Digital Signal Processing Reference
In-Depth Information
12
Recapitulation of Pade-optimized processing
of biomedical time signals
In this chapter, we shall discuss the main aspects and place of Pade ap
proximants in a general framework of rational functions in the mathematical
literature through a large branch known as theory of approximations. As
such, we shall recapitulate how one is unavoidably led to Pade approximants
as the optimal rational function for many fields, and especially for solving the
spectral analysis or quantification problem in various fields such as ICRMS,
NMR, MRS, etc. The salient features of the Pade methodology will be il
luminated, such as convergence acceleration of slowly converging series and
induced convergence of divergent series as well as sequences. Also summarized
will be the astounding precision of Pade approximants in reconstructing the
machine accurate, true values of all the input spectral parameters via complex
frequencies and amplitudes. The novel concept of exact signalnoise separa
tion will be highlighted, as well, with reference to Froissart doublets. Finally,
a special focus will be placed upon the relevance of the Padeguided MRS for
tumor diagnostics in clinical oncology. Not all these multifaceted issues will
be covered in this recapitulation with equal weight. Some pertinent aspects
are left for the next chapter with our concluding remarks and prospects for
future developments with emphasis on applications to diagnostics in medicine,
where Pade approximants find its new home.
12.1 The central role of rational functions in the theory
of approximations
General rational functions R(z
−1
) are defined as quotients of two other func
tions f(z
−1
) and g(z
−1
) via
1
R(z
−1
) =
f(z
−1
)
g(z
−1
)
.
(12.1)
1
Here, for convenience, the independent variable z
−
1
is chosen as the inverse of a general
complex variable z.
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