Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 211

number of physical resonances as explicitly shown in Figs. 6.1 -6.8 . It can
be seen that, upon stabilization via convergence of genuine resonances, the
number of spurious resonances is systematically about four times larger than
that of genuine resonances. Another common and interesting feature observed
in Figs. 6.1-6.8 is that genuine and spurious resonances seem to repel each
other as if they were some opposite electrostatic charges. The greater the
distance between these resonances, the less interaction between them. This
can best be seen for reconstructions from the noiseless FID as illustrated on
panel (i) in Fig. 6.2 in the FPT (+) , as well as in Fig. 6.6 in the FPT (−) .
Here, the distant wings of Froissart doublets appear quite unperturbed, but
stronger disturbances of these spurious structures begin to set in near the
range populated by genuine resonances, [0.985, 4.68] ppm. At larger signal
lengths, such as N/2 = 512 and N/2 = 1024 [6, 35], many Froissart doublets
penetrate into the region right above the genuine resonances. Still none of
these spurious harmonics ever enter inside the unit circle, where all the har
monics of genuine resonances are located, as per the FPT (+) . As such, in this
variant of the FPT, the borderline|z|= 1 of the unit circle C acts as a cut
which Froissart doublets cannot cross at all. This is a common occurrence at
any partial length, including N/4 = 256 concerning Figs. 6.1 and 6.3.
Specifically, polezero confluences in Froissart doublets are seen using Eu
ler polar coordinates for harmonic variables in Figs. 6.1 (noisefree) and 6.3
(noisecorrupted) for the FPT (+) . This is also the case in Figs. 6.5 (noisefree)
and 6.7 (noisecorrupted) for the FPT (−) . Useful complementary illustrations
for the formation of Froissart pairs through polezero coincidences can also
be observed by employing Descartes rectangular coordinates for linear fre
quencies on panel (i) in Figs. 6.2 (noisefree) and 6.4 (noisecorrupted) for
the FPT (+) . The corresponding configurations from the FPT (−) are shown
on panel (i) in Figs. 6.6 (noisefree) and 6.8 (noisecorrupted). On panel
(ii) in Figs. 6.2 (noisefree) and 6.4 (noisecorrupted), displayed are Froissart
zerovalued amplitudes for the FPT (+) . The related constellations of the am
plitudes for the FPT (−) are presented on panel (ii) in Figs. 6.6 (noisefree) and
6.8 (noisecorrupted). This twofold signature (polezero confluences and the
accompanying zero amplitudes) represents the unequivocal manner by which
the FPT (±) can disentangle genuine from spurious resonances. As mentioned
in chapters 2, 3 and 5, one of the easiest visual means to immediately spot
all Froissart doublets is to search for their zero amplitudes, as on panel (ii)
for both considered time signals: noisefree in Figs. 6.2 [FPT (+) ] and 6.6
[FPT (−) ], as well as noisecorrupted in Figs. 6.4 [FPT (+) ] and 6.8 [FPT (−) ].
A small subinterval of the whole Nyquist interval for linear frequencies recon
structed by the FPT (±) using the same noiseless FID as in Figs. 6.2 and 6.6
was presented and discussed earlier in chapter 3 via Fig. 3.19, as a preview
of the more detailed illustrations and analyses from the current chapter.
As Figs. 6.5-6.8 clearly show, the FPT (−) mixes the genuine and spurious
resonances in the same region|z|> 1 and Im(ν − ) > 0. Still, the clear pattern
of Froissart doublets for harmonic variables, linear frequencies and amplitudes