Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 184

This is the proof of the earlier stated result from (2.186). In an alternative
derivation, we could avoid dealing with the equalities in (5.19) and (5.20) by
defining P ± K and Q ± K as the monic polynomials. Recall that a polynomial is
defined as monic when its coe cient, as a multiplier of the highest power of
the expansion variable, is equal to unity. As such, P ± K and Q ± K can be monic
polynomials when all their expansion coe cients are divided by p ± K

and q ± K ,

respectively.
As they stand, the expressions from (5.21) directly transcend Froissart dou
blets via polezero cancellations. This cancellation, in turn, decreases the or
der of the FPT from K = K T to K T −K F = K G . Hence, the concept of

Froissart doublets via polezero cancellation represents an e cient procedure
for reduction of the order of the model for Padebased quantification in MRS
and elsewhere.
In fact, polezero cancellations diminish the dimensionality of the interim
problem, which would be of the order K T = K F +K G without the elimination
of the K F Froissart doublets. Thus, upon discarding the K F Froissart dou
blets, we are left with the order K G , which is then necessarily the exact order
of the original problem. From the physics viewpoint, this means that the
recovered order K G is indeed the exact number of genuine poles. This is the
way the true number of genuine resonances is unambiguously reconstructed
from the input FID by applying the FPT (±) .

5.6 Denoising Froissart filter

The critical point for retrieving the true number K G is the virtue of the
FPT (±) to unequivocally differentiate between the genuine (Pade) and spu
rious (Froissart) poles and zeros. This achievement is based on polezero
cancellations that effectively filter out all the spurious, i.e., Froissart poles
from the full solution of the quantification problem. Such a filtering leaves
us with the genuine poles alone, and this is vital to the ultimate solution
of the quantification problem. Hence, it is appropriate to coin the term the
denoising Froissart filter, i.e., the DFF, for the outlined procedure.
One of the direct meanings of the term 'denoising' is the 'noise reduction',
where Froissart doublets are considered as noise because of their spuriousness.
Such a terminology can be used for noisecorrupted as well as noisefree input
FIDs. Of course, in both these cases, the exact number K G is unknown in
advance of spectral analysis. This implies that any estimate K ′ = K G would
inevitably produce a nonzero difference FID(input)−FID(reconstructed by
using K ′ ). Such a difference is spurious and, thus, acts implicitly as noise for
both noiseless and noisy input FIDs.