Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 147

in MRS, the Pade approximant is, in point of fact, the exact theory. This con
sideration in itself further confirms that this processor should be considered
optimal for MRS.
Thus, even the Fourier analysis naturally leads us to the Padebased MRS.
Furthermore, these observations are completely consistent with an indepen
dent formulation of the FPT. Namely, the Maclaurin series or the z−transform,
the Green function and the Heaviside partial fraction representation contain
ing implicitly the Pade polynomial quotient, are all incorporated into the FPT
from the outset, and there is actually no reference to the Fourier integral or
Fourier analysis. Moreover, while these expansions for the exact spectrum
might provisionally be the starting point of the FPT, the end result of the
Pade spectral analysis is the set of fundamental frequencies and amplitudes
as the reconstructed attenuated harmonic constituents of the FID from the
time domain. Exactly the same results are gleaned if the analysis begins with
the FID rather than its spectrum. Thus, the process of estimation in the
FPT is performed by handling the time and frequency domain on the same
footing. Both time and frequency domain analysis generate exactly the same
characteristic equation or the eigenproblem, whose solutions are the spectral
parameters. Therefore, for the full quantitative information about the system,
it is su cient to specify these parameters as the fundamental frequencies and
the corresponding amplitudes without necessarily indicating the way in which
they might be used. Such spectral parameters can subsequently be employed
to generate spectra in various modes, to retrieve the FID or to extrapolate it
to unmeasured values of the time signal, etc.
A system will be deemed robust if it is capable of maintaining stability in
the face of external excitations or perturbations. In numerous systems across
a wide range interdisciplinary fields, including basic sciences and engineer
ing, optimal stability is modeled by rational response functions via polynomial
quotients as predicted by the Pade analysis. Thus, instead of defining these
rational polynomials via their frequency representations of the Heaviside par
tial fractions, it is su cient to simply give the defining parameters of these
functions as the fundamental frequencies and amplitudes. This is reminiscent
of an alternative definition of an ordinary single polynomial, not necessarily
through its development in powers of the independent variable. Rather, this is
done via the set of all its genuine parameters as the expansion coe cients. It
should also be pointed out that these general concepts concerning Padebased
MRS render the interface between the time and frequency domain analysis
rather loose. But this is as it should be. In contrast, from the very beginning,
fittings are defined either the time or frequency domain. These alternatives
depend upon whether the FID or its spectrum is used to adjust the free, fitting
parameters via some ad hoc models [79]-[83].
As mentioned, quantification in MRS is mathematically illconditioned.
This is due to the lack of a continuous dependence of the sought solution
upon the input data. The intrinsic di culty of this problem, irrespective of
the methods used, can yield large variations in the reconstructed spectral pa