Digital Signal Processing Reference
In-Depth Information
versatile and powerful capability of performing analytical continuations and
convergence accelerations. These two features of the PA are not limited to se
ries via power expansions as in (12.4), but are equally applicable to any given
sequence of numbers for which, e.g., the limiting values are sought when the
number of included terms becomes infinitely large.
The discussed two main features of the Pade functions via its two wings, the
PA (convergence rate enhancement of slowly convergent series or sequences),
and the PzT (forced convergence of originally divergent series) are jointly
embodied into the fast Pade transform, FPT. In the fast Pade transform,
the PzT and PA are relabeled as FPT
(+)
and FPT
(−)
, respectively, where
the superscripts±refer to the employed independent variables, z
+1
≡z and
z
−1
= 1/z, respectively. By definition, the FPT
(+)
accomplishes analytical
continuation through the forced convergence of divergent series. Likewise, the
FPT
(−)
achieves acceleration of slowly converging series or sequences. Given a
Maclaurin series (12.4), the FPT
(+)
and FPT
(−)
are aimed at approximating
the same function F(z
−1
)
F(z
−1
)≈R
(FPT)±
(z
±1
).
(12.19)
Functions R
(FPT)±
(z
±1
) are explicitly defined as rational polynomials
R
(FPT)±
(z
±1
)≡
P
±
L
(z
±1
)
Q
±
K
(z
±1
)
(12.20)
L
K
P
±
L
(z
±1
) =
p
±
r
z
±r
Q
±
K
(z
±1
) =
q
±
r
z
±s
(12.21)
r=1,0
s=0
where r = 0 and r = 1 correspond to P
−
L
(z
−1
) and P
L
(z), respectively. As
in (12.16), the qualities of the FPT
(±)
, i.e., the adequacy of the two approxi
mations in (12.19), are governed by the explicit definitions
∞
c
n
z
−n
−
P
±
L
(z
±1
)
=O(z
±(L+K+1)
).
(12.22)
Q
±
K
(z
±1
)
n=0
The remaindersO
±
(z
±(L+K+1)
) follow by developing R
(FPT)±
(z
±1
) as
∞
P
±
L
(z
±1
)
Q
±
K
(z
±1
)
b
±
n
z
±n
=
(12.23)
n=0
where the expansion coe
cients{b
±
n
}can be computed from the previously
extracted polynomial coe
cients{p
±
r
,q
±
s
}. Like the earlier reasoning with
(12.14) and (12.15), the explicit calculations and comparisons between (12.4)
and (12.23) reveal the significance of the error termsO(z
±(L+K+1)
). This
implies that both rational functions P
±
L
(z
±1
)/Q
K
(z
±1
) would be able to ex
actly reproduce the first L + K terms from the infinite set{c
n
}of the in
put Maclaurin series (12.19). According to (12.23), the rational polynomials
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