Digital Signal Processing Reference
In-Depth Information
Moreover, having obtained the exact delayed CF coe
cients{a
(s)
}from
(4.41), one could deduce the exact delayed continued fractions of a fixed order
as the explicit polynomial quotient, which is the Pade approximant. Thus,
the evenorder delayed CF is the Pade approximant of the form
n
P
CF(s)
(u)
n
G
CF(s)
2n
(u) = a
(s)
1
.
(4.46)
Q
CF(s)
(u)
n
Similarly, the delayed oddorder CF, which is labeled as G
CF(s)
2n−1
(u), is extracted
from the evenorder CF by putting a
(s)
2n
≡0
G
CF(s)
2n−1
(u)≡{G
CF(s)
(u)}
(n = 1, 2, 3,...).
(4.47)
2n
a
(s)
2n
=0
n
(u) and Q
CF(s
n
(u) from (4.46) can be introduced
by their general power series representations
P
CF(s)
Here, the polynomials
n−1
n
P
CF(s)
n
p
(s)
Q
CF(s)
n
q
(s)
n,n−r
u
r
n,n−r
u
r
.
(u) =
(u) =
(4.48)
r=0
r=0
The polynomial expansion coe
cients p
(s)
n,n−r
and q
(s)
n,n−r
are available as the
analytical expressions that were derived in protracted calculations by Belkic
[5, 17]
2(n−m+2)
2(n−m+3)
2n
p
(s)
n,m
= (−1)
m−1
a
(s)
r
1
a
(s)
r
2
a
(s)
r
m−1
(4.49)
r
1
=3
r
2
=r
1
+2
r
m−1
=r
m−2
+2
m−1 summations
2(n−m+1)
2(n−m+2)
2(n−m+3)
2n
q
(s)
n,m
= (−1)
m
a
(s)
r
1
a
(s)
r
2
a
(s)
r
3
a
(s)
r
m
r
1
=2
r
2
=r
1
+2
r
3
=r
2
+2
r
m
=r
m−1
+2
m summations
(4.50)
where n≥m. With these and the other listed closed expressions, the FPT
is established as the only parametric estimator by means of which signal pro
cessing can entirely be carried out from the explicit analytical formulae. This
overrides the mathematical illconditioning of the alternative numerical algo
rithms.
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