Digital Signal Processing Reference
In-Depth Information
initions as the Cauchy residues (2.183) or (2.192) of the associated Green
functions. There is also the uniqueness proof for these amplitudes. This
proof demonstrates that, for the same set of retrieved complex frequencies, all
the various ways of computing the associated amplitudes must yield identical
results [5].
The fundamental frequencies for the genuine resonances reconstructed via
the HLSVD and FPT under the same conditions should therefore coincide
with each other. Consequently, despite the fact that in the HLSVD, the
amplitudes are computed by solving a system of linear equations, rather than
through the analytical expression of the FPT, the results from both methods
must be the same. This should serve as a check of the numerical solutions
for the amplitudes in the HLSVD. Obviously, the procedure in the FPT is
optimal for the amplitudes of resonances encountered in MRS, since these
Pade residues are available in the closed, analytical formula (2.183) or (2.192).
As stated in chapter 1, it has frequently been contended in MRS that the so
called prior information from fittings should be used in nonfitting algorithms,
as well. Moreover, it is claimed that the overall quality of quantification is
predetermined by some prior information. However, the alleged 'biochemical'
prior information used in MRS is most often a preassigned relationship, e.g.,
fixed phases or amplitude ratios or quotients of peak areas of some resonances
(or imposed connections among widths of several peaks in a selected part of
the spectrum or throughout the Nyquist interval, etc.), as done in VARPRO,
AMARES and LCModel. These unsubstantiated claims imply that such prior
information should be viewed as a requirement for successful quantifications
in MRS. This is incorrect, since the introduction of this type of prior infor
mation actually represents an attempt to mitigate the effect of subjectivity
of fittings. These attempts via certain imposed constraints germane only to
fitting are, in fact, aimed at reducing the error from under or overestimation
of subjectively preselected metabolites in an expansion set. Thus, for exam
ple, underfitting would result in an overestimation of the actual values of
the spectral parameters (amplitudes) because of the lack of su
cient number
of resonances in an expansion sets. Such an outcome is due to some attempts
to minimize the LS residuals. This is particularly unwelcome in medical diag
nostics via MRS, since such minimizations could easily overestimate the peak
areas and this, in turn, would undermine the estimates for the related concen
trations. The fitted peak areas are affected by attempts to minimize the error
from the missing genuine resonances in a chosen basis set by overweighting
the relative contribution from the included resonances.
These outlined fundamental deficiencies of fitting approaches to quantifi
cations in MRS also indicate the need for better estimations of spectral pa
rameters. These nonfitting strategies should not contain any free parameters
and should provide unique solutions for restoration of the unknown harmonic
components from FIDs in MRS.
The FPT is wellsuited to fulfill this task of quantification in MRS. This
can be appreciated by noting that the FPT is the exact filter for all the
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