Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 6

ing the powerful concept of Froissart doublets [44], a phenomenon unique
to the Pade polynomial quotient, which is the form of the spectrum in the
FPT. To make a link to the field of signal processing, it should be noted
that the equivalent name “polezero cancellation” for Froissart doublets was
independently used in studies on discrete time systems described by the PzT
[45]. Froissart reported his finding in 1969 in a Symposium Proceeding in
Mathematics [44]. Unluckily, his results have not appeared subsequently as
an in extenso paper in a regular journal to be accessible to a wider audi
ence. Fortunately, already in 1970 - 1972 his method was cited and further
analyzed by Basdevant [46, 47], as well as in 1973/1974 by Gammel and Nut
tall [48]-[50]. These initial studies dealt with Froissart doublets within the
Pade approximant (PA) [51] based upon some Taylor series having the ex
pansion coe cients perturbed by random noise. Subsequently, from 1978 to
the current times, the most thorough investigations on Froissart doublets,
or equivalently, Froissart “noise functions” as well as on the socalled Frois
sart polynomials, were accomplished by Gilewicz [52]-[57] who developed a
more formal mathematical basis for the whole subject. Froissart doublets are
within the realm of nonanalytic or quasianalytic “noise functions” [58]-[61]
because they closely mimic random noise, which is encountered in all compu
tations with finite precision numerical arithmetic (roundofferrors), as well
as in experimental measurements. This aspect was examined by Barone [62]
and illustrated with the help of Monte Carlo simulations. Froissart doublets
within the decimated Pade approximant [5, 14] were used by O'Sullivan et al.
[63, 64] in spectral analysis of experimentally measured noisy time signals from
reverberant acoustic environments, including recorded music (the sonamed
music transposition) where the measured FIDs are reliably modeled by sums
of complex damped harmonics with stationary amplitudes. A multivariate
version of Froissart phenomena was formulated by Becuwe and Cuyt [65] and
numerically illustrated in the special case of two dimensions.
Within MRS, we have systematized the concept of Froissart doublets in two
variants of the Pade approximant, the FPT (+) and FPT (−) , by establishing
a novel and longawaited strategy of exact signalnoise separation (SNS), as
an unequivocal twofold signature through the extreme closeness of spurious
poles to zeros and, simultaneously, via the smallness of the corresponding am
plitudes [11, 24, 30, 31, 34]. All these mentioned studies on Froissart doublets
represent only a selected small part of the otherwise large bibliography which
can be found in Ref. [6]. Froissart doublets will be revisited in depth with
detailed illustrations in chapters 3, 5 and 6.
Noise is ubiquitous and, to reemphasize, it appears in time signals that
are measured experimentally or generated theoretically. As stated, the ori
gin of noise in computations is in finite arithmetics, and it manifests itself
through random roundofferrors. Noise in measured FIDs is usually due to
many factors that can be systematic errors, statistical uncertainties, random
perturbations, etc. Irrespective of its multifaceted origin, for reliable interpre
tation of measured time signals and their spectra, it is of prime importance to