Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 7

devise an adequate method for distinguishing, with certainty, the true from
false information which are mixed together in all experiments. Failure to do
precisely this is one of the major causes for potentially incorrect conclusions,
even when meticulous attention has been paid to perform the given mea
surement. Froissart doublets within the Pade approximant come to rescue
the situation by providing the most robust method to date for unequivocally
distinguishing genuine from spurious information, as has systematically been
demonstrated within MRS [11, 24, 30, 31, 34]. Given that the Pade methodol
ogy is unprecedentedly spread over many areas, as reviewed recently in Refs.
[5, 34], wider applications of the powerful and versatile Froissart concept of
signalnoise separation are anticipated across interdisciplinary research fields
of basic and applied sciences as well as technologies.
Fourier analysis, being linear, treats noise and physical signal on the same
footing and, therefore, cannot distinguish one from the other. Fourier grid
frequencies, being real, are all located precisely on the circumference|z|= 1
of the unit circle in the complex plane of the harmonic variable z. The same
locations along|z|= 1 represent the natural boundary of noise to which noise
frequencies tend to distribute themselves with probability 1, except at most
for a nearly zeroarea set (i.e., an open set of points of measure smaller than
δ, where δ is an infinitesimally small positive number) [58, 59, 61]. Thus,
maximal mixing of noise and physical signal occurs when using the FFT.
This explains why the Fourier analysis fails for time signals with poor signal
tonoise ratio (SNR). By implication, such a feature automatically advocates
against using maximally noisepolluted total shape spectra from the FFT for
postprocessing via fitting as an attempt to solve the quantification problem
in MRS and elsewhere.
On the other hand, the nonlinearity of the Pade analysis helps distinguish
the genuine from noisy part of the signal, such that the latter can be sup
pressed. The mechanism for this is simple and direct for, e.g., the FPT (+)
which pushes all the noise poles to the unit disc boundary|z|= 1 (which thus
acts as a noise attractor), and simultaneously keeps all the physical poles
strictly in the interior of the unit circle |z|< 1. Hence the cleanest signal
noise separation. Specifically, using the concept of Froissart doublets in the
FPT, nonphysical poles are identified by their very tight pairing with the
corresponding zeros (strong polezero couplings), and the ensuing negligibly
small amplitudes. Discarding such spurious Froissart doublets as defective
resonances de facto purifies the genuine information, and this is how noise
suppression is achieved in the FPT. In general, should some “defect poles”
(“stray poles”) be found by the FPT, they will automatically be accompa
nied by the corresponding “defect zeros” (“stray zeros”) which would annul
the ensuing spurious Froissart resonances through the vanishingly small am
plitudes/residues. Thus, in the Pade spectrum, any defective part stemming
from overestimating the exact number K of resonances disappears altogether
in the end through either implicit or explicit cancellations of the found defect
poles and defect zeros.