Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 194

any systematic reliability for processing insu ciently accurate FIDs corrupted
with noise as encoded via MRS. Additionally, and most importantly, in the
illustrations which follow, we will also spectrally analyze simulated time sig
nals embedded in random Gaussian distributed noise in order to see whether
it would still be possible to unambiguously reconstruct all the genuine reso
nances, including the weakest ones as well as those that are tightly overlapped
and practically degenerate. Thereby, it will be demonstrated that the FPT
is also a highly reliable method for quantifying noisepolluted time signals
reminiscent of those encoded via MRS in clinical diagnostics.
Nevertheless, this comprehensiveness in the envisaged testing would still be
incomplete without being supplemented by the critical demand for unequivo
cal separation of genuine from spurious information in the output data. Such
a demanding list of prerequisites for adequacy in signal processing might be
approached by many methods [146]-[205], but with varying and usually insuf
ficient success in parametric estimations. Based upon our thorough analysis
from chapter 3 with accuracy predetermined by 4digit input data, and rely
ing upon the conclusions on a number of our earlier studies [5, 10, 11, 24, 34],
we shall opt for the fast Pade transform to accomplish the formulated task of
paramount significance for quantification in MRS.
From a theoretical viewpoint, all the sought complex frequencies and am
plitudes can unambiguously and exactly be retrieved from a given input time
signal by the FPT. In its role of the response or system function, given by the
unique quotient of two polynomials, the FPT is the most frequently applied
estimator in various research fields. In principle, the FPT can achieve the
socalled spectral convergence, which represents the exponential convergence
rate as a function of the signal length for a fixed bandwidth. This unique prop
erty equips the FPT with unprecedented highresolution capabilities that are,
in fact, theoretically unrestricted. In the present chapter, we will illustrate
these features by the exact reconstruction (within machine accuracy) of all the
spectral parameters from an input time signal containing 25 harmonics that
are complex, attenuated exponentials. To make the already di cult task even
more challenging, we shall include two practically degenerate resonances with
chemical shifts differing by a remarkably small fraction of only 10 −11 ppm.
We will show that by employing merely a quarter of the full signal length,
the FPT is capable of exactly reconstructing all the input spectral parame
ters defined with 12 digits of accuracy. Hence, achieving 12digit accuracy in
reconstructions by using 12digit accurate input data proves that the FPT is
robust even against roundofferrors.
In particular, we shall demonstrate that when the FPT is near the conver
gence region, an unparalleled “phase transition” takes place, because literally
two additional signal points are su cient to attain the full 12 digit accuracy
with the exponentially fast rate of convergence. Such an achievement repre
sents the crucial proofofprinciple for the highresolution power of the FPT
for machine accurate input time signals. Finally, we shall demonstrate that
the FPT can unambiguously separate genuine from spurious resonances using