Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 192

5.10 Disentangling genuine from spurious resonances
Once both the FPT (+) and FPT (−) have attained convergence, they yield the
same spectral parameters for the genuine resonances
z k,P

9
=

= z − k,P z k,Q = z − k,Q

ν k,P = ν − k,P ν k,Q = ν − k,Q k∈K : genuine resonances. (5.47)

G

;

d k = d − k

By contrast, the remaining spectral parameters for all Froissart resonances
never converge because of their spuriousness. Even the slightest alteration in
the signal length can change the distributions of the parameters of spurious
resonances in the complex planes. Furthermore, Froissart harmonic variables
and Froissart frequencies differ when passing from the FPT (+) to FPT (−)

9
=

z k,P = z − k,P z k,Q = z − k,Q

ν k,P = ν − k,P ν k,Q = ν − k,Q

k∈K : Froissart resonances. (5.48)

F

;

d k = 0 = d − k

According to (5.48), both sets of Froissart amplitudes{d ± k

}(k∈K F ) in the
FPT (+) and FPT (−) are equal to zero, in agreement with (5.32).
In practice, the most e cient manner to spot Froissart doublets while in
specting tables containing the output list of all the spectral parameters re
constructed by the FPT (±) for noisefree (noisecorrupted) input FIDs is to
look for zero or near zero amplitudes, d ± k

= 0 or d ± k

≈0, respectively. An
even simpler way is to inspect the corresponding panels in the Argand plots
of frequency distributions in the complex ν ± −planes mounted on top of the
absolute values of amplitudes as a function of chemical shifts. This would lead
to an easy observation of Froissart doublets through zero or near zero ampli
tudes associated with polezero coincidences or nearcoincidences in the case
of noiseless or noisy time signals, respectively. We have already encountered
a part of such a graphic inspection in Fig. 3.19 for a noisefree input FID. A
related monitoring of Froissart doublets for the corresponding noisecorrupted
FID will be presented in chapter 6.
Robust disentangling of genuine from spurious resonances is a part of the
SNS concept, which also underlies the determination of the exact or optimally
accurate number K of resonances for noiseless or noisepolluted FIDs, respec
tively. In section 2.8 of chapter 2, several procedures were analyzed for finding
the true number K. One of the discussed procedures is based upon Froissart
doublets for determining the exact number K of genuine resonances, and this
will be illustrated in chapter 6 for both noiseless and noisy time signals.