Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 177

5
Signal-noise separation via Froissart doublets
The FPT converges upon reaching constancy or stabilization of the whole set
of the retrieved frequencies and amplitudes of all the physical resonances. Fur
thermore, such a stabilization is a true signature of the reconstruction of the
exact number of resonances. Namely, once the stage at which full convergence
is achieved, with any further augmentation of the partial signal length N P to
wards the full signal length N, all the fundamental frequencies and amplitudes
will “stay put”, i.e., they shall still remain constant [11, 34]. Critically, we
shall see in the illustrations in chapter 6 that this stability is maintained
within machine accuracy at any value N P between the step when convergence
has first been reached up to the full signal length N. Moreover, during re
construction of the exact input data, the convergence stage itself is reached
with an exponential convergence rate [188]. This accomplishment establishes
the FPT as a spectral analyzer which is capable of yielding an exponentially
accurate approximation for time signals from MRS and other related fields [5].
Both the stability of all the spectral parameters for every genuine resonance
and the constancy of the estimate of the true number of resonances can be
established by the concept of Froissart doublets [44] or polezero cancellation
[45]. As discussed, by this phenomenon, all the additional poles and zeros
of the Pade spectrum P ± K+m /Q ± K+m for m > 1, beyond the stabilized num
ber K of resonances, will automatically cancel each other via (2.186). With
such a feature, the FPT becomes safeguarded against contamination of the
final results by spurious resonances, because all the poles due to unphysical
resonances originating from the denominator polynomial become identical to
the associated zeros of the numerator polynomial. Evidently, this yields pole
zero cancellation by way of the polynomial quotient in the FPT from (2.186).
Advantageously, polezero cancellations can be exploited to separate spurious
from genuine contents of the signal. Because these unphysical poles and zeros
always appear as pairs in the FPT, they are considered as doublets. Moreover,
they are known as Froissart doublets after Froissart [44] who discovered by
empirical means this remarkable phenomenon, which is unique to the Pade
approximant precisely because of modeling via the polynomial quotient.
By its nature, noise represents spurious information which corrupts the
genuine part of the signal. However, polezero cancellations could be used to
discriminate between noise and the physical information in the investigated
time signal. This is the most important application of Froissart doublets in
MRS [11, 24, 30, 31, 34] and also in other similar fields where the FPT could be
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