Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 8

Defect poles and zeros are also encountered in nonspectral problems, e.g.,
in approximations of a given function by the Pade approximant. This can
best be seen when using a finiteorder (say K ′ ) diagonal Pade approximant
[K ′ /K ′ ] f (z) to represent a function f(z) which has an infinite number of
known true poles and zeros. Invariably, a number L ′ < K ′ of Padedetected
poles and zeros will be close to their original counterparts from f(z), whereas
the remaining M ′ ≡K ′ −L ′ > 0 ones will be fake. Good agreement will be
found between the PA and the original function in the domain containing L ′
true poles and zeros. Near the fake pairs, [K ′ /K ′ ] f (z) will considerably de
viate from f(z). Such deviations will widely and unpredictably change when
the order is increased from K ′ to K ′′ > K ′ . The number of true poles and
zeros will increase from L ′ to L ′′ > L ′ , and so will the number M ′′ of the
fake ones (M ′′ ≡K ′′ −L ′ > 0). In general, the numerical values of true poles
and zeros will improve for K ′′ relative to K ′ . Simultaneously, the distances
between new fake poles and zeros will diminish, as this is precisely how the
concept of “convergence in measure” manifests itself [61]. Eventually, with
the Pade order becoming infinitely large, the PA will converge almost every
where to f(z) except at a countable number of tightly packed fake poles and
zeros. However, even in this simple application of the PA aimed at obtaining
the approximation [K/K] f (z)≈f(z), the found fake poles and zeros are not
a totally useless defect. Quite the contrary, we may say that their existence is
even necessary for any finite order K <∞. It is precisely through these fake
poles and zeros, appearing as Froissart pairs/doublets, that the Pade approx
imant [K/K] f (z) (K <∞), despite its finite number of poles and zeros, is
nevertheless able to mimic quite well the function f(z) which has an infinite
number of poles and zeros. Such fake poles and zeros, i.e., Froissart doublets,
being artefacts, cannot be found in f(z) and, moreover, they induce random
ness which is the noisy part of the approximation [K/K] f (z). Yet, we see that
such random artefacts can be useful. In some other examples, it is precisely
through lining up of Froissart poles and zeros that the PA is able to closely
mimic functions with cuts and branch points. Thus, in general, it is because of
Froissart doublets that the PA, as a meromorphic function, having poles and
zeros as its only singularities, is capable of providing good approximations to
nonmeromorphic functions with cuts and/or branch points.
In signal processing, by increasing the running order K ′ of the diagonal PA,
one always finds a number of genuine and spurious (Froissart) resonances for
a time signal which has exactly K true harmonics. Further, one can illustrate,
as will be done in the present topic, that within finite arithmetics, the optimal
Pade spectral analysis is achieved in reconstructing a large number of input
parameters for all the genuine resonances if K ′ is considerably larger than
K. The resulting K ′ −K > 0 Froissart doublets help achieve machine accu
racy (e.g., 12digit input, 12digit output for the retrieved true resonances),
while simultaneously annihilating themselves in the final absorption spectra
through the polezero cancellations. By contrast, if one could be able to
perform spectral analysis within exact arithmetics, the K th order Pade ap