Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 90

Viewed together, the major properties of the FPT (+) and FPT (−) are com
plementary. Thus, each of these two versions of the FPT must independently
arrive at a tradeoff in order to perform optimally. The outlined differences
between the FPT (+) and FPT (−) justify the use of both versions of the FPT
even for noiseless time signals. In such a case, the genuine resonances are
known exactly, so that spurious poles can be identifiable with fidelity. A fa
vorable result from the presentlydesigned testing on a typical synthesized
FID will provide an impetus to apply the FPT to distinguish genuine and
spurious resonances in encoded time signals from MRS.
Since our aim is quantification of FIDs from MRS, the synthesized time
signals are chosen to possess the main physical features of importance to
clinical diagnostics. We achieved this similarity by using the numerical values
of the complex fundamental frequencies and the amplitudes in the synthesized
FID to be reminiscent of the tabulated data as reported in the MRS literature.
Furthermore, the FID generated theoretically with these spectral parameters
yields absorption total shape spectra similar to a highly resolved spectrum
computed from an FID encoded via MRS from a healthy human brain at
short echo time (20 ms) at the magnetic field strength B 0 =1.5T [88].
The findings and the detailed interpretation from this chapter are presented
in seven sections 3.1 -3.7 . These results are obtained using a sequence of partial
signal lengths N/M (M = 2−32) as well as the full length N with M = 1.
•The first section 3.1 presents the tabular input and reconstructed data
of all the spectral parameters. It also gives graphic illustrations of the input
data and shows the superiority of parametric signal processing in MRS over
nonparametric estimations. This includes 4 subsections. Three subsections
are with tabular data: 3.1.1 for the input data ( Table 3.1 ) as well as sub
sections 3.1.2 and 3.1.3 for convergence of the numerical values of all the
spectral parameters reconstructed by the FPT (−) ( Tables 3.2 a nd 3.3 ) . Sub
section 3.1.4 g ives the physical interpretations of the graphs of the input data
( Fig. 3.1 ) . It also shows the clinical advantages of Pade reconstruction of the
component shape spectra and the associated metabolite concentrations that
are completely lacking from the corresponding total shape spectra as the only
outcome of the Fourier analysis ( Fig. 3.2 ) .
•The second section 3.2 has subsection 3.2.1 with the absorption total
shape spectra ( Fig. 3.3 ) and subsection 3.2.2 comparing the convergence
rates of these spectra in the FPT (−) and FFT for varying N/M ( Figs. 3.4

and 3.5 ).
•The third section 3.3 continues the analyses in subsections 3.3.1 and 3.3.2
for the residual or error spectra ( Figs. 3.6 and 3.7 ) and subsections 3.3.3 and
3.3.4 with the consecutive difference spectra ( Figs. 3.8 and 3.9 ) . Section 3.3
is a part of the intrinsic error analysis in the FPT (+) and FPT (−) without

recourse to the FFT or to any other estimator.
•The fourth section 3.4 c ontains the absorption component and total shape
spectra ( Figs. 3.10 - 3.12 ) . The absorption component shape spectra display
each individual resonance. The sum of all these elementary, constituent spec