Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 199

•(ii) two variants of the fast Pade transform for crossvalidations of quan
tifications via the FPT (+) and FPT (−) for the output data,
•(iii) four partial signal lengths in both versions of the FPT to monitor
the convergence rate for the fixed bandwidth,
•(iv) structured and unstructured partial N P to exhibit the advantages of
nonFourier partial signal lengths; the former is of the FFT type, N P = N/4 =
2 8 = 2 m (m = 256 at convergence, and the latter of the nonFFT type N P

positive integers) via N P = 180, 220 and 260 near convergence,
•(v) two types of graphs to take advantage of different distributions of the
same quantities: Argand plots (real versus imaginary numbers) and physical
quantities as a function of chemical shifts,
•(vi) two coordinate systems, polar (Euler) and rectangular (Descartes), in
complex planes to illuminate the complementary interpretations of harmonic
variables (signal poles and zeros) and fundamental frequencies.
Admittedly, Figs. 6.1 -6.14 from section 6.5 will appear to be quite involved.
This is due to their said multilevel informational content, intended to be
conveyed in a way which is both thorough and concise. To simultaneously
achieve thoroughness and conciseness, as well as to make these figures as
selfexplanatory as possible, we equipped them with the following utilities:
•the minimum pertinent text and formulae, e.g., for spectral parameters
such as harmonic variables, linear frequencies and amplitudes that are sought
via reconstructions in quantifications,
•the canonical representations of the FPT (±) with signal poles and zeros,
•a selection of the main features for the two subrepresentations of the
fast Pade transform via pFPT (±) (“poles of the FPT”) and zFPT (±) (“zeros
of the FPT”), with the understanding that the complete FPT representation
involves all poles and all zeros,
•the characteristic equations and the formulae for the amplitudes in the
FPT (±) to recall the source of the reconstructed spectral parameters,
•the expressions for the signatures of Froissart doublets (polezero conflu
ences and the accompanying zero amplitudes),
•the way in which the exact number K G of genuine resonances is recon
structed, as the difference between the total number K of all found resonances
and the number K F of Froissart doublets, K G = K−K F .
Despite such elaborated texts and formulae appearing in these graphs either
through the main titles, subtitles on subplots or within a number of panels,
the overall visual and informational display is nevertheless free from clutter 1 .
These additions to Figs. 6.1-6.14 are meant to be both a facilitator (for an
easier followup of all the graphic illustrations) and, most importantly, a re
minder to the corresponding theoretical analyses from, e.g., chapter 5 for an
indepth connection to the physical/mathematical meaning of the displayed

1 The figures in chapter 3 were also aided by the appropriate texts, titles, sub-titles and,
occasionally, formulae, all of which were aimed at helping the reader to grasp the main
messages from those graphic illustrations.