Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 220

in the FPT (+) at N P = 180 and 260 are depicted in Fig. 6.10 via the Argand
plots in Descartes rectangular coordinates. Likewise, Fig. 6.13 is devoted to
{ν − k }in the FPT (−) at N P = 180 and 220. Finally, Fig. 6.11 displays the

absolute values of amplitudes{d k }in the FPT (+) at N P = 180 and 260 as

a function of chemical shifts in the whole Nyquist range. The same type of
graph for the corresponding complete set{d − k

}retrieved by the FPT (−)

at

N P = 180 and 220 is illustrated in Fig. 6.14 .
In both versions of the FPT, there is a striking difference between the
Argand plots in Euler and Descartes coordinates when passing from the stage
prior to convergence to full convergence of genuine harmonic variables and
linear frequencies. Thus, in Fig. 6.9 for the FPT (+) , the only perceivable
difference to the naked eye between panels (i) and (ii) of the ArgandEuler
plots for{z k

}at N P = 180 and 260, respectively, is the absence and presence
of the data for the 11th harmonic as indicated by a small star symbol in
the first quadrant near the circumference of the unit circle |z| = 1 in the
direction of about 45 ◦ from the center of the circle C. Other poles and zeros
do not seem to differ much (only visually, of course) when comparing the
top (N P = 180) and bottom (N P = 260) subplots in Fig. 6.9. A similar

6.12 for{z − k observation is made while looking at Fig. }at N P = 180 and

220 in the FPT (−) , when comparing panels (i) and (ii), respectively. Here, of
course, all genuine harmonic variables{z − k

}are in the 4th quadrant, and the
approximate location of the mentioned small star symbol near k = 11 & 12 is
now placed at∼360 ◦ −45 ◦ = 315 ◦ relative to the center of the unit circle C).
Except for the 11th resonance, the remaining possible differences in the
Argand plots in Euler polar coordinates at and near convergence of genuine
resonances are hardly visible because of the logarithmic scale which is in
herent in the exponential harmonic variables{z ± k }. Thus, it is expected that

more perceivable differences between genuine resonances at and prior to con
vergence will be observed using a linear scale. Indeed, this is precisely the
case, as can be seen by inspecting the Argand plots in Descartes rectangular
coordinates for linear frequencies{ν k }at N P = 180 and 260 for the FPT (+)

on panels (i) and (ii) in Fig. 6.10. Significant differences are obvious among
a number of genuine harmonic variables from panels (i) and (ii) of this figure.
As to Froissart doublets, it is seen that they do not converge at all. As more
spurious poles and zeros emerge in passing from N P = 180 to N P = 260,
they redistribute themselves along two major lines or wings that, via a care
ful inspection, can be seen to differ from each other throughout the entire
Nyquist range 3 . A like situation is encountered in the FPT (−) when com
paring the corresponding panels (i) and (ii) at N P = 180 and 220 in Fig.
6.13. Similar to linear frequencies, more pronounced differences between the
reconstructions at and prior to convergence are also anticipated on the level

3 This is even more evident when the corresponding tabular data for all the retrieved reso-
nances are analyzed, as done in Refs. [6, 35]).