Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 196

associated resonances/metabolites for Im(ν ± k ) > 0. Such highly illustrative
types of graphs will also be plotted in the present chapter, with important
implications related to the concept of Froissart doublets [44] through pole
zero coincidences [45] using noiseless and noisecorrupted time signals that
are theoretically generated (synthesized, simulated).

6.3 The goals and plan for presentation of results

The results from this chapter on quantification by the FPT (±) for synthesized
time signals are given in the two subsequent sections. In the first section 6.4 ,
we deal with machine accurate input and output data for a noiseless FID.
Machine accuracy is defined here to be the precision of 12 digits. This should
not be considered as a limitation, since other higher accuracies could also be
considered with equal success. The second section 6.5 is devoted to graphic
presentations of all the findings. Here, we use a noisefree as well as a noise
corrupted simulated time signal. Accuracy of the spectral parameters of both
of these FIDs is restricted to 4digits to mimic the realistic, MRencoded data
in clinical scanners that operate at B 0 = 1.5T [88]. In section 6.5 , 12digit
accuracy could also be used for the input parameters to generate a noiseless
FID, but it would be pointless to add noise to this latter time signal to ob
tain the corresponding noisy FID. The reason being that the added noise itself
would affect, say the 4th digit. Therefore, it is more reasonable to consistently
use 4digit accuracy for both the input and output data when dealing with
noisecorrupted FIDs. Moreover, for an unambiguous interpretation of any
departure of distributions of noisy reconstructed spectral parameters in com
plex planes from their noiseless counterparts, it is necessary that the noisefree
input data themselves from section 6.5 are also given with 4digit accuracy.
Further, for the same reason of avoiding potential influence of other uncon
trollable factors that could lead to ambiguities in our comparative analyses
of the results from the noiseless and noisy FIDs, we choose the noiseless time
signal{c n }in section 6.5 t o be identical to the noiseless part of the noisy FID,

which is of the form{c n + r n

}, where r n is a given random noise.
The reason for not performing comparisons of the results of quantification
for a noiseless c ′ n and noisy{c n + r n }with{c ′ n }={c n }is that even very

small differences between{c ′ n

}and{c n }could lead to different distributions
of poles and zeros of the response functions in complex planes. This would
impinge upon a clearcut interpretation because any detected displacements
of these poles and zeros for the noisy{c n + r n }relative to the noiseless case
{c ′ n }for{c ′ n

}={c n }would stem from the two inseparable origins. One origin
could be a small difference (say, in the 4th digit) between{c ′ n }and{c n }, i.e.,

|c ′ n −c n |∼10 −4 . The other origin could be due to noise r n itself in{c n + r n }