Digital Signal Processing Reference
In-Depth Information
This property makes the first n terms of series in (2.79) equal to zero and this
reduces π
n
(z) to
∞
∞
n,m
z
−m−1
=
n,n+m
z
−n−m−1
.
π
n
(z) =
(2.82)
m=n
m=0
The expansion coe
cients
n,m
from (2.82) can be computed recursively via
β
n+1
n+1,m
=
n,m+1
−α
n
n,m
−β
n
n−1,m
0,0
= 1
(2.83)
where the parameters{α
n
}and{β
n
}are the same Lanczos coupling constants
from (2.49). Employing (2.55) and (2.61), we can rewrite (2.81) as
n
q
r,n−r
r+m
= 0
(0≤m < n)
(2.84)
r=0
or in the alternative matrix counterpart
0
1
0
1
0
1
0
1
2
n−1
q
0,n
q
1,n−1
q
2,n−2
.
q
n−1,1
n
n+1
n+2
.
2n−1
@
A
@
A
@
A
1
2
3
n
n+1
. . .
.
.
. .
n−1
n
n+1
2n−2
2
3
4
=−
.
(2.85)
It is seen that the n×n matrix on the lhs of (2.85) represents the usual Hankel
matrix H
n
(
0
)≡{
i+j
}(0≤i≤n−1 , 0≤j≤n−1). Given the first 2n
moments{
r
}, the system (2.85) of n linear inhomogeneous equations can
be solved by taking q
n,n−r
as the unknown. In particular, the solution of
the system (2.85) exists and it is unique for the nonzero value of the Hankel
determinant
det H
n
(
0
) = 0.
(2.86)
The relation (2.86) can be proven by supposing that, e.g., the opposite is valid,
det H
n
(
0
) = 0, and establishing the contradiction afterwards. The relation
det H
n
(
0
) = 0 means that there exists a linear dependence among the rows
of det H
n
(
0
).
Linear dependence signifies that there is a nonzero column
vector y ={y
r
}(0≤r≤n−1) such that H
n
(
0
)y = 0
0
@
1
A
0
@
1
A
0
@
1
A
0
1
2
n−1
1
2
3
n
2
3
4
y
0
y
1
y
2
.
y
n−1
0
0
0
.
0
n+1
=
(2.87)
.
.
.
.
.
.
.
n−1
n
n+1
2n−2
or alternatively
y
T
H
n
(
0
)
y = 0
(2.88)
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