Digital Signal Processing Reference
In-Depth Information
2.8.4 Froissart doublet spuriousness in the frequency domain
for finding K
In the frequency domain, the exact Padereconstruction of the true number
K works with the help of the concept of Froissart doublets
z
±
k
= z
±
k
(2.225)
where z
±
k
and z
±
k
are the poles and zeros of the GreenPade transfer func
tion G
±
K
′
(z
±1
) = P
±
K
(z
±1
)/Q
±
K
(z
±1
). Because the number K is unknown in
advance, the computation in the FPT is carried out by a gradual and system
atic augmentation of the order K
′
in the GreenPade functions G
±
K
′
(z
±1
). The
changes in K
′
cause fluctuations in the retrieved functions. However, K
′
will
saturate, i.e., it will reach a value after which such fluctuations in G
±
K
′
shall
disappear within a prescribed level of accuracy. Such a stabilized value of
the order K
′
is the sought K. This phenomenon will be manifested in (2.191)
through cancellation of all the terms in the Pade numerator and denominator
polynomials if the computation is continued beyond the stabilized value of
the order in the GreenPade functions
P
±
K+m
(z
±1
)
Q
±
K+m
(z
±1
)
=
P
±
K
(z
±1
)
Q
±
K
(z
±1
)
(m = 1, 2, 3,...).
(2.226)
By means of (2.192) and (2.225), Froissart amplitudes are found to be zero
d
±
k
z
±
k
= z
±
k
.
= 0
for
(2.227)
Moreover, polezero cancellations also take place if the computations discover
multiplicities in{z
±
k
}even for an FID whose spectrum is known to be non
degenerate. In this case, the same degeneracies also appear in{z
±
k
}, and this
gives the degenerate Froissart doublets whose automatic elimination by means
of polezero cancellations finally gives a nondegenerate spectrum, as should
be from the outset. Polezero cancellations can also happen prior to detection
of the true K for all the spurious poles that are canceled by the associated
spurious zeros. However, what counts is not so much that spurious resonances
uncontrollably appear, but rather that in the FPT they can unequivocally be
identified through polezero pairings as Froissart doublets of zero amplitudes.
In sharp contrast, genuine resonances never have zero amplitudes, due to the
absence of polezero coincidences.
In this way, Froissart doublets emerge as a reliable procedure for deciding
whether a reconstructed resonance is genuine or spurious. It should be em
phasized that spurious resonances can arise in spectral analysis for both noise
free and noisecorrupted (synthesized or encoded) FIDs. Thus, the concept of
Froissart doublets could advantageously be employed as a reliable and robust
procedure for separating the physical from nonphysical (noisy) information.
At the end of the computations, all the Froissart doublets are dropped. There
fore, the list of the reconstructed spectral parameters{ω
±
k
,d
±
k
}will contain
only the physical information through the genuine resonances.
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