Digital Signal Processing Reference
In-Depth Information
respectively. Each element of the set{a
(s)
}can be obtained from the equality
between the expansion coe
cients of the series of the rhs of (4.17) developed
in powers of u
−1
n
}from (4.11). Such a procedure
is similar to the one from Ref. [17] for nondelayed time signals and Green
functions when s = 0. Thus, it su
ces here to give only some of the main
results that will be needed in the analysis which follows. For example, the
expressions for the general CF expansion coe
cients are given by
and the signal points{c
n+s
H
n
(c
s+1
)H
n−1
(c
s
)
H
n−1
(c
s+1
)H
n
(c
s
)
H
n−1
(c
s+1
)H
n+1
(c
s
)
H
n
(c
s
)H
n
(c
s+1
)
a
(s)
2n
a
(s)
=
2n+1
=
(4.20)
α
(s)
= a
(s)
2n+1
+ a
(s)
[β
(s
n
]
2
= a
(s)
2n
a
(s)
(n≥1)
(4.21)
n
2n+2
2n+1
where the Hankel determinant H
n
(c
s
) is given in (2.200). The parameters
α
(s
n
and [β
(s
n
]
2
are the Lanczos coupling constants in the nearest neighbor
approximation [5]. Using (4.20) and (4.21), a recursive algorithm can also be
established for computations of all the coe
cients{a
(s)
}. Thus, with the help
n
of the alternative notation
a
(s)
2n
a
(s)
2n+1
≡q
(s)
n
≡e
(s)
n
(4.22)
the product of q
(s)
with e
(s)
becomes
n
n
H
n−1
(c
s
)H
n+1
(c
s
)
H
n
(c
s
)
q
(s
n
e
(s)
=
.
(4.23)
n
Similarly, the product of q
(s)
n+1
and e
(s)
reads as
n
q
(s)
n+1
e
(s)
= q
(s+1)
n
e
(s+1)
n
s≥0.
(4.24)
n
Further, employing the following identity of the Hankel determinants
[H
n
(c
s
)]
2
= H
n
(c
s−1
)H
n
(c
s+1
)−H
n+1
(c
s−1
)H
n−1
(c
s+1
)
(4.25)
we can find the sum of q
(s+1)
and e
(s+1)
n−1
in the form
n
+ e
(s+1)
q
(s)
n
+ e
(s)
n
= q
(s+1)
n
n−1
.
(4.26)
The relationships (4.24) and (4.26) for the delayed continued fraction coe
cients represent the Rutishauser quotientdifference (QD) algorithm [156]
9
=
;
.
e
(s)
= e
(s+1)
n−1
+ q
(s+1)
−q
(s)
n
n
n
e
(s+1)
q
(s)
n+1
= q
(s+1)
n
e
(s)
(4.27)
n
n
=
c
s+1
c
s
e
(s)
0
q
(s)
1
= 0
(s≥1)
(s≥0)
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