Recapitulation of Pad´e-optimized processing of biomedical time signals - Signal Processing in MRS with Biomedical Applications - page 376

However, and this is what sets the PA apart from other methods, this “built
in” Pade variance is not an adjustable function, which can be used as a fitting
recipe for producing anything which one subjectively decides to be “su
ciently good”. Rather, the variance in the PA is free from any adjustable
parameter and, as such, it is fully and objectively controlled solely by the
structure of the input data F(z −1 ).
The possibility of explicitly computing the difference termO − (z −L−K−1 )
in the Pade estimate F(z −1 )≈P − L (z −1 )/Q − K (z −1 ) is the basis of the error
analysis of proven validity in the PA. If desired, such an evaluated variance
type remainderO − (z −L−K−1 ) may be fed back into the PA which, in turn,
can undergo iterative and systematic improvements with the possibility of
reproducing the original, input function F(z −1 ) with any prescribed accu
racy. However, this is not even necessary because with, e.g., the diagonal
PA, the numerically exact relationship

n=0 c n z −n = P − K (z −1 )/Q − K (z −1 ) to
literally hundreds of decimal places, if an application of this unprecedented
accuracy would ever be needed, can be achieved by systematically increasing
the common degree K in polynomials P − K (z −1 ) and Q − K (z −1 ).
Pade approximants can be computed through many different numerical al
gorithms, including the most stable numerical computations via continued
fractions. Moreover, unlike any other related method, for the known F(z −1 ),
both Pade polynomials P − L (z −1 ) and Q − K (z −1 ) in the PA can be extracted by
purely analytical means in their simple and concise closed forms [5, 17]. This
represents the gold standard against which all the corresponding numerical
algorithms should be benchmarked for their stability and robustness.
Outside mathematics, per se, theoretical physicists are most appreciative
of the power and usefulness of the PA, which they began to use more than
half a century ago in many problems ranging from the BrillouinWigner per
turbation series to divergent expansions in quantum chromodynamics in the
theory of strong interactions of elementary particles. The reason for such a
widespread usage of this method in theoretical physics is that, in fact, the
most interesting and also the most important series expansions emanating
from realistic problems are divergent. Other frequently encountered series,
although convergent in principle, often converge so slowly that they become
virtually impractical in any exhaustive application. Here, the PA comes to
rescue the situation in both cases by converting divergent into convergent
series and accelerating slowly converging series.
The reason that the same method is able to tackle these diametrically op
posing di culties is in the nonlinearity of the PA, as is obvious from the
definition (12.16). The condition for linearity of a given function F(z −1 ) is,
for example, F(az −1
1

∞

+bz −1
2

) = aF(z −1
1

)+bF(z −1
2

) for any two constants{a,b}

and any two values{z − 1 ,z −1 }of independent variable z −1 from the domain

of definition of F(z −1 ). In general, rational functions including the PA from
(12.2) do not satisfy this latter linearity condition. Therefore, general rational
functions R(z −1 ) from (12.1) as well as the Pade approximant, R (PA) (z −1 ),
from (12.2) are nonlinear. It is due to its nonlinearity that the PA gains its

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