomial taken at the found nodal frequencies{ω k }are the sought

amplitudes{d k }whose absolute values and phases yield metabolite concen

trations and phases of the associated time signal components. From these
remarks, particularly in MRS as the central theme of the present book, it is
clear how theory (via mathematics, physics, chemistry, biology) comes into
play. It is the experimental measurement which encodes the time signal, as
a response of the examined system to external perturbations. The theory,
however, explains and interprets the measured experimental data. Therefore,
theories driven by experiments are an indispensable and inseparable part of
measurements. Especially, spectral analysis, as an inverse problem, starts
from the encoded time signal and builds the whole theory of measurement.
In this topic, we show both through a theoretical development and illus
trations as they apply to clinical diagnostics in the realm of MRS, how Pade
approximants can be used to find the unique solutions{ω k ,d k }to the dif
ficult quantification problem of critical importance in medicine. Moreover,
within the Pade approximant, we present a novel and long awaited procedure
in signal processing via the denoising Froissart filter for exact signalnoise
separation. This is implemented through the powerful concept of Froissart
doublets, manifested as polezero coincidences in the system response func
tion. Such a denoising filter can clearly distinguish genuine (physical) from
spurious (unphysical) resonances by reliance upon the stability criterion. We
can vary different quantities, e.g., signal length, or we may change the de
grees of the numerator and denominator Pade polynomials. These variations
would leave a certain number of pairs of spectral parameters{ω k ,d k

}stable,
once they converged, and they indeed do, as soon as the Pade approximant
exactly reconstructs the true number of metabolites. Such stable resonances
represent genuine components from the input time signal. However, the same
variations of the polynomial degrees would never lead to convergence of the
remaining subset from the whole retrieved collection{ω k ,d k }. These non

converged resonances fluctuate irregularly with the changes in the degrees
of Pade polynomials and they never stabilize. Hence, they are categorized
as spurious, as they are absent from the input time signal. These spurious
resonances are easily spotted through their unmistakable twofold signature
identified as polezero confluences and the associated zero amplitudes in the
Pade response function. Because they appear as pairs, such spurious struc
tures are called Froissart doublets after Froissart who 40 years ago discovered
this phenomenon by computer experiments when adding random noise to a
deterministic synthesized signal with a single genuine component.
We significantly expand and generalize the concept of Froissart doublets as
the main part of the comprehensive strategy of signalnoise separation, which