Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 121

component shape spectrum from panel (i) in the same figure.
Herein, we can therefore conclude that while obtaining the residual or error
spectra at the level of background noise may be a necessary condition, this is
not su cient for judging the reliability of estimation. Therefore, it is recom
mended to pass beyond the point where full convergence of the total shape
spectra has been reached for the first time (in this case above N P = 180) in
order to verify that anomalies as seen on panels (i) and (iv) of Fig. 3.11 do
not occur in the final results. Such final results are obtained for N P = 220
and 260 displayed on panels (ii) and (iii) for the components as well as panels
(v) and (vi) for the envelopes. Clearly, monitoring the stability of the com
ponent spectra should be done in concert with inspection of the constancy of
the reconstructed genuine spectral parameters.

3.5 Distributions of reconstructed spectral parameters

in the complex plane

Distributions of spectral parameters in FPT (+)

3.5.1

Figure 3.13 reveals further insights into the exact quantification within the
FPT (+) . As was previously the case in Figs. 3.3 and 3.10 , all the obtained
results are for N/4 = 256. The absorption total shape spectrum is shown on
panel (iv) in Fig. 3.13, where the individual numbers of resonances are located
near the related peaks. Thus each wellresolved isolated resonance is marked
by the corresponding separate number, e.g., k = 8, 9, etc. Similarly, the
overlapped, tightly overlapped and nearly degenerate resonances are labeled
as the sum of the pertinent peak numbers, e.g., k = 1 + 2 or k = 5 + 6 + 7,
etc. On panel (v) in Fig. 3.13 the absorption component shape spectra
of the constituent resonances k = 1−25 are shown, and all the individual
numbers are indicated for an easier comparison with panel (iv) on the same
figure. Thereby, the hidden structures are welldelineated in these component
shape spectra. These hidden resonances are those that are overlapped (k =
1 + 2, 3 + 4, 5 + 6 + 7, 14 + 15, 16 + 17), tightly overlapped (k = 22 + 23) and
nearly degenerate (k = 11 + 12).
As previously, the resonance positions Re(ν k ) as chemical shifts are in
descending order on the abscissa when proceeding from left to right. With
respect to the ordinate, if only panel (vi) for the distribution of resonance
frequencies in the corresponding complex plane is analyzed alone, without
regard to the other panels in Fig. 3.13, there does not appear to be a need
for inverting the imaginary frequency Im(ν k ). However, panel (vi) is most
informative if it is seen not only independently of the other panels, but also
in its relation to the rest of the panels in Fig. 3.13. In particular, panels (v)