Digital Signal Processing Reference
In-Depth Information
of the reconstructed spectral parameters is predicted by, e.g., the HaziTaylor
stabilization method from resonant scattering theory [140]. According to that
method, the computed eigenenergies will have constant values when the di
mension of the finiterank Hamiltonian matrix is increased by systematically
augmenting the size of the box in which the examined physical system is en
closed. This procedure can be easily implemented in the FPT in order to
separate the physical portion of the FID from its noise. As a consequence,
the stable transients are distinguished from those transients that are unstable.
The former and the latter are identified as true (physical) and spurious (non
physical, noisy or noiselike), respectively. This procedure has been validated
using the FPT in ICRMS [13, 14] and NMR [15].
As will be presented in chapter 7, the FPT has been tested using encoded
FIDs from MRS. Therein, the superior resolving power and convergence rate
of the FPT for total shape spectra are established in relation to the FFT.
Comparative analyses within the FPT and FFT are seen to be important
when the FFT spectra with the full FID are of high quality so that they
could be considered as the gold standard in their own category of envelope
spectra. This was the case in the studies from Refs. [8, 9] and [18][23] using
FIDs of high SNR as encoded via MRS [141] at magnetic field strengths of 4T
and 7T from the occipital region of healthy human brain. Among the major
conclusions that emerged was that the FPT attains at least two times superior
resolution compared to the FFT for the same fraction N/M of the full signal
length N. Of particular note, this was true even for M = 2 when N/2 signal
points were used in both the FPT and FFT. These findings essentially mean
that the FPT can reach the same resolution as the FFT by using only the
first half N/2 of the full signal length N. Such a remarkable superiority of the
FPT relative to the FFT for total shape spectra computed from time signals
encoded with high SNR is expected to be even greater for synthesized FIDs.
Including the envelope spectra in the present chapter helps to link the re
sults reported here with those from chapter 7 as well as from Refs. [8, 9]. This,
in turn, would allow another consistency check of our previous conclusions. It
should be emphasized, however, that it is necessary to proceed beyond com
parisons between the Pade and Fourier total shape spectra. The results of
quantification need to be explicitly reported. These results of the FPT for
the numerical values of the spectral parameters reconstructed from in vivo
MRS FIDs were implicitly present in the studies of the total shape spectra.
This was the case because the reported total shape spectra [8, 9] were built
using the Heaviside partial fraction expansions
K
k=1
d
±
k
z
±1
/(z
±1
−z
±
k
) with
z
±1
= exp (±iωτ) and z
±
k
= exp (±iω
±
k
τ). These fractions can only be com
puted after reconstruction of the complex frequencies{ω
±
k
}and amplitudes
{d
±
k
}for all the physical resonances. However, the numerical values of the
spectral parameters were not explicitly reported in our examination of total
shape spectra [8, 9], since the focus was on a comparison of the convergence
rates of the FPT and FFT. Such comparisons are only possible using total
Search WWH ::
Custom Search