Digital Signal Processing Reference
In-Depth Information
They represent the leading class of functions from mathematics that find rich
areas of applications in many research fields, ranging from physics to life sci
ences. This is primarily due to the main features of rational approximations,
as they apply to analysis and interpretation of data that can come from either
experimental measurements or from theory by means of numerical computa
tions. Such features are interpolation and extrapolation. By interpolation,
one attempts to reliably generate the values of the observables in certain
ranges or points where measured data are unavailable. By extrapolation,
one tries to faithfully predict the values that could have been measured had
the experiment continued beyond the last recorded data point of the studied
physical quantity. Both cases are of great practical importance, since a reli
able method would save extra measurements or possibly very time consuming
numerical computations. Yet, the goal of an optimal theory is not to try in
vain to achieve physically adequate interpolation and extrapolation by simple
minded fitting with its nonuniqueness, subjectivity and bias. Rather, the aim
is to ingrain these two features into adequate mathematical models without
adjustable parameters for physics theories. The most suitable framework for
solving this challenging simultaneous interpolationextrapolation problem is
provided by rational functions of the general type (12.1).
12.2 The dominant role of Pade approximant among all
rational functions
The simplest, and crucially, the most powerful rational functions, are Pade
approximants R
(PA)
(z
−1
), introduced by ratios of two polynomials P
−
L
(z
−1
)
and Q
−
K
(z
−1
) of degrees L and K
R
(PA)
(z
−1
) =
P
−
L
(z
−1
)
Q
−
K
(z
−1
)
(12.2)
L
K
P
−
L
(z
−1
) =
Q
−
K
(z
−1
) =
p
−
r
z
−r
q
−
r
z
−s
(12.3)
r=0
s=0
}are the expansion coe
cients of P
−
L
(z
−1
) and Q
−
K
(z
−1
). The
most stable are the diagonal and paradiagonal PA as obtained for L = K and
L−1 = K, respectively. The polynomial ratio from (12.2) becomes unique if
it is taken to approximate a given Maclaurin series
where{p
−
r
,q
−
s
∞
F(z
−1
) =
c
n
z
−n
(12.4)
n=0
where the elements c
n
of infinite set{c
n
}are the known expansion coe
cients.
In applications to signal processing in many different fields, the expansion
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