Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 95

In addition, overlapped resonances will appear corresponding to the following
11 peaks: k = 1, 2 (0.985 ppm, 1.112 ppm), k = 3, 4 (1.548 ppm, 1.689 ppm),
k = 5, 6, 7 (1.959 ppm, 2.065 ppm, 2.145 ppm), k = 14, 15 (3.009 ppm, 3.067
ppm) and k = 16, 17 (3.239 ppm, 3.301 ppm). The very closely overlapped
resonances will appear as the following 2 peaks: k = 22, 23 (3.944 ppm, 3.965
ppm). These latter resonances are separated from each other by 0.021 ppm.
Finally, there will be almost degenerate resonances that are comprised of 2
peaks k = 11, 12 (2.675 ppm, 2.676 ppm) separated by a mere 0.001 ppm. In
the absorption total shape spectrum these two nearly coincident peaks will be
completely unresolved and will appear as a single resonance.
It is expected that in the k = 11 & 12 peaks of near degeneracy all the
conventional fitting techniques would fail to detect the smaller peak which
is k = 11. In fact, there would be no justifiable reason to initialize fitting
two peaks for a resonance which appears to be a single structure. Even if a
fitting procedure were to begin with the two preassigned modeling resonances,
disregarding the appearance of a bellshaped single peak, there would be no
justifiable reason for not choosing even a larger number, e.g., 3 or 4 or more
small peaks below the dominant resonance k = 12 in the k = 11 & 12 peaks.
This illustrates the fundamental ambiguity of fittings from MRS. This in
cludes such programs as VARPRO, AMARES, LCModel, etc. Specifically,
a key weakness of these fittings is their nonuniqueness in handling the in
verse quantification problem in MRS. This neardegeneracy k = 11 & 12 in
the presently synthesized FID substantially increases the overall challenge for
spectral analysis by any estimator, including the FPT.
3.1.2 Numerical values of the reconstructed spectral param-
eters at six signal lengths, N/M (N = 1024,M = 1−32)
Table 3.2 presents the detailed convergence rates of the numerical values of
the complex frequencies and amplitudes for the reconstructed resonances by
the FPT (−) using 6 partial signal lengths (N/32 = 32,N/16 = 64,N/8 =
128,N/4 = 256,N/2 = 512) as well as the full FID (N = 1024). In panels (i),
(ii) and (iii) of Table 3.2, the spectral parameters of the detected 10, 14 and 20
resonances are shown at N/32 = 32,N/16 = 64 and N/8 = 128, respectively.
Clearly, these latter findings are approximate values for the corresponding ex
act input data for the parameters. This occurs because the number of signal
points (N/M≤128) is not su cient. In contrast, panels (iv), (v) and (vi)
of Table 3.2 display the spectral parameters found at N/4 = 256,N/2 = 512
and N = 1024, respectively. These latter results should be compared with
the corresponding input data from Table 3.1 . It thereby can be seen that
from, e.g., panel (iv) of Table 3.2 for a quarter (N/4 = 256) of the full sig
nal length N that the FPT (−) has retrieved the entire set of 25 resonances
with all their exact values for all of the spectral parameters. Furthermore,
it is remarkable that these reconstructed spectral parameters remain totally
unchanged, even after attaining full convergence, when further signal points