Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 195

both noisefree and noisecorrupted FIDs. All of the conclusions reached in
this chapter about the performance of the FPT in MRS are also valid in, e.g,
analytical chemistry when using ICRMS or NMR spectroscopy, and in other
applied sciences and technologies with signal processing for data analysis [5].

6.2 The key factors for high resolution in quantification

Following the outlined task, we shall apply the FPT to provide machine ac
curate quantification, and illustrations of signalnoise separation. Machine
accuracy is sought to benchmark the FPT on the noisefree FIDs. Passing
this most stringent test is far from being academic or formal. Quite the con
trary, those estimators that cannot achieve a comparable accuracy as the FPT
for fully controlled noiseless FIDs have no guarantee whatsoever for reliable
performance in the case of uncontrolled FIDs such as those which are encoded
in vivo by means of MRS. The maximal sought accuracy which is possible for
noisecorrupted synthesized or encoded FIDs is 34 stable digits in the recon
structed concentrations of the genuine, clinically relevant metabolites. This
level of requested accuracy can readily be achieved by the FPT for synthesized
noisecorrupted FIDs as well as encoded in vivo time signals.
Before proceeding further, we must enumerate the key factors that influence
the overall performance of estimators in general parametric reconstructions,
especially when aiming at machine accuracy. First of all, the resolving power
and convergence rate of a given estimator depend on certain obvious char
acteristics of time signals, such as SNR and the total acquisition time T. In
addition, however, there are more subtle aspects of signal processing that can
be of critical importance with respect to accuracy and robustness of the signal
processor under examination. Among these key features are the configurations
of poles and zeros in the complex plane, the smallest distance among poles
on the one hand and zeros on the other, particularly when such distances
are compared to the level of noise, the density of signal poles and zeros in
the chosen portion or throughout the Nyquist range, interseparations among
poles and zeros, their distance from the real frequency axis and the smallest
imaginary frequencies (the longest lifetimes of transients) in the spectrum.
As thoroughly presented and analyzed in chapter 3, Argand plots provide
a remarkably convenient mathematical tool for scrutinizing the enumerated
subtle features of signal processing. These plots display the imaginary part as
a function of the corresponding real part of a given complexvalued quantity,
such as the harmonic variables z ± k , the fundamental frequencies ν ± k and the
corresponding amplitudes d ± k . It is also useful to depict the dependence of
the absolute values of the amplitudes,|d ± k |/Im(ν ± k ). As

discussed in chapter 3, these are proportional to the concentrations of the

|, or peak heights|d ± k