Digital Signal Processing Reference
In-Depth Information
The general matrix element λ
(s)
n,m
can be introduced by the 2×2 determinant
λ
(s)
1,m−2
λ
(s)
1,m−1
λ
(s)
n,m
≡−
.
(4.31)
λ
(s)
n+1,m−2
λ
(s)
n+1,m−1
The alternative form of this expression is the following recursion
n,m
= λ
(s)
1,m−1
λ
(s)
−λ
(s)
1,m−2
λ
(s)
λ
(s)
(4.32)
n+1,m−2
n+1,m−1
which is initialized by the elements
λ
(s)
λ
(s)
λ
(s)
n,3
n,2
= (−1)
n+1
c
n+s−1
= (−1)
n+1
c
n+s
n,1
= δ
n,1
(4.33)
Upon generation of the arrays{λ
(s)
i,j
where δ
n,1
is the Kronecker symbol.
},
one can compute all the coe
cients{a
(s)
}of the delayed continued fractions
n
(4.17) by means of the formula
λ
(s)
1,n+1
λ
(s)
a
(s)
n
=
(n = 1, 2, 3,...).
(4.34)
1,n−1
λ
(s)
1,n
Inserting (4.34) into (4.22), we arrive at
λ
(s)
1,2n+1
λ
(s)
1,2n+2
q
(s)
n
e
(s)
n
=
=
.
(4.35)
λ
(s)
1,2n−1
λ
(s)
λ
(s)
1,2n
λ
(s)
1,2n
1,2n+1
The Lanczos coupling parameters{α
(s
n
, [β
(s
n
]
2
}are deduced by substituting
the string{a
(s)
}into (4.21). The dependence of the pair{α
(s
n
, [β
(s
n
]
2
}upon
the matrix elements{λ
(s)
1,n
n
}can be made explicit by means of (4.21) and (4.34)
that yields the expression
[λ
(s)
1,2n+2
]
2
+ λ
(s)
1,2n
λ
(s)
λ
(s)
1,2n+2
1,2n+3
α
(s)
n
[β
(s)
n
]
2
=
=
1,2n
]
2
. (4.36)
λ
(s)
1,2n
λ
(s)
1,2n+1
λ
(s)
λ
(s)
1,2n−1
[λ
(s)
1,2n+2
Thus, the recursion (4.31) of the vectors{λ
(s)
}invokes only their prod
ucts and differences, but no divisions. This gives the name for the 'product
difference' algorithm. Originally, the PD algorithm for nondelayed signals or
moments (s = 0) was proposed by Gordon [158]. The generalization of the PD
algorithm to delayed time signals or moments{c
n+s
n,m
}was given by
Belkic [5, 17]. It follows from (4.34) that the PD algorithm performs the divi
sion only once at the end of the computations while obtaining the delayed CF
coe
cients{a
(s)
}={
n+s
}. Because of this advantageous feature, the PD algorithm is
errorfree for those signal points{c
n+s
}that are integers. In practice, integer
data matrices{c
n+s
}are measured experimentally in MRS, NMR, ICRMS,
n
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