Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 135

computation and, therefore, it is essential to reduce their original dimension
without information loss. This is important even without the obvious concern
for computational demands because capturing the essence of the investigated
large system by an adequate extraction of a relatively small number of the
main parametrizing characteristics allows simple and yet reliable descriptions.
To achieve this goal, the parametric version of the FPT does not need a spe
cially designed procedure, since Froissart doublets solve the dimensionality
reduction problem en route while separating genuine from spurious informa
tion. This can be accomplished either in the whole Nyquist range or in one or
more subintervals (windows) using the original time signal of the full length
N. Alternatively, the Nyquist interval can be split into any number of sub
intervals and the original time signal could be beamspaced, i.e., decimated
to a much shorter length N D ≪N with no information loss in any given

window. Such a method, called decimated Pade approximant (DPA), was in
troduced by Belkic et al. [13] and explicitly combined with Froissart doublets
by O'Sullivan et al. [63, 64].
Using the parametric variant of the FPT, both the polezero stabilization
leading to the true K, and the exact reconstruction of all the fundamental
frequencies, as well as the corresponding amplitudes are illustrated together in
Fig. 3.19 . Only a small number of all the obtained Froissart doublets appears
in the shown frequency window in Fig. 3.19. The selected subinterval 0 - 6
ppm is important because all the MRdetectable brain metabolites lie within
this chemical shift domain of the full Nyquist range. Froissart doublets as
spurious resonances are detected by the confluence of poles and zeros in the
list of the Padereconstructed spectral parameters, as per (2.186). It is seen
on panel (i) in Fig. 3.19 that the FPT (+) disentangles the physical from
unphysical resonances by the opposite signs of their imaginary frequencies.
In other words, the FPT (+) provides the exact separation of the genuine
from any spurious, i.e., noiselike content of the investigated time signal. By
contrast, in the FPT (−) depicted on panel (ii) in Fig. 3.19, genuine and spuri
ous resonances are mixed together, since they all have the same positive sign of
their imaginary frequencies. Nevertheless, the emergence of Froissart doublets
also remains evidently clear in the FPT (−) via coincidence of poles and zeros,
with the ensuing unambiguous identification of spurious resonances. Precisely
due to polezero coincidences, each Froissart doublet has zerovalued ampli
tudes, as seen on panel (iii) in Fig. 3.19, as per (2.227). This result, as another
signature of Froissart doublets, represents a further check of consistency and
fidelity of separation of genuine from spurious resonances and this is the basis
of the SNS concept mentioned in chapter 1. Note that the full auxiliary lines
on each subplot in Fig. 3.19 are drawn merely to transparently delineate the
areas with Froissart doublets.
Once the Froissart doublets are identified and discarded from the whole set
of the results, only the reconstructed parameters of the genuine resonances will
remain in the output data. Crucially, however, the latter set of Paderetrieved
spectral parameters also contains the exact number K of true resonances as