Digital Signal Processing Reference
In-Depth Information
the difference between the total number of all the found resonances and the
the diagonal forms of the FPT
(±)
spectra, P
±
K
(z
±1
)/Q
±
K
(z
±1
), there are only
24 genuine zeros as opposed to 25 genuine poles. The missing zero (located far
away from the shown interval of the linear frequency ν =−i(ln z)/(2πτ) is the
socalled “ghost” zero of the numerator polynomials P
±
K
(z
±1
) and corresponds
to the harmonic variables z = 0 in the FPT
(+)
or z =∞in the FPT
(−)
. The
∞
n=0
c
n
z
−n
is not defined at the origin, since the
value z = 0 of the harmonic variable z = exp (iωτ) for Im(ω) > 0 is the
singularity point
2
.
Overall, Froissart doublets simultaneously achieve three important goals:
(i) noise reduction, (ii) dimensionality reduction and (iii) stability enhance
ment. Stability against perturbations of the physical time signal under study
is critical to the reliability of spectral analysis. The main contributor to in
stability of systems is its spurious information. Being inherently unstable,
spuriousness is unambiguously identified by the twofold signature of Froissart
doublets (polezero coincidences and zerovalued amplitudes) and, as such,
discarded from the output data in the FPT. What is left is genuine informa
tion which is stable.
In practice, due to various reasons (finite arithmetics, computational round
offerrors, uncertainties in measured signal points, etc.), Froissart doublets
do not elicit exact polezero matchings and, consequently, the corresponding
amplitudes do not reduce to zero exactly, although they are very small. In
such a case, the common factors in the Pade numerator and denominators,
P
±
K
(z
±1
)/Q
±
K
(z
±1
), will be the product of the quotient terms of the type
[z
±1
−(z
±1
k
original Maclaurin series
+ ε
±
k
)]/[z
±1
−(z
±1
+ ε
±
k
)] where|ε
±
k
|≪1 and|ε
±
k
|≪1 as well as
k
ε
±
k
= ε
±
k
. Nevertheless, the concept of Froissart doublets will be preserved in
this case, as well, albeit in an approximate form, with the following rationale.
The locations of poles{z
±1
k
+ε
±
k
}and zeros{z
±1
k
+ ε
±
k
}will still be very close
to each other due to|ε
±
k
|≪1 and|ε
±
k
|≪1. To find the qualitative behavior of
the corresponding amplitudes{d
±
k
}, it su
ces to mention that the amplitudes
are the residues of the Pade polynomial quotients. Here, the Cauchy notion
of a residue can be conceived as the residual or remainder from a difference
between the given pole and zero.
Thus, in general, d
±
k
−z
±1
k
where certain nonzero constants of
proportionality need not be given for the present purpose (for more details on
this, see chapter 6). When poles z
±1
k
∝z
±1
k
and zeros z
±1
k
in these amplitudes are
replaced by the mentioned terms z
±1
k
+ε
±
k
and z
±1
k
+ ε
±
k
, we have d
±
k
∝(z
±1
k
+
ε
±
k
)−(z
±1
+ε
±
k
), so that d
±
k
∝ε
±
k
−ε
±
k
. It then follows from here, on the account
k
of|ε
±
k
|≪1 and|ε
±
k
|≪1 that|d
±
k
|≪1. Therefore, for approximate pole
zero coincidences, the ensuing amplitudes are indeed very small. Hence the
2
Being located literally far away from both genuine and Froissart poles and zeros, one could
use the term “outliers” (from statistical analysis of data) for “ghost” zeros and “ghost” poles
(for more details, see chapter 5)
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