Conclusion and outlooks - Signal Processing in MRS with Biomedical Applications - page 392

lation from their environment and/or when they are perturbed by external
excitations or fields. These major characteristics of general physical, chem
ical or biological systems of small as well as large dimensions or degrees of
freedom are universally embodied in their spectra that can be comprehended
and adequately interpreted in terms of a relatively limited number of leading
parameters. Such parameters are the two sets of physically measurable quan
tities (observables) called the characteristic numbers or eigenenergies{E k }

}of the system's operator ˆ , and the corresponding or eigenfrequencies{ω k

intensities{I k }of the spectral lines or lineshapes.

The socalled system's operator ˆ

}or amplitudes{d k

is the generator of the system interactive
dynamics. In quantum physics ˆ = H where H is the system's Hamiltonian
operator consisting of kinetic energy operators for describing kinematics, and
potential operators responsible for interactions. Physically, eigenfrequencies
{ω k

}are the fundamental or nodal frequencies of the system intrinsic os
cillations that are supposed to be determined while applying any external
perturbations. These latter disturbances can be, e.g., a radiofrequency pulse
combined with a static and a gradient magnetic field, as implemented in MRS.
The studied system first absorbs the external radiation or impinging parti
cle, and then dispenses the received excess of energy through some discrete
internal transitions among the quantummechanically possible energy levels
leading to, e.g., excitation, particle emission, etc. We infer about such intrin
sic transitions only later when they are completed and this can be manifested
in various ways, as can be exemplified as follows.
•(i) Emission spectra of one or more ejected particles can be measured in
experiments for the case of a decay of an unstable transient system comprised
of the target and an external perturbation via the projectile beam. There are
many examples of this mechanism throughout physics, e.g., nuclear transmu
tations, involving nuclei as projectiles and targets. Here, a large number of
peaks is routinely observed in the measured cross sections. This is interpreted
as formation of a transient compound system (projectile plus target) with a
number of positive energies{E k }that produce metastable states of this de

caying aggregate. If the impact energy E i is swept across these quasibound
energies{E k

}of the compound system, enhanced cross sections are detected

whenever E i ≈E k . This leads to resonance peaks in measured cross section

spectra. The mathematical form of these socalled BreitWigner resonances
coincides with the PA, which then can be used to determine, e.g., positions,
lifetimes and areas of such peaks. The found peak areas are proportional to
peak heights (resonance amplitudes) and these reconstructed data are directly
related to the number of projectiles that undergo resonant collisions with the
target. Such resonance spectra are usually plotted as counts per channel,
where the ordinate gives sticks proportional to the number of the incident
particles (counts) that hit the detector at the given resonant energy (chan
nel). We can see here how the Pade approximant is perfectly suited to describe
the wellknown BreitWigner mechanism of formation and subsequent decay
of compound systems comprised of the projectile and target particles. It is