Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 216

allows the exact solution of the quantification problem by the FPT (−) . On the
other hand, the FPT (+) can be seen in Figs. 6.1 -6.4 to sharply separate all
the physical from every nonphysical resonance in the two disjoint regions,
inside|z|< 1 and outside|z|> 1 the unit circle for harmonic variables, as
well as for frequencies via Im(ν + ) > 0 and Im(ν + ) < 0. This represents an
unprecedented separation of genuine from the spurious (noisy or noiselike)
informational contents of the investigated FIDs by using the FPT (+) . Such a
signalnoise separation is expected to play a major role in optimally reliable
spectral analysis, not only for quantifications in MRS, but also in other areas
of signal processing in many different branches of basic as well as applied
sciences, including research and implementations in engineering, technology
and industry [5, 6].

6.5.2 Zooming near convergence for Pade genuine resonances
and instability of non-converged configurations of Frois-
sart doublets in FPT (±)
As discussed, there is yet another grouping of the figures from this chapter
depending on the employed partial signal length N P . Thus, the analysis from
subsection 6.5.1 is entirely devoted to quantification by the FPT (±) at a quar
ter of the whole signal length (N P = N/4 = 256). At this value of the partial
length N P , full convergence is obtained in the FPT (±) for all the genuine
resonances. However, it is also important to give a graphic display of the con
figurations of spectral parameters in complex planes prior to convergence 2 .
For this reason, in the present subsection, we shall zoom into a relatively
small partial length interval (N P = 180, 220, 260) around a quarter of the
full signal length from the preceding analyses. Here, in dealing with these
three nonFFT type partial signal lengths, we shall restrict our discussion to
the noiseless FID. This is deemed su cient in the present graphic illustra
tions of the main trend in changes of the mentioned configurations during the
process of convergence towards the stabilized results for genuine resonances.
Simultaneously, this should also su ce to illuminate the corresponding typical
patterns of movements of Froissart doublets in complex planes with increased
order of the diagonal FPT (±) , i.e., with the augmented common degree of
numerator and denominator Pade polynomials.
Thus, the output data obtained using the partial signal lengths N P = 180−
260 of the same noiseless FID from subsection 6.5.1 are given in Figs. 6.9-
6.11 for the FPT (+) and in Figs. 6.12 -6.14 for the FPT (−) . In Fig. 6.9 for
the FPT (+) , shown are the Argand plots in the Euler polar coordinates for
harmonic variables{z k

}at N P = 180 and 260. Figure 6.12 deals with similar

graphs for{z − k }at N P = 180 and 220 in the FPT (−) . Linear frequencies{ν k }

2 Recall that a similar line of thought was also present in sub-section 6.4.2 of section 6.4 for
the tabular data.