Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 106

exp (±iω ± k τ) reconstructed with N/4 = 256 where ω k = 2πν k and ω ± k = 2πν ± k .
On panel (iii) the locations of the 1st and the 25th damped harmonics for lipid
and water are denoted by Lip and H 2 O, respectively. These represent the two
endpoints of the complex harmonic variable interval within which all the 25
studied resonances reside. To avoid clutter, numbers for the remaining 23
resonances on both sides of the circumference|z|= 1 are not written on panel
(iii). These numbers will be shown in Figs. 3.13 and 3.14 , whereas the corre
sponding acronyms for the metabolites shall be depicted in Fig. 3.10 . Panels
(iv) and (v) in Fig. 3.3 display the two Pade absorption total shape spectra
from the Heaviside partial fractions of the FPT (+) and FPT (−) , respectively,
computed using a quarter signal length (N/4 = 256). These results are iden
tical to those obtained with N/2 = 512 and N = 1024, as is anticipated from
the corresponding numerical values of the spectral parameters from panels
(iv) - (vi) in Table 3.2 .
Panel (vi) in Fig. 3.3 presents the Fourier absorption total shape spectrum
evaluated via the FFT using the full FID with N = 1024. A comparison of
panels (iv)-(vi) in Fig. 3.3 reveals that zerovalued spectra are obtained from
the difference between any two selected pairs of these spectra. These differ
ences are termed residual or error spectra and are explicitly shown in Figs.
3.6 and 3.7 . Clearly, when the FFT is used in such residuals, subtraction of
the Fourier from the Pade spectra is possible only if the FPT is also computed
at the Fourier grid points for chemical shifts.
The present finding concerning the error spectra sharpens the corresponding
conclusion to be presented in chapter 7 where one half of the entire FID was
needed for the FPT to obtain negligible residuals. Herein, the FPT used only
the first quarter of the full FID to achieve the resolution of the FFT (which
requires the entire time signal) and thereby yields zerovalued residual spectra.
It should nevertheless be reemphasized that in chapter 7, we used encoded
FIDs that, despite their excellent SNR, inherently contain noise, whereas the
present chapter is dealing with a theoretically constructed noiseless FID.
On all the panels from the right column in Fig. 3.3, the abscissae are the
chemical shifts, i.e., Re(ν k ) (in ppm), and the ordinates are the intensities
(in au) of the structures in the total shape spectra. The Fourier and the
Pade absorption total shape spectra are denoted by Re(F k ) with 0≤k≤
N−1 and Re(P ± K /Q ± K ), respectively. as the real parts of the corresponding
complexvalued spectra, F≡F k and P ± K /Q ± K ≡P ± K (z ±1 )/Q ± K (z ±1 ). The

Fourier spectrum F k can be obtained from (2.162), provided that G N (z −1 ) is
evaluated at the Fourier grid points, F k = G N (exp (−2iπk/N)).
As per the convention in NMR and MRS, chemical shifts in ppm are dis
played graphically in descending order when passing from left to right on the
abscissa. In the present case, this implies that the largest resonant frequency
(water at 4.68 ppm) for resonance k = 25 is situated on the extreme left corner
of the entire interval for chemical shifts on the abscissa. Such conventions,
notations, nomenclature and units will also apply to other figures throughout
this topic.