Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 225

with each other, so that they are canceled from the canonical representation
of P K /Q K (see chapter 5). We have also shown that the same saturation
P K ex +m /Q K ex +m = P K ex /Q K ex (m = 1, 2,...) occurs in the equivalent Heavi
side partial fraction representation of the Pade polynomial quotient through
zero amplitudes for any K > K ex . Furthermore, polezero confluences can also
appear at any K≤K ex . However, the corresponding amplitudes are always
found to be equal to zero. We have hereby demonstrated that all zerovalued
amplitudes and the associated polezero coincidences represent the unambigu
ous signatures of nonphysical information (noise and noiselike) that appears
during spectral analysis. This is the essential feature of exact signalnoise
separation by polezero cancellations in the FPT. It can certainly be antici
pated that the presentlydescribed new type of signal denoising through the
Froissart filter will have important and broad applications throughout various
fields within signal processing.
We have presented several illustrations concerning machine accurate recon
structions of all the fundamental frequencies and amplitudes, including the
unambiguous retrieval of the exact number of resonances. Convergence under
these stringentlyimposed conditions (exact 12 digit output for exact 12 digit
input data) has been accomplished. This further confirms the robustness of
the FPT even with respect to roundofferrors. We achieve this by using two
regular computational routines from MATLAB for solving a system of linear
equations and rooting the characteristic equation. Thus, the robustness of the
FPT is mainly due to the rational model for the response function, and not to
some speciallydesigned algorithm. The Pade model of polynomial quotients
is intrinsically robust. Above all, this is expected from the physics vantage
point, as dictated by quantum mechanics through Green functions which in
variably reduce to a ratio of two polynomials. The subsequent computations
translate this solid theoretical basis into the exact numerical results.
Resonance is a special type of phenomenon in which phase transitions play a
critical role. If we examine phase spectra, this is evidenced in the appearance
of marked jumps by π at each resonant frequency, as the Levinson theorem
prescribes. When there are numerous resonances in a spectrum, such as those
studied herein, reconstruction of four machine accurate spectral parameters
for each resonance (frequencies and amplitudes both complexvalued) becomes
a daunting numerical challenge. Specifically, the 25 resonances need a numeri
cally exact solution for 100 spectral parameters. We should thus consider this
optimization problem as searching for the global minimum of an objective
function in a hyperspace of 100 dimensions.
The density of states is one of the key determinants of resolution and con
vergence rate. The average number of fundamental frequencies in the window
of interest defines the resolution of the FPT. This is also true for all other
parametric estimators. In contrast, in the FFT, resolution is defined by the
distance between the two adjacent frequencies. As is known from the Lanczos
algorithm, convergence is first reached for the outermost frequencies in the
range of investigation. The innermost and tightly congested frequencies con