Digital Signal Processing Reference
In-Depth Information
Of course, the original function (to be represented byG
FPT(s)+
K
(u) in the form
of a rational polynomial) is the same Green functionG
(s)
N
(u
−1
) as it was in
(4.81)
(u) =
A
(s)+
(u)
: G
FPT(s)+
K
FPT
(s)+
K
.
(4.99)
B
(s)+
K
(u)
The polynomials A
(s)+
K
(u) and B
(s)+
K
(u) of the same degree K are given by
K
K
A
(s)+
K
B
(s)+
K
a
(s)+
r
u
r
b
(s)+
r
u
r
.
(u) =
(u) =
(4.100)
r=1
r=0
As in the sum over k in (4.97), the variable of the numerator and denominator
polynomials from (4.99) is set to be u. This is to be contrasted to variable
u
−1
, which appears in the original sum (4.80).
The corresponding delayed diagonal fast Pade transform with respect to
the truncated Green functionG
(s)
N
(u
−1
) from (4.79) reads as
(u) = u
−1
A
(s)+
(u)
G
FPT(s)+
K
(u) = u
−1
G
FPT(s)+
K
K
.
(4.101)
B
(s)+
K
(u)
The numerator polynomial A
(s)+
K
(u) from (4.99) or (4.101) is seen not to have
the free term, a
(s)
0
= 0, i.e., the sum over r begins with r = 1, such that the
first term is given by a
(s)
1
u. Otherwise, the convergence range of G
FPT(s)+
K
(u)
is located inside the unit circle (|u|< 1) where the original sumG
(s)
(u
−1
) from
(4.78) is divergent. We can extract the polynomials A
(s)+
K
(u) and B
(s)+
K
(u)
from the condition
(u)≡
N−1
c
n+s
u
−n
=
A
(s)+
(u)
G
(s)
N
K
.
(4.102)
B
(s)+
K
(u)
n=0
Specifically, multiplying (4.102) by B
(s)+
K
(u) we arrive at
9
=
;
.
N−1
B
(s)+
K
c
n+s
u
−n
= A
(s)+
(u)
(u)
K
n=0
(4.103)
K
N−1
K
b
(s)
r
u
r
c
n+s
u
−n
a
(s)
r
u
r
=
r=0
n=0
r=1
Here, performing the indicated multiplications of the two sums and equating
the coe
cients of the same powers of the expansion variable yields
K
b
(s)+
0
b
(s)
r
c
n+s+r
= 0
c
n+s
+
(n = 1, 2,...,M). (4.104)
r=1
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