Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 143

FIDs built from attenuated complex exponentials with stationary and non
stationary amplitudes that yield Lorentzian and nonLorentzian spectra in
MRS. The term 'filter' denotes an operation applied to the time signal with the
result given by a baseline constant c ∞ . This constant is the limiting stationary
value of the time signal from which all the harmonics (transients) have been
filtered out in a transformation whose outcome is equivalent to taking the limit
of infinite times (nτ−→∞) in the considered FID, i.e., c n . This exact filter
is the Shanks transform, i.e., the ST which is, as noted, the FPT applied
directly in the time domain [5]. The implicit time limiting process t−→
∞, which goes hand in hand with the ST, well illustrates the extrapolation
features of the timedomain version of the FPT. The ST is defined by (2.198)
as a quotient of two Hankel determinants. In the case of higher dimensions,
these determinants are not practical for extensive numerical computations.
Nevertheless, the ST is salvaged by the Wynn ε−algorithm (2.201) which
computes the ratios of the two Hankel determinant recursively. This recursion
is actually another computational algorithm in the FPT. In order to confirm
that these statements about filtering of harmonics from an FID are plausible,
it should be recalled that there is a simpler and more familiar accelerator
called the Aitken extrapolation, or equivalently, the 2 −process [5, 77]. If an
FID has only one harmonic (K = 1), then the Aitken transformation is the
exact filter in the abovedefined sense. A direct generalization of the Aitken
transform to an FID with any higher number of harmonics (K > 1) represents
the ST as the multiexponential version of the singleexponential time signal.
The Pade methodology is not just another spectral analyzer added to the
other already existing ones. In addition to its practical advantages of unique
ness and exactness for solving the quantification problem in MRS, the most
fundamental characteristic which distinguishes the Pade methodology from
the conventional signal processors is that it is much more than a powerful
computational framework. Namely, the Pade methodology is the cornerstone
of the very formulation of the Schwinger variational principles, the Heisen
berg scattering matrix, the Green function, to name only a few of the leading
strategies in the most successful physics theory - quantum mechanics [5].
Overall, in this chapter, we have focused upon the possibility of obtaining
the exact solution of the quantification problem for noiseless synthesized FIDs
from theoretical modelings within MRS. We have employed the two equivalent
variants of the fast Pade transform, i.e., the FPT (+) and FPT (−) defined with
their initial convergence regions that reside inside and outside the unit circle,
respectively. Quantification in MRS entails mathematically illconditioned
harmonic inversion. As stated, other terminologies used to describe this prob
lem include: spectral analysis, reconstruction, spectral decomposition, etc. In
this problem, the FID is given, but the damped fundamental harmonics from
which it is built are not known. By means of the FPT, we succeeded to exactly
reconstruct the spectral parameters that are the complex frequencies and the
associated complex amplitudes. These extracted quantities can be interpreted
to yield the physical and biochemical information about the metabolites in