Digital Signal Processing Reference
In-Depth Information
the case of all genuine resonances from synthesized time signals, as well as op
timal accuracy for noisecorrupted input data that can be either theoretically
generated or experimentally measured,
•superresolution; the highest resolution beyond the RayleighFourier bound
2π/T with T being the duration of the investigated time signal, and
•signalnoise separation; unequivocal disentangling of genuine from spuri
ous information by means of Froissart doublets (polezero coincidences).
Of paramount importance is to single out the milestone achievement of
the FPT in solving the noise problem which evaded an exact treatment for
more than half a century. The FPT provides the first exact separation of
genuine (physical) from spurious (unphysical, noise and/or noiselike) infor
mation encountered either in theory or measurements involving time signals.
This is accomplished by means of Froissart doublets that are coincident pairs
of poles z
±
k
≡z
±
k,Q
and zeros z
±
k,P
in the response functions P
±
K
(z
±1
)/Q
±
K
(z
±1
)
from the FPT
(±)
. Here, z
±
k,P
and z
±
k,Q
are the solutions of the numerator and
denominator characteristic or secular equations
P
±
K
(z
±
k,P
) = 0 z
±
k,P
= e
±iω
±
k,P
τ
Q
±
K
(z
±
k,Q
) = 0 z
±
k,Q
= e
±iω
±
k,Q
τ
.
(12.24)
Froissart polezero confluences are synchronized with the corresponding zero
values obtained for Froissart amplitudes
Spurious : z
±
k,Q
= z
±
k,P
{d
±
k
∴
}
= 0.
(12.25)
z
±
k,Q
=z
±
k,P
By changing the degree K of the polynomials in the diagonal FPT
(±)
from
P
±
K
(z
±1
)/Q
±
K
(z
±1
), Froissart doublets unpredictably and uncontrollably alter
their positions in the complex z
±1
−planes. They never converge (stabilize)
even when the whole signal length is exhausted. Therefore, these latter res
onances that roam around in the complex planes are considered as spurious
or unphysical. As such, unstable resonances are identified by their twofold
signature: polezero coincidences and zero amplitudes for noisefree time sig
nals. The same type of signature is also operative for noisecorrupted time
signals (theoretically generated or experimentally measured), but with the
approximations z
±
k,Q
≈z
±
k,P
and d
±
k
≈0. Crucially, however, although Frois
sart doublets are unstable against even the smallest external perturbation
(e.g., altering the degree of the Pade polynomial, adding noise, etc.), they
nevertheless consistently preserve the relationships in (12.25).
By contrast, there are the retrieved resonances with spectral parameters
that converge, and these are viewed as stable, genuine or physical resonances.
The signatures of all such genuine resonances are
Genuine : z
±
k,Q
= z
±
k,P
{d
±
k
∴
}
= 0.
(12.26)
z
±
k,Q
=z
±
k,P
Amplitudes d
±
k
are the Cauchy residues of quotients P
±
K
(z
±1
)/Q
±
K
(z
±1
) taken
at the poles z
±
k,Q
. Here, the word residues has the transparent meaning of the
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