Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 89

shape spectra, because these are the only spectra that can be obtained by the
FFT. In a total shape spectrum generated through the FPT via the Heaviside
representation, the sum is carried out over all the spectral parameters. Thus,
the information on the inherently performed quantification is not readily ap
parent. As a check, the total shape spectrum in the FPT could be computed
directly, as well, via a frequencybyfrequency evaluation of the defining poly
nomial quotient without prior reconstructions of the spectral parameters.
In this chapter, we go beyond presenting the total shape spectra, by giving
the explicit numerical values of the reconstructed spectral parameters. In
this respect, the most stringent criterion is the exact retrieval of the known
complex frequencies and amplitudes. This is possible for a synthesized FID. As
mentioned, such time signals are considered in this chapter by choosing their
spectral parameters to be consistent with those FIDs that are encoded via
MRS. The quantification problems for these synthesized FIDs will presently
be solved using the two variants of the FPT, that are initially defined inside
and outside the unit circle, namely the FPT (+) and FPT (−) , respectively.
Strictly speaking, the theory foresees that for synthesized noisefree input
FIDs, these two versions of the FPT should generate the same results once
convergence has been attained in both variants for all the spectral parameters,
as per (2.187). Such a hypothesis is tested herein. This testing is important
in light of the fact that with noisy FIDs encoded via MRS, as those stud
ied in Refs. [8, 9], the use of both the FPT (+) and FPT (−) is needed as
an intrinsic crossvalidation of estimations. As per this crossvalidation, the
only acceptable reconstructions will be those found by both the FPT (+) and

FPT (−) .
It should be pointed out that these two versions of the FPT have different
algebraic structure. These are designed to work in the two complementary
regions of the complex plane for the harmonic variable z. The FPT (+) and
FPT (−) operate in the first and the fourth quadrant of the complex z−plane,
respectively. These two opposite regions are very different with respect to
the expected locations of spurious poles. This is the case for those from
simulations as well as for encoded FIDs. Thus, for the FPT (+) , we expect
that there will be a clear separation of the physical poles (inside the unit
circle,|z|< 1) from the nonphysical ones (outside the unit circle,|z|> 1).
This separation should facilitate robust and stable estimations by the FPT (+)
whenever there is noise in the examined FIDs. This postulate should be
verified, particularly since the FPT (+) must carry out the numerically di cult
and illconditioned task of analytical continuation by numerical means by
inducing convergence into the input Maclaurin sum (2.162) which diverges
inside the unit circle when N is increased.
On the other hand, the FPT (−) accelerates the already convergent input
sum (2.162). This is an easier task from the computational vantage point com
pared to analytical continuation. Instead, the FPT (−) has another challenge,
namely to separate the genuine from the spurious poles that are intermixed
within the same region outside the unit circle.