Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 200

quantities. In this way, the mathematical physics from the preceding theoret
ical expositions becomes fully ingrained into Figs. 6.1 -6.14 , that are followed
by their pertinent practical interpretations in section 6.6 , thus establishing
one whole - comprehensive data analysis of the exact solution of the quantifi
cation problem in MRS. To adhere to the main theme of the present book,
these specific illustrations are for biomedical FIDs, but the entire powerful and
versatile PadeFroissart methodology remains valid for a multitude of other
applications across interdisciplinary research dealing with time signals in the
form of sums of damped complex exponentials [5, 6].

6.4 Numerical presentation of the spectral parameters
6.4.1 Input spectral parameters with 12-digit accuracy
Table 6.1 gives the 12digit input data for the quantification problem under
examination. As mentioned, these data are the complex fundamental fre
quencies and the corresponding amplitudes from a synthesized noiseless FID,
whose associated spectrum is comprised of a total of 25 resonances, some of
which are individual although tightly packed peaks, while others are closely
overlapped or nearly degenerate. As in chapter 3, the first 4 digits of all the
12digit values of the spectral parameters were chosen to closely correspond
to the typical frequencies and amplitudes found in proton MR time signals as
encoded in vivo from a healthy human brain at 1.5T [88].
Furthermore, all the input parameters from Table 6.1 are set to be in ex
act 4digit mutual agreement with the corresponding data from Table 3.1
in chapter 3. The remaining 8 digits in the 12digit data are chosen in an
arbitrary manner without any special ordering/rule, except for care being
taken to avoid the situations where rounding in Table 6.1 would preclude the
imposed exact agreement with the 4digit input data from Table 3.1. The
columns in Table 6.1 of the input fundamental transients are headed by labels
n ◦ k , Re(ν k ) (ppm) , Im(ν k ) (ppm) , |d k

|(au) and M k that represent the run
ning number, real and imaginary frequencies (both in ppm), absolute values
(moduli) of amplitudes (in arbitrary units) and the metabolite assignments,
respectively. For convenience, we shall interchangeably use the notations k,
# and n ◦ k for the running number of resonances. Of particular note are the
crossings of the 2nd column with the 11th and 12th rows where the two
chemical shifts Re(ν 11 ) = 2.67602157683 ppm and Re(ν 12 ) = 2.67602157684
ppm are separated by an extraordinarily small splitting Re(ν 12 )−Re(ν 11 ) =
1×10 −11 ppm. This maximally sharpens the corresponding earlier splitting
Re(ν 12 )−Re(ν 11 ) = 1×10 −4 ppm for the 4digit data in Table 3.1 from
chapter 3. Within the first common 4 digits, all the other remaining tight
separations among several metabolites stay unaltered relative to Table 3.1