Digital Signal Processing Reference
In-Depth Information
where π is the projection operator π =|Υ
k
)(Υ
k
|. Here, positive integer K can
be finite or infinite depending on the number of the eigenvalues u
k
of
U. Thus,
K
k=1
(Φ
0
|{u1−U}
−1
|Φ
0
) =
|{u1−U}
−1
|Υ
k
)(Υ
k
it followsR(u) = (Φ
0
|Φ
0
) =
K
k=1
(Φ
0
K
k=1
(Φ
0
}
−1
|Υ
k
)(Υ
k
|{u−u
k
|Φ
0
) =
|Υ
k
)(Υ
k
|Φ
0
)/(u−u
k
) so that
K
d
k
u−u
k
|{u1−U}
−1
|Φ
0
) =
R(u) = (Φ
0
(2.129)
k=1
|Υ
k
)
2
with d
k
= (Φ
0
|Φ
0
) is employed.
It follows from (2.129) thatR(u) is the scalar PA of the order [(K−1)/K].
This is expected, since the corresponding resolvent (2.128) is the operator PA,
i.e., 1(u1−U)
−1
= (u1−U)
−1
1. Thus, the generating fraction
2
of the auto
correlation functions or signal points c
n
is the PA of the order [(K−1)/K]
where the symmetry (Φ
0
|Υ
k
) = (Υ
k
∞
A
K−1
(z)
B
K
(z)
1
u
c
n
z
n
=
z =
(2.130)
n=0
∞
a
0
+ a
1
z + a
2
z
2
++ a
K−1
z
K−1
b
K
+ b
K−1
z + b
K−2
z
2
++ b
0
z
K
c
n
z
n
=
(2.131)
n=0
K−1
K
a
r
z
r
b
K−r
z
r
.
A
K−1
(z) =
B
K
(z) =
(2.132)
r=0
r=0
In (2.125), the given expansion coe
cients U
n
(x) of the Taylor series in y
retrieves exactly the generating function f(x,y), where all the coe
cients
of the denominator polynomial are known. However, in (2.131) when the
c
n
's are given, the form of the generating fraction is known to be the PA
A
K−1
(z)/B
K
(z), but the coe
cients{a
r
}and{b
r
}of the numerator and
denominator polynomials are unknown. Of course, they can be uniquely de
termined through multiplication of (2.131) by B
K
(z) and via the subsequent
equating the coe
cients of the like powers. We reemphasize that the gener
ating fraction of the signal points always has the degree of the denominator
polynomial by one unit higher than that of the numerator, as per (2.131).
We shall now consider the inverse problem of reconstructing the auto
correlation functions{c
n
}when their generating fraction A
K−1
(z)/B
K
(z)
from the lhs of (2.129) is given. No Fourier integral will be used. Instead, only
the purely algebraic method originated by Prony [138] will be employed. With
2
The term 'generating function' is customarily used for ordinary polynomials, but for ratio-
nal polynomials, the corresponding nomenclature 'generating fraction' of Prony [138] seems
to be more transparent.
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