Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 107

3.2.2 Pade and Fourier convergence rates of absorption total

shape spectra

Figures 3.4 and 3.5 compare the convergence rates of the absorption total
shape spectra from the FFT and FPT (−) using the FID of the varying length
N/M where M = 1−32 (N = 1024). We first analyze Fig. 3.4 which is
concerned with 3 partial signal lengths N/32 = 32,N/16 = 64 and N/8 = 128.
The left and the right columns of this figure are for the Fourier and the Pade
spectra via panels (i)-(iii) and (iv)-(vi), respectively. In examining the Fourier
panels (i) and (ii) together with the Pade panels (iv) and (v) in Fig. 3.4,
particularly striking differences are noted between the FFT and FPT (−) at
the two shortest investigated signal lengths N/32 = 32 and N/16 = 64. The
FFT does not provide any spectroscopic information whatsoever on panels (i)
and (ii). These two Fourier spectra show merely wide, uninterpretable bumps
near 2 ppm and 3 ppm. In sharp contrast, the FPT (−) on panel (iv) with
only N/32 = 32 shows several physical resonances, such as the lipid complex
between 1 ppm and 2 ppm, NAA near 2 ppm, Cr around 3 ppm as well as
close to 4 ppm, PCho in the vicinity of 4.3 ppm and H 2 O near 4.7 ppm. Of
particular note is that with only 32 signal points out of 1024, these Pade
reconstructed resonance profiles on panel (iv) in Fig. 3.4 are actually quite
reasonable. Admittedly, some of these lineshapes in the FPT are rather
broad. This is the case because 32 signal points do not provide su cient
accuracy in the estimates for the exact values of the spectral parameters.
However, by increasing the input information with N/16 = 64 signal points,
the performance of the FPT (−) is improved, as seen on panel (v) in Fig. 3.4.
Thus, higher accuracy is achieved in reconstructing the spectral parameters by
the FPT at N/16 = 64 than at N/32 = 32, as is also clear from the associated
numerical values listed on panel (ii) in Table 3.2 . Furthermore, panel (v)
on Fig. 3.4 displays four more resonances that are wellreconstructed by the
FPT (−) . These are the peaks corresponding to NAA near 2.7 ppm, Cho close
to 3.2 ppm, mIns around 3.6 ppm and Glu near 3.8 ppm. Especially in
comparison to the spectra from the FFT on panels (i) and (ii) in Fig. 3.4,
the results of the FPT (−) from panels (iv) and (v), albeit not fully complete,
display the evidently superior convergence rate in the Pade analysis relative to
the FFT. This trend is further displayed on panels (iii) and (vi) at N/8 = 128
in Fig. 3.4 for the FFT and FPT (−) , respectively.
Figure 3.5 continues with comparisons of the total shape spectra from Fig.
3.4 using a larger number of signal points: N/4 = 256,N/2 = 512,N = 1024.
These produce the results presented on panels (i), (ii), (iii) for the FFT and
(iv), (v), (vi) for the FPT (−) . Using a quarter (N/4 = 256) of the full FID,
the FFT on panel (i) on Fig. 3.5 has still not generated adequate estimates
for NAA, Cr and Cho that are three diagnostically informative metabolites.
This is still the case even with one half (N/2 = 512) of the full FID as seen
on panel (ii) in Fig. 3.5, since the FFT underestimates the actual height of
NAA. It is only at the full signal length (N = 1024) that the correct estimates