Digital Signal Processing Reference
In-Depth Information
Formally, the recursions (4.68) and (4.69) for P
(s
n
(u) and Q
(s
n
(u) are the
same, with the exception of the different initializations.
The derivation from Ref. [5, 17] shows that the Green functions G
LCF(s)
(u)
e,n
and G
LCF(s)
o,n
(u) are linked to the delayed PadeLanczos approximant (4.66)
by the expression
G
PLA(s)
n
(u) = G
LCF(s)
e,n
(u)
(n = 1, 2, 3,...).
(4.74)
Therefore, the delayed PadeLanczos approximant G
PLA(s)
(u) and the even
n
part of the delayed Lanczos continued fraction G
LCF(s)
(u) coincide with each
e,n
other for any order n.
Of course, it is important to investigate whether the odd part of the delayed
LCF, i.e., G
LCF(s)
(u), could also be located within the PadeLanczos general
o,n
table for G
PLA(s)
(u).
To this end, by considering, e.g., the diagonal case
n,m
(L = K + 1) in (4.66)
P
(s)
n+1
(u)
Q
(s
n
(u)
c
s
β
(s)
1
G
PLA(s)
n+1,n
(u)≡G
PLA(s)
n
(u) =
(4.75)
it can be shown that [5, 17]
G
PLA(s)
n
(u) = G
LCF(s)
o,n
(u)
(n = 1, 2,...).
(4.76)
Moreover, in general, there are no integers n and m for which G
PLA(s)
(u)
n,m
can be equal to G
LCF(s)
(u) [5, 17]. This occurs because the denominator in
o,n
G
LCF(s)
o,n
(u) is a polynomial γ
0
u + γ
1
u
2
++ γ
n
u
n+1
without the constant
free term∝u
0
≡1. The additional term u in the denominator of G
LCF(s)
(u)
o,n
relative to G
LCF(s)
(u) implies that G
LCF(s)
(u) might originate from the Pade
e,n
o,n
approximant in the variable u
−1
instead of u.
This is proven in the next
section.
4.10
The fast Pade transform FPT
(−)
outside the unit
circle
In this section, the analysis begins with the exact delayed Green function
(4.11). The expansion (4.11) is the Maclaurin series in powers of the variable
u
−1
≡z = exp(iωτ) and, hence, convergent for|u|> 1, i.e., outside the unit
circle in the u−complex plane.
It will prove convenient to rewrite G
(s)
in
terms of an auxiliary functionG
(s)
as follows
∞
G
(s)
(u) =
c
n+s
u
−n−1
= u
−1
G
(s)
(u
−1
)
(4.77)
n=0
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