Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 118

3.10 , that only a quarter N/4 = 256 of the full FID is necessary for both the
FPT (+) and FPT (−) to fully resolve all the individual resonances, including
the peaks that are isolated (k = 8, 9, 10, 13, 18, 19, 20, 21, 24, 25), overlapped
(k = 1, 2; k = 3, 4; k = 5, 6, 7; k = 14, 15; k = 16, 17), tightly overlapped
(k = 22, 23) as well as nearly degenerate (k = 11, 12). Furthermore, panels
(iii) and (vi) of this figure show that the component shape spectra coincide in
the FPT (+) and FPT (−) . This observation is consistent with the conclusion
from Fig. 3.3 , where the corresponding envelope spectra on panels (iv) and
(v) were seen to be the same in the FPT (+) and FPT (−) . This is anticipated
from the definition of the total shape spectrum in the FPT as the sum of the
component shape spectra of all the constituent resonances. Obviously, this
will be true only for unique reconstructions such as those achieved by means
of the FPT.
In sharp contrast is the nonuniqueness of all fittings. In such reconstruc
tions, a given absorption total shape spectrum is fitted by some preassigned
component shape spectra that might very well differ markedly in nearly all
the essential details from the corresponding exact counterparts.
3.4.2 Absorption component spectra and envelope spectra
near full convergence
In Fig. 3.11 we see the absorption component shape spectra (left column)
and total shape spectra (right column) from the FPT (−) computed near full
convergence at 3 partial signal lengths N P = 180, 220, 260. The three panels
on the right column for the total shape spectra have all reached full conver
gence. However, on the left column for the corresponding component shape
spectra, full convergence is achieved at N P = 220, 260. On panel (i) for
the component shape spectra at N P = 180, peak k = 11 is absent, and
peak k = 12 is overestimated. Furthermore, the area of the 12th peak is
overestimated by the amount of the area of the absent 11th peak. As a
consequence of this latter compensation, the total shape spectrum has not
reflected that either shortcoming had occurred. Namely, the total shape spec
trum on panel (iv) for N P = 180 reached complete convergence even though
peak k = 11 was missing and peak k = 12 was overestimated. This full con
vergence is also seen in the corresponding zerovalued spectra for the residual
Re(P − K /Q − K )[1024]−Re(P − K /Q − K )[180] (N = 1024,N P = 180) and consecu
tive difference Re(P − K /Q − K )[220]−Re(P − K /Q − K )[180] (N P = 220, 180) shown
on panel (iv) i n Figs. 3.7 and 3.12 , respectively.
By way of recapitulation, the left column in Fig. 3.12 redisplays the Pade
absorption component and total shape spectra, but this time superimposed on
top of each other at N P = 180, 220, 260. This is particularly illuminating when
such shape spectra are juxtaposed to the corresponding three consecutive dif
ference spectra on the right column in Fig. 3.12. Three consecutive difference
spectra, built from the corresponding total shape spectra, are seen as identical
on panels (iv), (v) and (vi) in Fig. 3.12, despite the lack of convergence of the