The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 82

2.8.5 Froissart doublets in exact analytical computations

The majority of previous studies consider Froissart doublets exclusively as
noise functions. This presumes that Froissart doublets are due only to noise
perturbations in the input data. However, spuriousness of Froissart doublets
manifested through polezero coincidences can also arise in ideal, noiseless
data [11, 34]. We already mentioned in chapter 1 that one of the reasons
for this is finite arithmetics in numerical computations because of roundoff
errors. As discussed, roundofferrors are random and, therefore, they too
represent computational noise. Hence, again, noise appears as a cause leading
to Froissart doublets even for noiseless input data if finite arithmetics is used
on computers.
The key issue to address, however, is whether computational noise is the
only cause for Froissart doublets in the case of noiseless input data? In other
words, could such doublets occur even when carrying out numerical compu
tations with no roundofferrors using infinite precision arithmetics? If this
would be feasible, then computational noise would not be the only reason for
Froissart doublets for noisefree data. The answer to this question is in the
a rmative, as can be readily shown using symbolic numerical computations
with the input data given by integers. However, a cause for spuriousness with
such infinite arithmetics might be di cult (if not impossible) to pinpoint by
numerical means.
If Froissart doublets are bound to happen in both finite and infinite precision
numerical computations, could there be any difference, after plotting, at least
in their locations relative to the unit circle? There would be a difference which
could be made visible after zooming into a minuscule segment of the plot to
within machine accuracy (the socalled machine ǫ) of finite arithmetics. Then
Froissart doublets from finite arithmetics would be seen as being displaced a
bit at the ǫ distance from their counterparts from infinite arithmetics.
Moreover, if Froissart doublets appear despite using infinite precision nu
merical computations, then they should also appear in purely analytical calcu
lations with no errors whatsoever. This happens to be true for both noisefree
and noisecorrupted synthesized time signals, as has recently been demon
strated by Belkic [35] who showed by purely analytical means that in both
cases Froissart doublets are due to linear dependence of the system of equa
tions for the Pade denominator polynomial.
As such, Froissart doublets need not necessarily be produced by noise alone
in noiseperturbed input data, since these spurious pairs can also appear even
when employing the noisefree input data. The only difference one could ex
pect between the results with noiseless and noisecorrupted signals is that,
as opposed to the former, the latter Froissart doublets will not be aligned
along the unit circle on a regular and smooth arc, but rather they will chaot
ically bifurcate relative to that noiseless arc, depending on the level of the
perturbation. Explicit numerical computations in Refs. [11, 34, 35], as well
as those reported in the present book (chapters 3 and 6) have confirmed this