Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 206

intermediate graphs prior to complete convergence in the FPT (±) . For the
latter cases, to be able to zoom into a narrow interval around the value of
N P for which convergence has been reached, we need to choose partial sig
nal lengths that are not equal to integer powers of 2 (i.e., not the structured
numbers from the FFT). This will be encountered in subsection 6.5.2 .
Subsection 6.5.1 deals only with converged reconstructions in the FPT (±)
as illustrated at the same N P , which is a quarter of the full signal length,
N P = N/4 = 256. Figures 6.1 -6.4 and 6.5 -6.8 present the results from the
FPT (+) and FPT (−) , respectively. In the FPT (+) , the noiseless data are given
in Figs. 6.1 and 6.2, whereas the noisy cases are shown in Figs. 6.3 and 6.4.
Likewise, in the FPT (−) , the noiseless cases are displayed in Figs. 6.5 and 6.6,
whereas the noisy data are plotted in Figs. 6.7 and 6.8.
Figures 6.1 and 6.3 [FPT (+) ] as well as Figs. 6.5 and 6.7 [FPT (−) ] display
the Argand plots for all the harmonic variables via the distributions of the
reconstructed poles z ± k,Q and zeros z ± k,P with respect to the unit circle in Euler
polar coordinates. Similarly, Figs. 6.2 and 6.4 [FPT (+) ] as well as Figs. 6.6
and 6.8 [FPT (−) ] on panel (i) depict the Argand plots of linear frequencies
of all the retrieved poles ν ± k,Q and zeros ν ± k,P in Descartes rectangular coordi
nates. Shown on panel (ii) in Figs. 6.2 and 6.4 [FPT (+) ] as well as in Figs.
6.6 and 6.8 [FPT (−) ] are the absolute values|d ± k

|of the recovered amplitudes

d ± k . The input data{ν k ,|d k |}are given in Figs. 6.2, 6.4, 6.6 and 6.8.

In Figs. 6.1-6.8, Froissart doublets from the FPT (±) are seen as being
distributed along circles (or arcs) and lines (or wings) in the polar and rect
angular coordinates, respectively. Away from the relatively narrow range of
genuine spectral parameters, the distributions of Froissart doublets are con
figured in quite a regular way for the noisefree time signals in Figs. 6.1 and
6.2 for the FPT (+) , as well as in Figs. 6.5 and 6.6 for the FPT (−) . The re
constructed data for the noisecorrupted FID show a totally different pattern
for constellations of Froissart doublets which are very irregularly configured
in Figs. 6.3 and 6.4 for the FPT (+) , as well as in Figs. 6.7 and 6.8 for the
FPT (−) . Such sharply distinct behaviors of Froissart doublets generated from
noisefree and noisecorrupted FIDs are due a marked instability of spurious
spectral structures to even the slightest changes in the input time signals. In
the noisy time signal{c n +r n

}of the correspond
ing noiseless counterpart{c n }cause random ruptures or bifurcations in the
distributions of Froissart doublets along deformed circular arcs and chains for
harmonic variables and linear frequencies, respectively. However, the most
important observation here is that for both noiseless and noisy FIDs, pole
zero coincidences are seen to occur in a remarkably systematic manner. This
permits a perfectly clear distinction of all spurious from true resonances for
unperturbed as well as perturbed FIDs. From this unequivocal differentiation
between noisefree and noisecorrupted MR time signals, all the correct values
for the genuine spectral parameters can be exactly reconstructed, including
the fundamental frequencies, the corresponding amplitudes and the original

}, random perturbations{r n