Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 101

mimics a typical, realistic situation seen in the customary presentations of
MRgenerated spectra of human brain metabolites from experimental encod
ing after water suppression. The corresponding original FIDs encoded from
the brain always generate a single giant resonance due to the fact that brain
tissue contains about 75% water. All the other brain metabolites are hardly
visible, since they appear as very tiny peaks superimposed on the tail of the
exponentially decaying curve of the huge water peak. While water is of pri
mary concern to MRI, which is aimed at detecting the sought proton spin
density distributions to depict the anatomy/morphology of the scanned tis
sue, this molecule is not usually in the focus of MRS. Therefore, regarding the
output data, there is every interest to reduce the concentration of the water
molecule as much as possible in order to allow the emergence of the other main
metabolites that are clinically more informative from the MRS perspective.
Customary water suppression can be accomplished in three different ways in
clinical scanners [7, 143], and invariably the resulting spectra contain a much
smaller, socalled residual water peak, which is simulated by the component
n k = 25 on panel (iv). As a result of successful water suppression at the end of
the experimental encoding, many metabolites pop out from the background,
and this is simulated on panel (iv) by a realistic distribution of the input
amplitudes{|d k |, Re(ν k )}for metabolites n k = 1−24 (Lip - PCho) in the
interval Re(ν k )∈[0.985, 4.271] ppm. The absolute values|d k |of the ampli
tudes d k are the key spectral parameters for determining the concentrations
of metabolites for the given Lorentzian lineshapes, as seen from (3.15).
Argand plots for the input harmonic variables z k (signal poles) and their
inverses z − k are shown on panels (ii) and (v) in Fig. 3.1 . The former and
the latter variables are all lying inside and outside the unit circle,|z|< 1 and
|z|> 1, respectively. As will be displayed later on, the complex quantities z k
and z − k are clustered in the 1st and 4th quadrant, respectively. This is not
apparent in Fig. 3.1 in which, for convenience, the values on the abscissae and
ordinates are used in descending order. The signal poles z k (or their inverses
z − k ) are tightly packed together and lie close to the inner (or outer) boundary
of the unit circle, respectively. They are more or less aligned along the inner
or outer arcs. The real and imaginary parts of the FID are seen in panels
(iii) and (vi) as heavily oscillating cosinusoids that are exponentially damped
with increased time. The earliest sampled signal points exhibit the largest
intensities. However, the intensities at larger times quickly become immersed
into the zerovalued background for this noiseless FID. Such a situation favors
those signal processors, like the FPT, that are capable of reconstructing all
the signal components by using the first, relatively short part of the FID with
sizable intensities. Of course, this is most relevant for noisy FIDs, since noise
dominates the physical signal at larger times.
Further, we present Fig. 3.2 t o illustrate the reason for which mathematical
methods are absolutely indispensable for MRS and many other fields that rely
upon signal processing. The top panel (i) in this figure depicts the time signal
from the input data that are encoded via MRS. As discussed, the shown free