Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 226

verge last. The approach to the global minimum can indirectly be followed
by looking for constancy of spectral parameters as a function of the truncated
signal length at a fixed bandwidth. When all the spectral parameters stabi
lize such that the results remain unchanged with the addition of more signal
points, the exact results for all the 100 parameters are obtained to within ma
chine accuracy. However, this does not take place in a steady process, i.e., by
obtaining the exact outer frequencies first and then awaiting for the remain
ing frequencies to attain their exact values one by one. Rather, the contrary
actually occurs. While the physical outermost frequencies do appear first,
this does not initially occur to machine accuracy. Machine accurate values
for the outermost frequencies are obtained simultaneously with the machine
accurate results for all the other genuine resonances. A very sharp transition
occurs at that point. Even when extremely close to the global minimum of the
objective function, all the spectral parameters still fluctuate around their true
values. At convergence, all the parameters reach their exact values and for
this to take place, it is su cient that the running order K of the FPT changes
by only one unit, i.e., two more signal points are needed. Subsequent to this
stage, any further addition of signal points does not affect the results within
the required 12 digit accuracy for noiseless FIDs. Thus, this process is seen to
be a veritable phase transition in which all the complexvalued quantities are
needed, with their phases being critical to the emergence of simultaneous res
onance for all the fundamental harmonics. We can see this most dramatically
if we fix the phases of the input amplitudes of each fundamental harmonics
to zero, as done in the present analysis. The reconstructed phases fluctuate
around zero, but no fully exact value of the remaining spectral parameters
for any resonance is reached until all the machine accurate zerovalued phases
are obtained. This is the signature of the algorithm's optimality attained in
the simultaneous retrieval of 100 spectral parameters through undergoing a
dynamic phase transition at which point all the sought quantities collectively
synchronize their machine accurate numerical values.
The distributions of poles and zeros in complex frequency planes are essen
tial to spectral analysis. Argand plots are particularly helpful for visualizing
polezero cancellations of complexvalued frequencies in the polar as well as
the rectangular planes. Zerovalued spurious amplitudes for nonphysical res
onances as a function of chemical shift are also displayed together with the
positive values of the amplitudes for true resonances. Illustrations are given
for both the FPT (+) and FPT (−) . In the FPT (−) physical and nonphysical
resonances are intermixed outside the unit circles. Nevertheless, the denois
ing Froissart filter completely disentangles one from the other according to the
twofold signature: polezero coincidences and zerovalued amplitudes. In the
FPT (+) , on the other hand, the true and spurious resonances are completely
separated from each other in two disjoint portions of the complex frequency
plane. This remarkable separation of genuine from spurious resonances rep
resents a key feature for signal processing in biomedical applications and for
other disciplines, as well.