Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 198

dealing with the 4digit input data, the 11th and 12th chemical shifts differ in
the 4th decimal place only by 1 unit, i.e., 1×10 −4 ppm, and all the phases of
d k are defined as 4digit zeros. Thus, in both the 4 and 12digit input data
considered in the present chapter, we have|d k

|= d k due to zerovalued phases

φ k ≡Arg(d k ) for 1≤k≤25. The chemical shift splittings between the 11th

and 12th fundamental harmonics in both accuracies from sections 6.4 a nd 6.5
mimic the near degeneracy. The FPT can indeed handle the exact degener
acy, too, as we verified in a check computation by entering the 12 identical
digits for the 11th and 12th chemical shifts. This feature of the FPT remains
valid for any other pairs or larger groups of resonances. Moreover, not only
chemical shifts, but the other three spectral parameters can also be selected to
coincide among the chosen transients. Such a capability of the FPT to treat
nondegenerate, neardegenerate and degenerate harmonics in quantification
problems on the same footing is very important, since MRencoded spectra
are abundant with tightly overlapping and nearly degenerate peaks.
As stated, the data are presented in two sections 6.4 and 6.5 each of which
is split into two subsections. All the tables deal only with genuine spectral
parameters, whereas every figure contains genuine and spurious reconstructed
data. The tables for noiseless cases are in section 6.4 with subsections 6.4.1
for the input data and 6.4.2 f or the output data in the FPT (−) . The graphs for
noiseless and noisy data are in section 6.5 with subsections 6.5.1 and 6.5.2
for reconstructions by the FPT (±) at full convergence (N P = N/4 = 256)
and near convergence (N P = 180, 220, 260) of genuine resonances. As always,
when convergence is addressed, this refers to data for genuine resonances,
since spectral parameters for spurious resonances never converge.
Section 6.4 with tabular data contains the 12digit accurate input data
in subsection 6.4.1 given in Table 6.1 . The output data reconstructed by
the FPT (−) are presented in subsection 6.4.2 via Table 6.2 for two partial
signal lengths of the nonFFT type (N P

= 2 m , m positive integers), e.g.,
N P = 180 (prior to convergence with accuracy 2 - 7 digits) and N P = 220
(full convergence with 12 digit accuracy throughout). Also reported in sub
section 6.4.2 is Table 6.3 for 12digit accurate reconstruction of all the spectral
parameters with the FFTtype signal lengths (N P = 2 m , m positive integers)
using the specifications N P = 2 8 = 256 (a quarter of the full FID) and N =
2 10 = 1024 (the whole FID). The output data for N P = 180 and N P = 220
are given to prove the exponential convergence rate of the FPT (−) around
the convergence point which occurs first at N P = 210 [6, 35]. This zooming
would not be possible if the FPT (−) were limited to the FFTtype structured
number 2 m (m positive integers) for signal lengths. The FFTtype partial
(N P = 2 8 = 256) and full (N = 2 10 = 1024) signal lengths are used to show
the constancy of machine accurate Pade reconstructions.
Section 6.5 presents Figs. 6.1 -6.14 that are multifaceted in conveying six
different spectral aspects using the input and output data:
•(i) two time signals, a noisefree and a noisecorrupted input FID, to test
the FPT in two clearcut situations, with and without external perturbations,