Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 5

cle (|z|> 1), it will automatically diverge inside the unit circle (|z|< 1).
An analytical continuator renders the Green function welldefined inside the
unit circle and, therefore, throughout the complex frequency plane with the
exception of singular points (poles). In addition to being a convergence ac
celerator, the FPT is also an analytical continuator. In this topic, we present
several illustrations with FIDs from MRS as crossvalidation, to demonstrate
that both convergence acceleration and analytical continuation by the FPT
yield exactly the same true spectral parameters.
Prior to quantification of FIDs from MRS and MRSI, clearly, the total
number of resonances is unknown. This parameter should actually be added
to the usual list of unknown quantities that are the complex frequencies and
amplitudes [5]. Similarly to the sought fundamental frequencies and ampli
tudes, the total number of resonances needs to be reliably determined from
the spectrally analyzed time signal. Thus far, in signal processing within
MRS and MRSI, the usual practice has been to surmise ad hoc about the
number of resonances. This is done by fitting the Fourierreconstructed spec
tral lineshapes via preassigned expansion functions or some model in vitro
spectra. These fittings are neither unique nor objective. The consequence is
either under or overfitting by under or overestimating the true number of
physical resonances. In other words, either some genuine resonances could
easily be missed, or some spurious ones might be registered. Neither of these
contingencies is acceptable, particularly for medical diagnostics.
The total number of resonances is handled by the FPT on the same footing
as the fundamental complex frequencies and amplitudes. Here, all three quan
tities are considered to be the unknowns of the quantification problem. The
total number of resonances will not be subject to guessing any longer nor need
this parameter be defined in advance. Thus, with Padebased quantification
in MRS and MRSI, the exact number of genuine resonances is determined
in a completely reliable manner, as in the case of the reconstructed frequen
cies and amplitudes. This remarkable feature of the FPT in providing exact
retrieval of the true number of physical resonances is due to the fact that
this method is the only exact filter for FIDs built from transient harmonics
in the form of damped complex exponentials with stationary and/or non
stationary polynomialtype amplitudes. These FIDs are precisely predicted
by the quantummechanical description of magnetic resonance phenomena.
Moreover, such data are encoded via MRS/MRSI when stringent experimental
conditions are fulfilled. These conditions are, for instance, proper suppression
of the giant water resonance in, e.g., neurodiagnostics, adequate shimming
of the static magnetic field to reduce inhomogeneities, etc. To unequivocally
prove these statements, it is essential to apply the FPT to exactly solvable
model problems with all controllable, i.e., precisely known input data. There
fore, using synthesized FIDs, we provide illustrations on precisely how the
FPT unambiguously determines the exact total number of resonances as well
as all the fundamental frequencies and amplitudes.
More specifically, the exact number of resonances is reconstructed employ