Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 146

As discussed, the FPT does not need a total shape spectrum such as the
one from the FFT or any other processor in order to reconstruct the spectral
parameters. On the contrary, the FPT begins by extracting the complex fre
quencies and the corresponding amplitudes directly from the input raw data.
The spectral analysis could actually finish at that very point. Of course,
when all the spectral parameters are faithfully reconstructed, the shape spec
tra in any mode can be subsequently constructed in a straightforward manner
through the available explicit formulae.
The computational work needed for solving the quantification problem via
the FPT is reasonably straightforward. The complex resonance frequencies
are reconstructed by determining the roots of the denominator polynomial
Q K . The optimal method for rooting a polynomial of arbitrarily large order
is to solve the equivalent eigenvalue problem of the Hessenberg matrix, which
is extraordinarily sparse (the polynomial coe cients on the first row, unity
on the main diagonal and zeros elsewhere) [13, 77]. The Pade denominator
polynomial Q K is actually the characteristic or secular polynomial from quan
tum mechanics and linear algebra in numerical analysis. After determining
the fundamental frequencies, the corresponding unique amplitudes are taken
from the analytical expression for the Cauchy residue of the quotient P K /Q K .
This is in contrast to the procedures needed in, e.g., the HLSVD in which a
system of linear equations must be solved.
We emphasize here that these features are convenient, so that the FPT quite
naturally arrives at the quantifications in MRS. A robust algorithmic imple
mentation of the FPT, as done in the present illustrations using exclusively
the standard MATLAB routines, requires unprecedentedly minimal compu
tational effort. Furthermore, the results are numerically exact in the face
of the typical finite precision arithmetic with its attendant roundoff errors.
Herein we explicitly showed that by using less than a quarter of the full time
signal length, the FPT (−) can reconstruct exactly all the spectral parameters
of the investigated 25 resonances. This includes isolated, overlapped, tightly
overlapped as well as nearly degenerate peaks.
It is of interest to note that the Fourier analysis itself actually corroborates
the success of Pade analysis. It should be recalled that from the Fourier analy
sis, the exact onesided Fourier integral from zero to infinity over a time signal
from MRS, modeled by a linear combination of damped complex exponentials
with constant amplitudes, is expressed exactly by the corresponding sum of
Lorentzians. This, in turn, is the frequency spectrum. Moreover, continuing
within the Fourier analytical framework, this latter sum is expressed explicitly
in the representation of the Heaviside partial fractions. When the sum over
these partial fractions is performed, the result is invariably the ratio of two
unique polynomials, and this is how the standard Pade approximant is ob
tained. However, the Fourier analysis stops short before this latter statement
becomes explicit. Nevertheless, with a deeper insight it can be seen that the
exact spectrum for these FIDs is a rational function given by a polynomial
quotient. Thus, for this class of functions that describe spectra as encountered