Digital Signal Processing Reference
In-Depth Information
the origin in a descending order with the aid of the expressions
c
s
=−
1
9
=
;
.
b
0
(b
1
c
s+1
+ b
2
c
s+2
++ b
K
c
s+K
(2.160)
c
s−1
=−
1
b
0
(b
1
c
s
+ b
2
c
s+1
++ b
K
c
s+K−1
.
This recursion recovers the preceding c
n
's and the general formula is
K
c
s−m
=−
1
b
0
(recovering the old c
n
′
s).
b
r
c
r+s−m
(2.161)
r=1
With such an established pattern, the set{c
n
}is said to form a forward
or backward selfrecurring sequence whose general term c
n
is given by the
geometric progression (2.138). The sum and the product of the two recurrent
sequences of the order K and K
′
also represent a recurrent sequence of the
order K +K
′
and KK
′
, respectively. Importantly, all the recurrent sequences
are summable, i.e., they are convergent to the correct limit.
The three main expressions (2.133), (2.159), (2.161) from this exposition
were derived by Prony [138] in 1790 and they are identical to those defining
the Pade approximant [51] from 1892. Based upon his course of Mathemati
cal Analysis via “Suite de Lecons d'Analyse de Prony” [138], Prony developed
in 1795 [139] a versatile method called later after his name for approximat
ing functions by linear combinations of exponentials. This is the wellknown
Prony method [139], which marks the birth of what later became the whole
field of signal processing. Rational approximations in the forms of polyno
mial quotients, known as the Pade approximants, were used by a number
of mathematical giants before Pade [51] dating back to the period 1730 -
1870 (Bernoulli, Stirling, Euler, Lambert, Lagrange, Jacobi, Hankel, Frobe
nius, Darboux, Laguerre and Chebyshev). Pade [51], through his doctoral
thesis, was widely credited to be the first to carry out the most systematic
study of these rational polynomials. Unfortunately, it does not seem to be
widely known that the published work of Prony [138, 139], who, due to his
great achievements became Baron de Prony, was also extremely systematic in
investigations of what later was termed the Pade approximant.
2.4 The fast Pade transform for quantum-mechanical
spectral analysis and signal processing
Irrespective of whether the analysis is carried out in the time or frequency do
main, the FPT appears as an optimal estimator especially in MRS. This is evi
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