Pad´e processing for MR spectra from in vivo time signals - Signal Processing in MRS with Biomedical Applications - page 231

Pade transform stem from the same input Maclaurin series which is divergent
for|z|< 1 and convergent for|z|> 1. Hence, we have seen that the FPT (+)
has a much more di cult task than FPT (−) . This is because FPT (+) must
induce convergence into the divergent series, while the FPT (−) ought to merely
increase the convergence rate of the already convergent Maclaurin series.
On the right side of Figs. 7.1 an d 7.2 , the absorption spectra are depicted for
the FPT (+) (top panels), FPT (−) (middle panels) and FFT (bottom panels)
all using the full signal length, N = 2048. It can be observed that the FPT (+)
and FPT (−) yield practically the same spectra. These two latter spectra are
seen in Figs. 7.1 and 7.2 to coincide with the corresponding spectrum from
the FFT to within the background noise. Such an excellent agreement among
these three methods for total shape spectra is expected, since the FFT can
serve as a gold standard for the investigated FID of high quality [141].

Convergence patterns of FPT (−) and FFT for absorp-
tion total shape spectra

7.3

Figures 7.3 -7.6 at 4T show the convergence rates of the absorption spec
tra computed by the FFT and the FPT (−) at 5 signal fractions (N/32 =
64,N/16 = 128,N/8 = 256,N/4 = 512,N/2 = 1024) as well as at the full
signal length N = 2048. Likewise, Figs. 7.7 -7.10 display the corresponding
spectra generated by the FFT and the FPT (−) at 7T. More specifically, Figs.
7.3, 7.4, 7.7 and 7.8 use separate graphs to show the convergence patterns per
tinent solely to the FFT or the FPT (−) . Additionally, with Figs. 7.5, 7.6, 7.9
and 7.10, that juxtapose the FFT and the FPT (−) on the same graphs at 4T
and 7T, respectively, we provide a direct inspection of the relative convergence
rates of these two signal processors.
As can be seen from Figs. 7.3 and 7.7, the FFT appears as a stable signal
processor with no undesirable surprises when the partial signal length N/M
is systematically augmented. This favorable feature of the FFT is amply
shared by the FPT (−) , which likewise produces no spikes or other spectral
deformations, as is clear from Figs. 7.4 and 7.8. Moreover, it is observed in
Figs. 7.4 and 7.8 that even with small fractions of the full time signal, the
FPT (−) can reconstruct the main metabolites.
For example, at N/32 = 64 with the FPT (−) , the NAA peak near 2.0 ppm,
as well as the creatine (Cre) peak near 3.0 ppm are clearly visible (top left
panels on Figs. 7.4 and 7.8). Over 60% of the NAA concentration (which
is proportional to the area under the peak) is already predicted at this very
short signal length. Furthermore, on the left middle panels in Figs. 7.4
and 7.8 (N/16 = 128), the FPT (−) is seen to yield nearly 90% of the NAA
concentration, as given by this most prominent peak (after water suppression).