Froissart doublets

Signal-noise separation (SNS) by Froissart doublets within the FPT<+) and FPT( ) is illustrated in Figures 4.10 and 4.11 for the noise-free and noise-corrupted time signals, respectively. Only a small number of all the obtained Froissart doublets appears in the shown frequency window in Figure 4.10. The selected subinterval 0-6 ppm is important because all the MR-detectable brain metabolites lie within this chemical shift domain of the full Nyquist range. Froissart doublets as spurious resonances are detected by the confluence of poles and zeros in the list of the Pade-reconstructed spectral parameters. 

The only difference between Froissart doublets for the noise-free c and noise-corrupted cn + rn time signals from Figures 4.10 and 4.11, respectively, is that the latter are more irregularly distributed than the former. This is expected due to the presence of the random perturbation rn in the noise-corrupted time signal. Flowever, this difference is irrelevant since the only concern to SNS is that noise-like or noisy information is readily identifiable by pole-zero coincidences. Note that the full auxiliary lines on each subplot in Figures 4.10 and 4.11 are drawn merely to transparently delineate the areas with Froissart doublets. 

GENUINE and SPURIOUS PARAMETERS in FAST PADE TRANSFORM NOISE-FREE FID FROISSART DOUBLETS (SPURIOUS RESONANCES) CONFLUENCE of PADE POLES PADE ZEROS GIVING NULL AMPLITUDES... 

Figure 4.10 A subset of the whole set of Froissart doublets in the FPT( ) at a quarter N/4 = 256 of the full length (N = 1024) of a noise-free time signal. On panel (i), the FPT(+) achieves a total separation of genuine from spurious resonances that are mixed together in the FPT( ) on panel (ii). Panel (iii) shows genuine and spurious amplitudes in the FPT( ). The reconstructed converged amplitudes are identical in the FPTf 1 > and the FPTf K All the spurious amplitudes are zero-valued.

The FPT provides the exact separation of genuine (physical) from spurious (unphysical, noise and /or noise-like) information encountered either in theory or measurements involving time signals. This is accomplished by means of Froissart doublets [16] that are coincident pairs of poles z = 2 q and zeros zfp in the response functions or complex-valued spectra... 

Thus, the distances between poles z q and zeros z p are proportional to the amplitudes df. Hence, df = 0 for the exact pole-zero coincidences in the Froissart doublets (19). It is vital to have full control over the locations of all... 

Here, in the numerator, it is permitted to have k = k, in which case every Froissart doublet from Eq. (19) would produce zero-valued terms (4q- Z)t,p)/ and thus the whole product in Eq. (23) will become zero. As mentioned earlier, this yields df = 0 for zfg = zfp, according to Eq. (19). It can also be seen that Eq. (21) and expression (23) are compatible with each other. In computations, expression (23) should not be used to obtain the amplitudes df in the FPT. This is because formula (23) employs the whole set of the reconstructed amplitudes to compute df for the kth resonance. Therefore, even the slightest inaccuracy, such as near cancellations of poles and zeros, rather than... 

Reconstructed resonances were assessed as to whether they were genuine via the concept of Froissart doublets. This was done throughout the entire Nyquist range, with special attention to the range between 1.3 and 3.3 ppm, that is, the range of interest. Of a total of 750 resonances, 741 were found to be spurious, that is, with zero amplitudes and the pole-zero coincidences. For each of the three time signals, there were nine true resonances. 

Dz. Belkic, Exact signal-noise separation by Froissart doublets in the fast Pade bansform for magnetic resonance spectroscopy, Adv. Quantum Chem. 56 (2009) 95. 

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