Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 3

is recognized as one of the main problems from quantum theory of resonances
and spectroscopy [5]. Thus, the general quantummechanical relaxation for
malism can be profitably utilized to solve the quantification problem from
MRS and MRSI.
To this aim, we employ quantummechanical signal processing and spectral
analysis via the Green function. This is implemented algorithmically through
the fast Pade transform (FPT) [5]-[37]. The frequency spectrum is thereby
obtained as the unique quotient of two polynomials. This form of a rational
quantum response function to the external perturbation (static and gradient
magnetic fields, radiofrequency pulses) is dictated both by the quantum origin
of MRS and MRSI, as well as by the resolvent form, i.e., the operator Pade
approximant of the Green operator/matrix of the system under study.
The FPT includes two different versions of the Green function, with the
outgoing and incoming boundary conditions inside and outside the unit circle
of the complex harmonic variable z. The former and the latter variants of the
FPT correspond, respectively, to the causal and anticausal Padez transform
from the signal processing literature, as well as from mathematical statistics
[5]. In this topic, we show inter alia how the FPT solves the quantification
problem exactly by extracting the sought spectral parameters. This is done
using synthesized noisefree and noisecontaminated FIDs generated theoret
ically, but based upon entirely realistic data as typically encoded via MRS.
Experimentally measured time signals from MRS are studied here, as well.
In both MRS as well as MRSI, it is very important to avoid encoding long
FIDs, particularly when employing clinical scanners. This is because the ex
ponentially damped envelopes of these FIDs become completely embedded in
noise at long total acquisition times T. We emphasize that the fast Fourier
transform (FFT) requires long T to augment its frequency resolution which is
equal to 2π/T. Therefore, reliance upon the FFT leads to a conundrum, with
two diametrically opposing requisites. Namely, attempts at resolution en
hancement entail long T, whereas this leads to increased noise which hampers
the sought improvement in spectral quality.
With the FFT, it is typically seen that attempts are made to solve this prob
lem via a “patching” procedure by using shorter bandwidths or window sizes.
These attempts engender further problems, such as spectral deformations via
Gibbs ringing as well as diminished resolution.
The more fundamental reason why these attempted strategies within the
FFT have been unsatisfactory lies in the fact that the Fourier analysis com
pletely ignores the actual structure of the encoded time signal. Namely, with
the FFT all time signals encoded with the same T have the same resolution
2π/T, regardless of their internal structure. As such, the FFT is limited
solely to shape estimations [5]. Yet, each FID has its structure and this is
quantifiable by specific parameters. Typically, neither this structure nor the
related parameters of the encoded time signal are known before the signal is
processed. Spectral synthesis must reveal or reconstruct these unknowns from
the encoded time signal. This is accomplished by performing spectral decom