Conclusion and outlooks - Signal Processing in MRS with Biomedical Applications - page 390

tally measured data. We also provide the optimally reliable solution of the
ubiquitous noise problem. This is done by unequivocal disentangling of the
genuine from spurious information using the concept of Froissart doublets. By
this strategy, spurious information is precisely identified by strong coupling of
unphysical poles and zeros through their strict coincidences. Such polezero
confluences are totally absent for physical, genuine resonances. Hence exact
noise separation or SNS as a novel paradigm in data analysis.

13.1 Leading role of Pade approximants in the theory of

rational functions and in MRS

General rational functions R(u) are defined as quotients R(u) = f(u)/g(u) of
two other functions f(u) and g(u) of a complexvalued independent variable
u. They play by far the most prominent role in the mathematical theory of
approximations. Importantly, it is this latter practical theory by which mathe
matics make their most significant and useful bridges towards other disciplines
across different research fields. The key mathematical features that determine
any function used in mathematical modeling across interdisciplinary applica
tions are the possible singularities (poles, cuts, branch points) and zeros. The
former and the latter are tightly connected, respectively, with the potential
existence of maximae (peaks) and minimae (valleys between adjacent peaks)
of the given function. The principal reason for the central role of rational func
tions of the general type R(u) = f(u)/g(u) in the theory of approximations
is in their mathematical form by which the numerator g(u) can provide ade
quate descriptions of singularities, whereas the denominator f(u) is suitable
for description of zeros. Poles and zeros can fully describe any system.
Notwithstanding its importance, generality does not necessarily always lead
to practical mathematical models for reallife systems. Thus, choosing some
very general form for f(u) in R(u) = f(u)/g(u) might not be so useful in
many practical situations, since f(u) besides having zeros could also possess
singularities on its own. In such a case, there will be a twofold source of singu
larities of R(u), one due to the denominator g(u) and the other stemming from
the numerator f(u). A more restricted, but also a more useful choice, would
be a subclass of general rational functions R(u) with a clearer separation of
the roles of the constituents f(u) and g(u). One such subclass would be the
set of pairs of functions{f(u),g(u)}for construction of R(u) = f(u)/g(u) in
which the only role of f(u) would be to describe zeros of R(u) whose singu
larities would then be left exclusively to g(u). In this way, we can be sure that
the sole singularities of R(u) are due to the denominator function g(u). This
choice would also imply that the only zeros of R(u) are those due to zeros of
the numerator f(u). For many applications, even this latter, restricted class