Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 110

are given by the FFT for the exact absorption total shape spectrum, (panel
(iii) in Fig. 3.5 ) . This compares poorly with the FPT (−) as seen on panel
(iv) in Fig. 3.5, where full convergence to the exact input result is attained
by using only a quarter N/4 = 256 of the full signal length N.
Within the FPT (−) the same conclusion also holds for one half N/2 and
the full length N of the input FID as seen on panels (v) and (vi). Taken
together, Figs. 3.4 a nd 3.5 provide a powerful demonstration that the FPT (−)
outperforms the FFT with respect to the convergence rate for varying partial
signal lengths N/M (M = 2−32). This can be seen by referring to panels
(i) and (ii) in Fig. 3.3 . There it can be observed that the examined FID
becomes negligible after exhausting the first N/4 = 256 data points, i.e.,
after T/4 = 256 ms. Therefore, an estimator with a resolution which is not
restricted by T = Nτ can be expected to yield accurate reconstructions by
utilizing only the first quarter of the FID. It is this expectation which is
fulfilled by the FPT (−) , as shown in Fig. 3.5. In contrast, the resolving power
2π/T in the FFT is prefixed only by T. Thus, the entire FID is necessary to
obtain a converged total shape spectrum.
The above conclusion regarding the FPT (−) also applies to the FPT (+) . As
shown in Ref. [19], at higher partial signal lengths, absorption total shape
spectra generated by the FPT (+) are either very close or identical to the cor
responding results of the FPT (−) from Figs. 3.4 and 3.5. This crossvalidation
between the two variants of the Padebased reconstructions enhances the over
all fidelity in the FPT.

3.3 Residual spectra and consecutive difference spectra

3.3.1 Residual or error absorption total shape spectra

Figure 3.6 displays the residual or error absorption total shape spectra in
the FPT (−) . These spectra are obtained by subtracting the absorption to
tal shape spectra for the full signal length from those for the partial lengths,
Re(P − K /Q − K )[N]−Re(P − K /Q − K )[N/M]. The label [L] indicates that altogether,
L signal points are used, where L is either N or N/M, with N being the full sig
nal length (N = 1024), and M is the truncation number (M = 2, 4, 8, 16, 32, 64).
Here, the number N/M < N represents the partial signal length, N P .
The stable and rapid convergence within the FPT (−) that was observed
previously in Figs. 3.4 and 3.5, is further reflected in the residual spec
tra from Fig. 3.6 as a systematic decline in the global error. Even at a
quarter N/4 = 256 of the full signal length, the error spectra become zero
throughout the entire frequency range. This continues to be the case for
N/M > 256. Thus, the partial length N/4 could justifiably be used rather
than N for computing these error spectra. Namely, the new differences