Digital Signal Processing Reference
In-Depth Information
0
@
a
(s)−
1
0
@
c
s
1
0
@
b
(s)−
1
0 0
0
a
(s)−
1
0
b
(s)−
1
A
A
A
c
s+1
c
s
0
=
.
(4.94)
.
.
.
.
.
.
.
a
(s)−
K
.
b
(s)−
K
c
K+s
c
K+s−1
c
s
Then, the explicit calculation from Refs. [5, 17] yields
G
FPT(s)−
n
(u
−1
) =G
LCF(s)
o,n
(u)
(u) =G
CF(s)
=G
CCF(s)
o,n
2n+1
(u)
(n = 1, 2, 3 ...)
(4.95)
so that
G
FPT(s)−
n
(u
−1
) = G
LCF(s)
o,n
(u)
(u) = G
CF(s)
= G
CCF(s)
o,n
2n+1
(u)
(n = 1, 2, 3 ...).
(4.96)
These results show that the delayed diagonal fast Pade transform, FPT
(s)−
,
with the convergence region outside the unit circle (|u|> 1) is the same as
the odd part of the delayed Lanczos continued fraction of order n. Further
more, G
FPT(s)
n
(u
−1
) and the original truncated Green function (4.79) are
convergent for|u|> 1 for N→∞. Thus, outside the unit circle, the delayed
diagonal fast Pade transform G
FPT(s)
n
(u
−1
) represents an accelerator of an
already convergent series which is the Green function (4.77).
4.11 The fast Pade transform FPT
(+)
inside the unit cir-
cle
There exists another version of the delayed diagonal Pade approximant for
the same functionG
(s
N
(u
−1
) from (4.79). This version can be obtained from
(4.14) which we recast as
∞
K
d
(s
k
u
G
(s)
(u) =
c
n+s
u
−n
=
u−u
k
.
(4.97)
n=0
k=1
The summation over k is an implicit ratio of two polynomials in the variable
u and, therefore, it represents the Pade approximant. A particular property
of (4.97) is that the numerator polynomial has no free term, i.e., a constant
independent of u. Hence, the sought variant of the rational function should be
taken as a ratio of two polynomials in u. This version of the delayed diagonal
Pade approximant, or equivalently, delayed diagonal fast Pade transform will
be denoted by G
PA(s)+
K
(u) or G
FPT(s)+
K
(u), respectively
G
PA(s)+
K
(u)≡G
FPT(s)+
K
(u).
(4.98)
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