Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 91

tra produces the corresponding absorption total shape spectrum. In Fig. 3.10
the numbers and acronyms are also indicated on a map of MRdetectable
metabolites from the human brain. Figures 3.11 and 3.12 address the inad
equacy of the usual practice in MRS to rely solely upon the residual spectra
in error analysis.
•The fifth section 3.5 is presented through 6 subsections. These are: sub
section 3.5.1 for the distributions of complex frequencies and amplitudes, as
well as poles, shown via the harmonic variables for the constituent resonances
in the associated complex planes as reconstructed by the FPT (+) after full
convergence ( Fig. 3.13 ) , subsection 3.5.2 which is the same as subsubsection
3.5.1 , except for using the FPT (−) ( Fig. 3.14 ) , subsections 3.5.3 and 3.5.4
with the convergence of the reconstructed complex frequencies ( Figs. 3.15
and 3.16 ) and subsections 3.5.5 and 3.5.6 for the convergence of the absolute
values of amplitudes ( Figs. 3.17 and 3.18 ) .
•The sixth section 3.6 contains a preview on Froissart doublets ( Fig. 3.19 )
in the FPT (±) . Here, the emphasis is in illustrating the main aspect of the
exact Froissartbased separation of genuine from spurious resonances in an
important part of the full Nyquist range with all the MRdetectable metabo
lites of the healthy human brain. More details on Froissart doublets are given
in chapters 5 and 6.
•The seventh section 3.7 p rovides an indepth overview of the results from
this chapter emphasizing the key importance of exact quantification for mag
netic resonance spectroscopy.
3.1 Input data (tabular & graphic) and reconstructed
tabular data
3.1.1 Input tabular data for the spectral parameters of 25
resonances
Table 3.1 presents the exact 4digit input data of all the spectral parameters
for 25 complex attenuated exponentials that constitute the presently synthe
sized FID from (3.1). These parameters are the fundamental frequencies and
the associated amplitudes. The realvalued frequencies Re(ν k ) are also termed
chemical shifts. The quantities|d k |represent the absolute values or moduli
of the corresponding amplitudes d k . All the individual phases φ k of the am
plitudes d k are chosen to have zero values, so that d k =|d k |exp (iφ k ) =|d k |.
These zero values for the phases are customary for spectral analysis of syn
thesized time signals in MRS [142]. Clearly, however, this is not a restriction
for the FPT which can process FIDs with nonzero phases φ k . The decay time
constants{λ k }that are proportional to the transverse relaxation times T 2k
are not explicitly given in Table 3.1. These can be deduced from the inverses