Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 113

Re(P − K /Q − K )[N/4]−Re(P − K /Q − K )[N/M] for M = 2, 8, 16, 32, 64 would appear
precisely the same as those on panels (i)(iii), (iv) and (vi). Clearly, in this
latter context, panel (v) is not mentioned, since in this subplot the error spec
trum would be zero by definition, Re(P − K /Q − K )[N/4]−Re(P − K /Q − K )[N/M]≡0
for M = 4. All told, the residual spectrum in the FPT (−) converges fully by
using at most the first quarter of the entire time signal, as expected from panel
(iv) in Fig. 3.5 . This finding also holds for the FPT (+) (not shown), as could
similarly be anticipated from Fig. 3.3 , where exactly the same absorption
total shape spectra are obtained in the FPT (+) and FPT (−)

at N/4.

3.3.2 Residual or error absorption total shape spectra near
full convergence
Figure 3.7 shows the residual absorption total shape spectra in the FPT (+)
(left column) and FPT (−) (right column) computed from the difference of two
absorption spectra Re(P − K /Q − K )[N]−Re(P − K /Q − K )[N P ] where N P = 180, 220,
260. This figure zooms into the range of the partial signal length near full
convergence. It is clear that these residual or error spectra in both variants of
the FPT are for all practical purposes equal to zero throughout the considered
frequency range. This confirms full convergence of all the total shape spectra
at N P = 180 even though the peak k = 11 is absent, as seen earlier on panels
(i) and (iv) of Table 3.3 .

3.3.3 Consecutive difference spectra for absorption envelope
spectra
On Fig. 3.8 we show the consecutive difference absorption envelope spec
tra obtained in the FPT (−) . These spectra are generated by subtracting
the absorption envelope spectra at the two consecutive signal lengths via
Re(P − K /Q − K )[N/M]−Re(P − K /Q − K )[N/(2M)]. Here, according to the notation
from Fig. 3.6 , the symbol [L] implies that L signal points are used. However,
in Fig. 3.8, the number L is either N/M or N/(2M), where, as previously,
N is the full signal length (N = 1024) and M is the truncation number
(M = 2, 4, 8, 16, 32, 64).
These consecutive difference spectra are found in Fig. 3.8 to be zero at
N/4 = 256. The same finding also extends to N/M > 256. Through this
alternative error analysis, the result once again corroborates that the FPT (−)
has reached full convergence with only a quarter of the entire FID. This is
also the case, as well, for the FPT (+) (not shown).
The main reason for showing the consecutive difference spectra in Fig. 3.8,
although the error spectra have already been analyzed in Fig. 3.6, is to display
the finer details of the convergence rate on the local level by using adjacent
values of the signal length. This is a helpful supplement to the estimates of
the global error available from the associated error spectra Re(P − K /Q − K )[N]−
Re(P − K /Q − K )[N/M] from Fig. 3.6. Clearly, the latter formula is also a differ