Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 4

position which, in turn, avoids the need for long T. In parametric estimation,
in contrast to Fourier nonparametric processing, resolution is not determined
solely by T.
As noted, the structural parameters are the fundamental frequencies and
the associated amplitudes, as the elements from which the attenuated har
monics from the investigated FID are built. These nodal frequencies are
complexvalued because external perturbations set the investigated system
into motion via exponentially damped harmonic oscillations. With the pas
sage of time, these oscillations must be attenuated thus exhibiting decreased
intensity of the signal from the system, since the total time duration of the
signal is always finite (T <∞) in any realistic experiment. In other words, no
process which generates time signals as the system's response to external exci
tations could possibly last forever, and this is the origin of the said attenuation
causing the system to be dissipative. Dissipative systems are associated with
nonHermitean “Hamiltonians” which have complex fundamental frequencies
{ω k }whose real and imaginary parts describe the cosinusoidal oscillations and
exponentially damped amplitudes, respectively. Gamow's [38]-[43] complex
energies E k or frequencies ω k are common in scattering theory physics for
transient (metastable, decaying) states such as resonances. Their real parts
have the customary meaning of energy or frequency observables (experimen
tally measurable quantities) as in Hermitean quantum mechanics, whereas the
imaginary parts are proportional to the inverses of the individual k th state's
halflifetime, as the measure of the Heisenberg uncertainty stemming from the
decaying nature of the considered state.
An important group of algorithms for spectral decomposition are those in
the category known as convergence accelerators and analytical continuators
of Fourier series and sequences. It should be pointed out that for a given
bandwidth, the FFT converges in a linear fashion, at a slow rate 1/N with
increasing signal length N. This convergence can be accelerated significantly.
The FPT is a convergence accelerator which has at least a quadratic con
vergence 1/N 2 with a systematically augmented N. Furthermore, near full
convergence as a function of the signal length, the FPT displays “spectral
resolving power”, which denotes a remarkable exponential convergence rate
to the exact values of the sought genuine parameters. This feature trans
lates directly into two simultaneous advantages: resolution enhancement and
shortening the FID. With this, the socalled Fourier dichotomy that “a resolu
tion enhancement cannot be obtained without prolonging T”, no longer is the
case. Operationally, this advantage of the FPT is related to its modeling of
the spectrum by a quotient of two unique polynomials, rather than the single
polynomial from the FFT.
The same result can alternatively be achieved via analytical continuation
rather than convergence acceleration. Cauchy introduced this powerful con
cept of analytical continuation, by which a divergent sequence is made to
converge. If, e.g., a series in powers of the harmonic variable z (as the Green
function representing the exact spectrum) is convergent outside the unit cir