Pad´e processing for MR spectra from in vivo time signals - Signal Processing in MRS with Biomedical Applications - page 241

rudimentary.
In Figs. 7.6 and 7.10 , we extend comparisons of the convergence patterns
of the FFT and the FPT (−) to M = 1−4. As stated earlier, full convergence
of the FPT (−) is obtained at half signal length (N/2 = 1024). This can also
be observed by comparing the spectra at half and full signal length of the
FPT (−) (the middle and bottom right panels in Figs. 7.6 and 7.10).
These two spectra differ only by the background noise and, therefore, the
total shape spectrum in the FPT (−) at half signal length can be considered
as fully converged. Notably, even the triplet of glutamineglutamate near
2.5 ppm is completely resolved by the FPT (−) at N/2. To achieve the same
resolution for this triplet, the FFT requires the full signal length N.
Overall, due to rapid convergence, the FPT (−) can extract vital informa
tion about shortlived metabolites that are conventionally detectable only at
very short echo times. The obtained unprecedented, steady and fast conver
gence of the FPT (−) sharply contrasts with most nonlinear estimators that
wildly oscillate before eventually stabilizing and only then perhaps converg
ing. Moreover, the steadiness and convergence rate are markedly better in
the FPT (−)

than in the FFT, as seen on Figs. 7.5 , 7.6, 7.9 a nd 7.10.

7.4 Error Analysis for encoded in vivo time signals

No parametric method is of much practical value without error analysis. When
the fully converged FFT does not qualify to be considered as a gold standard,
as in most clinical MRS data at 1.5T, we resort to intrinsic testing within
the two variants of the FPT and carry out the error analysis by computing
the residual or error spectra. These are defined as the difference between the
spectra in the FPT (+) and FPT (−) at a given signal length N. Such spectra
are used to help answer the question: where does one stop in the convergence
process within the fast Pade transform?
The first part of the answer is: one stops when the residual spectra as a func
tion of the signal length become indistinguishable from the background noise,
e.g., the RMS. The second part of the answer is: one stops when constancy is
reached in the values of all four spectral parameters (position, height, width
and phase) of each genuine, physical resonance for varying signal length.
Hence, these two components of the present strategy applied while varying
the signal length, i.e., (i) indistinguishability of the residual spectra from
the background noise, and (ii) stabilization of spectral parameters, constitute
error analysis of proven validity [5, 20].
Here, we shall begin our error analysis by using the FFT as a gold standard
for these spectra from the FIDs encoded at high magnetic field strengths.