Digital Signal Processing Reference
In-Depth Information
where
2
c
n
= c
n+1
−c
n
, c
n
= c
n+1
−c
n
and H
n
(c
s
) is the Hankel
determinant
.
c
s
c
s+1
c
s+2
c
s+n−1
c
s+1
c
s+2
c
s+3
c
s+n
c
s+2
c
s+3
c
s+4
c
s+n+1
.
H
n
(c
s
) =
(2.200)
.
.
.
.
.
.
c
s+n−1
c
s+n
c
s+n+1
c
s+2n−2
The Hankel determinants are not computed directly from their definition
(2.200) for large dimensions. This would be numerically impractical for higher
ranks due to the factorially growing number of multiplications. Rather, the
e
k
(c
n
)'s are computed recursively by the Wynn ε−algorithm [5]
1
= ε
(m−1)
n+1
ε
(m+1)
n
+
ε
(m)
n+1
−ε
(m)
n
ε
(−1)
n
ε
(0)
n
= 0
= c
n
(2.201)
where c
n
is present in ε
(m)
n
via the second initial condition, ε
(0
n
= c
n
. Upon
reaching convergence in (2.201), the ST can simply be extracted from the
relationship
e
m
(c
n
) = ε
(2m
n
.
(2.202)
The Wynn recursion (2.202) itself was established from a recursion which
links four neighboring elements in the table of Pade approximants of varying
order. As such, the recursive algorithm for the Shanks transform is an e
cient
algorithm of the general Pade methodology [5].
If a given FID is comprised of precisely K nondegenerate or degenerate
attenuated complex harmonics, (2.198) becomes
e
K
(c
0
) = 0.
(2.203)
This finding is the signature for detection of the exact K in the time domain
spectral analysis. Regarding the ST, the quotient form (2.198) itself serves
as the definition of the necessary and su
cient condition for the time signal
c
n
to possess exactly K fundamental transients{z
k
}(1≤k≤K). Such a
condition is the simultaneous fulfillment of the following two relations
H
K+1
(c
n
) = 0
H
K
(c
n
) = 0.
(2.204)
The remaining two sets of the system's parameters, that are the fundamental
harmonics and the corresponding amplitudes{z
k
,d
k
}, can also be found by
continuing the work on spectral analysis directly in the time domain.
The
explicit formulae for this purpose are [5]
z
k
=
e
k−1
(c
n+1
)
e
k−1
(c
n
)
n−→∞ (2.205)
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