Digital Signal Processing Reference
In-Depth Information
K
G
+K
F
−z
±
r
′
,P
) are equal to zero, because the following null
factors are always present in the product: [(z
±
k,Q
r
′
=K
G
+1,r
′
=k
(z
±
k,Q
−z
±
r
′
,P
)]
r
′
=k
= (z
±
k,Q
−z
±
k,P
) =
0 for z
±
k,Q
= z
±
k,P
F
, according to the definition (5.11) of Froissart
doublets as polezero cancellations.
where k∈K
Therefore, all the Froissart amplitudes
are found to be zerovalued
d
±
k
⇐⇒ z
±
k,Q
= z
±
k,P
.
= 0
(spurious)
k∈K
(5.30)
F
′
(z
±
k,Q
)
This also follows from the alternative formulae d
±
k
= P
±
K
(z
±
k,Q
)/Q
±
K
in (5.25) via (5.1) with P
±
K
(z
±
k,Q
) = 0 for z
±
k,P
= z
±
k,Q
and (5.8) where
Q
±
K
′
(z
±
k,Q
)
= 0. Using (5.13), we can decompose the whole sets{d
±
k
}into
two disjoint sets comprised of the genuine and spurious amplitudes
{d
±
k
={d
±
k
⊕{d
±
k
}
}
}
k∈K
F
.
(5.31)
k∈K
T
k∈K
G
The members of these two subsets from (5.31) are given by (5.28) and (5.30).
This can be recapitulated via
8
<
K
G
(z
±
k,Q
−z
±
r,P
)
p
±
K
q
±
K
r=1
k∈K
:
genuine
amplitudes
G
K
G
d
±
k
=
(z
±
k,Q
−z
±
s,Q
)
:
s=1,s=k
0
k∈K
:
Froissart
amplitudes.
(5.32)
F
To give another interpretation of the remarkable result (5.30), it is instructive
to return to the canonical formula (5.24) for the residues d
±
k
. In these expres
sions for d
±
k
, we shall single out the critical terms z
±
k,Q
−z
±
k,P
by factoring them
in front of the product symbol in (5.24), thus leaving certain complexvalued
nonzero remainders denoted by R
±
k
= 0
d
±
k
z
±
k,Q
−z
±
k,P
R
±
k
=
(5.33)
z
±
k,Q
−z
±
k
′
,P
K
=
p
±
K
q
±
K
R
±
k
R
±
k
= 0 .
(5.34)
z
±
k,Q
−z
±
k
′
,Q
k
′
=1
k
′
=k
The formula (5.33) states that the k th residues d
±
k
are proportional to the
distance between the poles z
±
k,Q
and zeros z
±
k,P
with the nonzero complex
constants of proportionality being R
±
k
. This is a plausible geometric interpre
tation of the residues in the complex plane of the harmonic variable z. Since
the remainders R
±
k
are always nonzero, they can act merely as modulators of
the complexvalued distance or metric∼(z
±
k,Q
−z
±
k,P
). In other words, if the
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