Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 164

etc. Likewise, the same infinite precision, i.e., without roundoff errors is also
possible in the PD algorithm for autocorrelation functions{c n }or power

moments{ n+s }expressed as rational numbers. For many circumstances in

physics, such as systems exposed to external fields (Zeeman effect, harmonic
oscillator, etc.), the role of signal points is taken by expansion coe cients that
are obtained exactly as rational numbers from quantummechanical perturba
tion theory. In such a case, one would operate directly with rational numbers
using symbolic language programming, e.g., MAPLE [185]-[187]. Regarding
e ciency, the computational complexity of the PD algorithm for the CF coef
ficients{a (s m }(1≤m≤n) is of the order of n 2 multiplications. This should
be compared with a direct computation of the Hankel determinant H n (c s ) of
the dimension n from the definition (4.20) for{a (s)

}where, e.g., the Cramer
rule would require the formidable n! multiplications that would prohibit any
meaningful application for large n.
As seen in section 4.6 , the QD algorithm (4.27) for the auxiliary double
array{q (s n ,e (s)

n

}performs divisions in each iteration. In a finiteprecision
arithmetic, this could produce roundoff errors that might cause the QD al
gorithm to break down for noninteger signal points. When the input data
{c n+s }are nonzero integers, divisions would yield rational numbers in the
course of the QD recursion. This also would be innocuous with errorfree
results if infinite precision arithmetic using rational numbers is employed by
means of, e.g., MAPLE [185]-[187].
Remarkably, it is possible to judiciously combine the vectors{λ (s)
1,m

n

}to

recursively generate Hankel determinants of arbitrary orders as

H n (c s ) = λ (s)

n

1,2n
λ (s)
2n−2

λ (s) λ (s)

n

H n−1 (c s ) =

1,m .

(4.37)

m=1

The recursion (4.37) can explicitly be solved by iterations and the result is
given by

λ (s)
1,2m
λ (s)
2m−2

n

H n (c s ) = c s H 1 (c s ) = c s (n = 2, 3,...). (4.38)

m=2

With this expression, the result of the generalorder Hankel determinant
H n (c s ) can e ciently be obtained from the recursively precomputed string
{λ (s)
1,m

}. A computationally very attractive property of the formula (4.38) is
that it effectively performs only the computation of the simplest 2×2 de
terminant (4.31). If the data{c n+s }are integers, the corresponding Hankel
determinant H n (c s ) will also be an integer number, say N (s n . In this case, the
result (4.38) for H n (c s ) would be a rational number. However, by definition
(4.37), the numerator in (4.38) reads as

n

m=2 λ (s)

= N (s)

n

m=2 λ (s)

2m−2 so
that H n (c s ) = N (s n , as it ought to be. To preserve such a feature in compu
tations, integer algebra should be used in which the integer numbers{λ (s)
i,j

n

1,2m

}