Digital Signal Processing Reference
In-Depth Information
are kept in their composite intermediate forms without performing the final
multiplications in (4.38). This would permit the exact cancellation of the de
nominator
n
m=2
λ
(s)
2m−2
by the associated portion of the numerator in H
n
(c
s
)
from (4.38) to give the exact result in the integer form H
n
(c
s
) = N
(s
n
.
The extension of the PD algorithm from its original nondelayed versions
of Gordon [158] to the delayed version of Belkic [5] is especially advantageous
with respect to the eigenvalues{u
k
}. When only the nondelayed CF coef
ficients{a
n
}≡{a
(0)
}are available as in Ref. [158], the eigenvalues{u
k
}
are generated either by solving the eigenproblem for the Jacobi matrix or by
rooting the corresponding characteristic polynomial Q
K
(u) = 0 [5]. On the
other hand, the delayed CF coe
cients{a
(s)
n
}can avoid altogether these two
latter conventional procedures and offer an alternative way of computing the
eigenvalues of data matrices from the following limiting procedure
n
λ
(s)
1,2k+1
s→∞
a
(s)
u
k
= lim
= lim
s→∞
.
(4.39)
2k
λ
(s)
1,2k−1
λ
(s)
1,2k
For checking purposes, it is also useful to apply the same limit s−→∞to
the string{a
(s)
2n+1
}and this gives
λ
(s)
1,2k+2
s→∞
a
(s)
lim
= lim
s→∞
= 0.
(4.40)
2k+1
λ
(s)
1,2k
λ
(s)
1,2k+1
To verify the accuracy of the results for the eigenvalues{u
k
}computed by
means of the relation u
k
= lim
s→∞
a
(s)
2k
from (4.39) within the delayed PD
algorithm we can use the analytical expression for a
(s)
2k
.
The exact closed
formula for the general delayed CF coe
cients a
(s)
has been obtained by
n
Belkic [5, 17] as
−σ
(s
n
π
(s)
−λ
(s
n
π
(s)
n+1
=
c
n+s
a
(s)
n−1
n−4
(4.41)
π
(s)
n
n
n
2
π
(s)
a
(s)
i
a
(s)
j
σ
(s)
n
=
=
(4.42)
n
i=1
j=2
n−3
j+1
k+1
λ
(s)
a
(s)
j
[ξ
(s)
j
ξ
(s)
j
a
(s)
k
a
(s)
ℓ
]
2
=
=
(4.43)
n
j=
[
n−
2
]
k=2
ℓ=2
π
(s)
σ
(s)
≡0
(n≤0)
≡0
(n≤3)
(4.44)
n
n
where the symbol [n/2] denotes the integer part of n/2. Once the CF coef
ficients{a
(s)
}have been generated, all the input time signal points{c
n+s
}
could be reconstructed exactly using the explicit expression
c
n+s
= π
(s)
n
n+1
+ σ
(s
n
π
(s)
n−1
+ λ
(s
n
π
(s)
n−4
.
(4.45)
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