Pad´e processing for MR spectra from in vivo time signals - Signal Processing in MRS with Biomedical Applications - page 245

7.4.2 Self-contained Pade error analysis: Consecutive differ-

ence spectra

Now we will present consecutive difference spectra within the FPT (−) in or
der to illustrate that selfcontained Pade error analysis can be achieved. In
Figs. 7.13 and 7.14 at 4T and 7T, respectively, six consecutive difference
spectra are displayed. The left upper panels begin with Re(P − K /Q − K )[N/32]−
Re(P − K /Q − K )[N/64]. Next follows Re(P − K /Q − K )[N/16]−Re(P − K /Q − K )[N/32] on
the middle left panels and then Re(P − K /Q − K )[N/8]−Re(P − K /Q − K )[N/16] on
the bottom left panels. The right upper panels show the subsequent differ
ence spectra: Re(P − K /Q − K )[N/4]−Re(P − K /Q − K )[N/8], then on the right middle
panels: Re(P − K /Q − K )[N/2]−Re(P − K /Q − K )[N/4] and then Re(P − K /Q − K )[N]−
Re(P − K /Q − K )[N/2] on the right lower panels.
Finally, we present Figs. 7.15 a n d 7.16 where the left and the right columns
convey two different types of information. Here, the left columns corre
spond to the right columns in Figs. 7.13 and 7.14, respectively, displaying
three consecutive difference spectra Re(P − K /Q − K )[N/4]−Re(P − K /Q − K )[N/8] ,
Re(P − K /Q − K )[N/2]−Re(P − K /Q − K )[N/4] , Re(P − K /Q − K )[N]−Re(P − K /Q − K )[N/2],
where N/4 = 512, N/2 = 1024 and N = 2048. These three subplots of intrin
sic error spectra demonstrate a steady decrease of the local error in the FPT (−)
when the number of signal points are systematically augmented. Of course,
error spectra that concern the FPT (−) can also be generated by reference to
other estimators, e.g., the FFT as done in Figs. 7.11 and 7.12 . Likewise, the
error spectra can be constructed by using the two variants of the FPT, i.e.,
the FPT (+) and FPT (−) , as done on the right columns of Figs. 7.15 and 7.16
showing the three residuals that are all related to the whole FIDs (N = 2048) :
ReF[N]−Re(P K /Q K )[N] (top panels), ReF[N]−Re(P − K /Q − K )[N] (middle
panels) and Re(P K /Q K )[N]−Re(P − K /Q − K )[N] (bottom panels). All the three
latter error spectra are seen to be indistinguishable from the background noise.
Importantly, the entirely negligible values of the whole residual spectrum
Re(P K /Q K )[N]−Re(P − K /Q − K )[N] are seen at any of the considered frequen
cies on the right bottom panels in Figs. 7.15 and 7.16. It is thereby shown that
the differences steadily diminish, and at Re(P K /Q K )[N]−Re(P − K /Q − K )[N],
become essentially indistinguishable from background noise levels.
This is a very important internal crossvalidation within the FPT itself. It
shows that the two variants, the FPT (+) and FPT (−) , are su cient for estab
lishing consistency of this processor without necessitating external checking
against, e.g., the FFT or other estimators. In nonparametric processing, the
FPT (+) and FPT (−) yield, respectively, the upper and lower bounds to en
velopes of the shape spectra separated from each other by residual values of
the order of the background noise, as evidenced in Figs. 7.15 and 7.16. Like
wise, in parametric processing via the fast Pade transform, the FPT (+) and
FPT (−) provide respectively the upper and lower variational bounds of the
estimated complex frequencies{ω ± k }and complex amplitudes{d ± k }. It should

be recalled that the FPT is a variational method [5].