Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 202

and, as such, need not be specially emphasized here.

6.4.2 Exponential convergence rates of Pade reconstructions

of spectral parameters with 12-digit accuracy

The results of the FPT (−) for machine accurate exact reconstructions of the
input spectral parameters are given in Tables 6.2 and 6.3 . In Table 6.2,
it can be seen that a varying level of accuracy is attained in the retrieved
spectral parameters from the FPT (−) near full convergence at 2 partial sig
nal lengths N P = 180, 220. As stated, neither length is compliant with the
FFTtype lengths, i.e., we have N P = 2 m (m positive integers). On panel (i)

at N P = 180, before full convergence is achieved, some 2 - 7 exactly recon
structed digits can be seen. Note that the 11th resonance is not detected here.
Its absence is marked by the sign “−” in the first column at the correspond
ing vacant location (n ◦ k = 11). However, at N P = 220 an enormous increase
in accuracy is observed on panel (ii), with all the 12 input digits exactly re
constructed for each spectral parameter of 25 resonances. Thus, with only
220 signal points out of 1024 entries from the full FID, the FPT (−) resolves
unequivocally the two near degenerate frequencies separated from each other
by an unprecedented chemical shift of merely 10 −11 ppm. This shows that
the FPT (−) has an exponential convergence rate (the spectral convergence)
[188] to the exact numerical values within machine accuracy of all the recon
structed fundamental frequencies and the associated amplitudes. Moreover,
these 12digit output results for the 12digit input data prove an indeed great
robustness of the FPT (−) even against roundofferrors.
In Table 6.3, we examine the accuracy when the partial signal length N P is
selected via a composite number 2 m (m positive integers), as customarily used
in the FFT. Here, on panels (i) and (ii), we show the results of the FPT (−)
at a quarter N/4 = 2 8 = 256, and the full signal length N = 2 10 = 1024.
It can be seen that these two panels yield exactly the same 12digit accurate
results throughout. We have also confirmed this to hold true for one half of
the full signal length, N/2 = 2 9 = 512 [6, 35]. As such, this constancy of
genuine spectral parameters persists at any signal length after retrieving the
true number of resonances. By contrast, spurious spectral parameters (not
tabulated) do not ever converge.
Taken together, the findings from Tables 6.2 and 6.3 demonstrate that the
FPT (−) continues to be stable by preserving 12digit accuracy after the point
at which full convergence is reached first at N P = 210 [6, 35]. In other words,
the addition of more signal points does not alter the stabilized results in any
way. Such an outstanding feature is of major importance with regard to the
stability of the FPT (−) in 12digit accurate quantification within MRS. This
is an extension of the previous conclusion on the stable reconstructions by the
FPT (−) reached in chapter 3 with 4digit accuracy, which is closer to the data
precision encountered with MRencoded in vivo FIDs.