Digital Signal Processing Reference
In-Depth Information
|z|> 1. According to the main feature of the Pade approximant for the input
power expansions of the form (2.162), both response functions or spectra in
the FPT
(±)
represent the unique rational polynomials P
±
L
(z
±1
)/Q
±
K
(z
±1
) in
their respective variables z
±1
G
N
(z
−1
)−G
±
L,K
(z
±1
) =O(z
±L+K±1
)
(2.163)
where
G
±
L,K
(z
±1
) =
P
±
L
(z
±1
)
L + K≤N.
(2.164)
Q
±
K
(z
±1
)
As before, the symbolsO(z
±L+K±1
) are remainders that are themselves the
Maclaurin series starting with the expansion terms z
±L+K±1
. The conven
tional meaning of theO(z
±L+K±1
) symbols from (2.163) becomes clear when
the GreenPade functions G
±
L,K
(z
±1
) are expanded in their own Maclaurin
series. Then the ensuing series agree exactly with the partial sum G
L+K
con
taining the first L+K≤N terms of the original Green function G
N
. However,
the remaining N−L−K terms from G
N
and G
±
L,K
are different and they
are all contained in the symbolsO(z
±L+K±1
) from (2.163).
Therefore, for
brevity, (2.163) can take the form
G
N
(z
−1
) = G
±
L,K
(z
±1
)
(2.165)
with the understanding that the left and the right sides are equal only up to
the neglected terms of the ordersO(z
±L+K±1
).
By construction, the rational polynomials G
L,K
(z) and G
−
L,K
(z
−1
) are the
Pade approximants to the finiterank Green function G
N
and, as such, they
are also called the GreenPade functions [5]. In such functions, P
±
L
(z
±1
) and
Q
±
K
(z
±1
) represent the numerator and denominator polynomial of the degree
L and K
L
P
±
L
(z
±1
) =
p
±
r
z
±r
(2.166)
r=r
±
K
Q
±
K
(z
±1
) =
q
±
s
z
±s
(2.167)
s=0
where r
+
= 1 and r
−
= 0. Because the Green functions G
N
(z
−1
) and
G
−
L,K
(z
−1
) possess the same independent variables z
−1
, it is evident that
the FPT
(−)
is the standard PA, which is denoted by the conventional sym
bol [L/K]
G
N
(z
−1
). On the other hand, in contrast to the input z−transform
G
N
(z
−1
), the complementary GreenPade function G
L,K
(z) is introduced in
terms of the variable z and, as such, the FPT
(+)
is recognized as the causal
Pade transform, PzT.
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