Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 116

ence spectrum for a varying partial length N/M which, however, only occurs
in the second term, whereas the total signal length N is always fixed in the
first term Re(P − K /Q − K )[N]. Obviously, some of the details on panels (i) - (iii)
on Figs. 3.6 and 3.8 are different. This is due to the lack of full conver
gence at N/M < 256. Even though convergence has not yet been reached at
N/8 = 128 on panel (iv), the data for Re(P − K /Q − K )[N]−Re(P − K /Q − K )[N/8] in
Fig. 3.6 and Re(P − K /Q − K )[N/4]−Re(P − K /Q − K )[N/8] in Fig. 3.8 are identical.
As discussed, this is because the first terms in the differences Re(P − K /Q − K )[N]
and Re(P − K /Q − K )[N/4] yield the same results.
3.3.4 Consecutive differences for absorption envelope spec-
tra near full convergence
In Fig. 3.9 the consecutive difference spectra are shown in the FPT (+)
(left column) and FPT (−) (right column). These were obtained by sub
tracting two absorption spectra Re(P − K /Q − K )[N P ]−Re(P − K /Q − K )[N ′ P ], where
N P = 180, 220, 260 and N ′ P = 140, 180, 220. This focuses upon a narrow re
gion around full convergence. Panels (i) and (iv), prior to full convergence of
the absorption total shape spectra, reveal some residual structures located in
the frequency interval 2 ppm - 4 ppm. However, all such residuals have dis
appeared on the middle and lower panels resulting in practically zerovalued
difference spectra.
3.4 Absorption component shape spectra of individual
resonances
3.4.1 Absorption component spectra and metabolite maps
A comparison is given in Fig. 3.10 of the results for the FPT (+) and FPT (−)
(left and right columns, respectively). The pertinent details with regard to
panels (i) and (iv) as well as (ii) and (v) of Fig. 3.10 have already been pre
sented in section 3.2 vi a Fig. 3.3 . Nevertheless, further important information
is presented on panels (ii) and (v) in Fig. 3.10 for the absorption total shape
spectra by displaying the usual acronyms that locate the positions of the major
MRdetectable metabolites associated with resonances stemming from FIDs
encoded via MRS from a healthy human brain. Here, the same acronyms
for several resonances (Cho, Glu, NAA) are seen at more than one chemical
shift. This is a consequence of the socalled Jcoupling [3]. On panels (iii) and
(vi) in Fig. 3.10, the absorption component shape spectra are presented for
each individual resonance. As explained previously, the sums of all of such
component shape spectra yield the associated total shape spectra from panels
(ii) and (v) in Fig. 3.10. Once again it is seen on panels (iii) and (vi) in Fig.