Digital Signal Processing Reference
In-Depth Information
the spuriousness in the set{z
±
k,P
}. Namely, the two types of spuriousness
from the two sources Q
±
K
(z
±1
) = 0 and P
±
K
(z
±1
) = 0 are strongly coupled to
gether. As a result, spurious Froissart poles{z
±
k,Q
}and zeros{z
±
k,P
}are always
born out as pairs. It is in this way that Froissart doublets manifest them
selves through polezero coincidences,{z
±
k,Q
}={z
±
k,P
}, as in (12.25). Such
an occurrence cancels the entire spuriousness from the polynomial quotients
P
±
K
(z
±1
)/Q
±
K
(z
±1
). This becomes particularly apparent when these ratios are
written in their canonical forms
K
(z
±1
−z
±
k,P
)
(z
±1
−z
±
k,Q
)
.
P
±
K
(z
±1
)
Q
±
K
(z
±1
)
=
p
±
K
q
±
K
(12.31)
k=1
If the running degree K is larger than the number of genuine resonances K
G
,
then all the terms (z
±1
−z
±
k,P
)/(z
±1
−z
±
k,Q
) from (12.31) for K−K
G
> 0
would contain spurious Froissart poles z
±
k,Q
and zeros z
±
k,P
. Hence polezero
cancellations leading to (z
±1
−z
±
k,P
)/(z
±1
−z
±
k,Q
) = 1 for Froissart doublets
z
±
k,Q
= z
±
k,P
, as per (12.25).
The ensuing consequence of these polezero cancellations onto the corre
sponding amplitudes of Froissart resonances can be seen at once from the
explicit formulae for d
±
k
in terms of all the recovered poles and zeros
K
(z
±
k,Q
−z
±
k
′
,P
)
=
p
±
K
q
±
K
d
±
k
.
(12.32)
(z
±
k,Q
−z
±
k
′
,Q
)
k
′
=k
k
′
=1
Here, in the numerator, it is permitted to have k
′
= k, in which case every
Froissart doublet from (12.25) would produce zerovalued terms (z
±
k,Q
−z
±
k,P
)
and, thus, the whole product in (12.32) will become zero. This yields d
±
k
= 0
for z
±
k,Q
= z
±
k,P
, as in (12.25). Note that in our computations, we never use
(12.32) to obtain the amplitudes d
±
k
in the FPT
(±)
. This is because formula
(12.32) employs the whole set of the reconstructed amplitudes to compute d
±
k
for the k th resonance. Therefore, even the slightest inaccuracy, such as near
cancellations of poles and zeros, rather than the theoretically exact cancella
tions, could spoil the precision of the sought d
±
k
for the given k. Instead, we
use the alternative expressions
P
±
K
(z
±
k,Q
)
Q
±
K
′
(z
±
k,Q
)
Q
±
K
′
(z
±
k,Q
) = 0
d
±
k
=
(12.33)
where the prime denotes the first derivative. Here, each k th amplitude on
the lhs depends only on one, i.e., the k th value of the rhs of Eq.(12.32) and,
hence, no other resonance can deteriorate the accuracy of the retrieved d
±
k
.
Overall, it is clear from these remarks that the FPT
(±)
possesses a very
elegant, simple and powerful solution for the exact identification of all spurious
Froissart resonances. When these are discarded, only genuine resonances are
left and this yields the exact solution of the quantification problem.
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