Digital Signal Processing Reference
In-Depth Information
∞
G
(s)
N
G
(s)
(u
−1
) =
c
n+s
u
−n
=
(u
−1
)
lim
N→∞
(4.78)
n=0
N−1
G
(s)
N
(u
−1
) =
c
n+s
u
−n
(4.79)
n=0
(u)≡
N−1
G
(s)
N
c
n+s
u
−n−1
= u
−1
G
(s)
(u
−1
).
(4.80)
N
n=0
With respect to the finiterank Green functionG
(s)
N
(u
−1
), we define the de
layed diagonal Pade approximant,G
PA(s)−
K
(u
−1
), or equivalently, the delayed
diagonal fast Pade transformG
FPT(s)−
K
(u
−1
)
G
PA(s)−
K
(u
−1
)≡G
FPT(s)−
K
(u
−1
)
(4.81)
as the following polynomial quotient
(u
−1
) =
A
(s)−
(u
−1
)
FPT
(s)−
: G
FPT(s)−
K
K
.
(4.82)
B
(s)−
K
(u
−1
)
In this expression, the numerator and denominator polynomials A
(s)−
K
(u
−1
)
and B
(s)−
K
(u
−1
) are given in the same variable u
−1
similarly to the function
G
(s)
K
K
(u
−1
) and B
(s)−
K
(u
−1
) have the same degree
K and can be written as sums of powers of variable u
−1
(u
−1
). The polynomials A
(s)−
via
K
K
A
(s)−
K
B
(s)−
K
(u
−1
) =
a
(s)−
r
u
−r
(u
−1
) =
b
(s)−
r
u
−r
.
(4.83)
r=0
r=0
The associated delayed diagonal fast Pade transform relative to G
(s)
N
(u) reads
(u
−1
) = u
−1
A
(s)−
(u
−1
)
G
FPT(s)−
K
(u
−1
)≡u
−1
G
FPT(s)−
K
K
.
(4.84)
B
(s)−
K
(u
−1
)
The unknown expansion coe
cients a
(s)−
and b
(s)−
from (4.83) can be found
r
r
by setting the equalityG
(s)
N
(u
−1
) =G
FPT(s)−
K
(u
−1
)
G
(s
N
(u)≡
N−1
c
n+s
u
−n
≈
A
(s)−
(u
−1
)
K
.
(4.85)
B
(s)−
K
(u
−1
)
n=0
Multiplication of (4.85) by B
(s)−
K
(u
−1
) yields the equations
9
=
;
.
N−1
B
(s)−
K
c
n+s
u
−n
= A
(s)−
K
(u
−1
)
(u
−1
)
n=0
(4.86)
K
N−1
K
b
(s)−
r
u
−r
c
n+s
u
−n
a
(s)−
r
u
−r
=
r=0
n=0
r=0
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