Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 126

or on panels (i) and (iv) in Fig. 3.10 . Furthermore, on panel (iii) in Fig. 3.14
all the resonances reconstructed by means of the FPT (−) are observed to lie

outside the unit circle (|z|> 1), as expected.
A careful inspection reveals that the k th heights h ± k

|/Im(ν ± k ) shown
on panels (ii) in Figs. 3.13 and Fig. 3.14 do not fully match the correspond
ing tops of the k th peaks in the component shape spectra d ± k z ±1 /(z−z ± k )
from panel (v). This is explained by the fact that the heights h ± k from their
definition in (2.186) are due to the lineshapes d ± k /(ω−ω ± k ) rather than to
the plotted spectra d ± k z ±1 /(z−z ± k ). The former and the latter lineshapes
are given in terms of the angular frequencies{ω,ω ± k

≡|d ± k

}and harmonic variables

{z ±1 ,z ±1
k

}, respectively, where z ±1 = exp (±iωτ) and z ±1
k

= exp (±iω k τ).

Convergence of fundamental frequencies in FPT (−)

3.5.3

Figure 3.15 shows how the complex fundamental frequencies retrieved by the
FPT (−) are configured in the complex frequency plane when utilizing six par
tial signal lengths (N/32 = 32,N/16 = 64,N/8 = 128,N/4 = 256,N/2 =
512) and the full FID (N = 1024). Consequently, panels (i), (ii) and (iii) show
the frequency distribution of the 10, 14 and 20 resonances that are retrieved
with N/32 = 32,N/16 = 64 and N/8 = 128, respectively.
It should be emphasized that the ordinate axis on panel (i) is enlarged
with respect to all other panels on Fig. 3.15. This adjustment is due to
a larger scatter of the two retrieved imaginary frequencies relative to the
corresponding exact values. This scatter surpasses the window 0 - 0.275 ppm
which is common to the remaining five ordinate axes on panels (ii) - (vi) in
Fig. 3.15. Panels (iv), (v) and (vi) in Fig. 3.15 show the complex frequencies
retrieved with N/4 = 256,N/2 = 512 and N = 1024, respectively. Note that
on panel (iv) for N/4 = 256, the exact frequencies (x's) and the associated
Padereconstructed values (circles) are in complete agreement with each other.
Furthermore, the plot on panel (iv) coincides with the plots on panels (v) and
(vi) for N/2 = 512 and N = 1024, respectively.
This graphic illustration of the full agreement between the complete set of
exact{ν k

}and reconstructed{ν − k

}frequencies, where the latter quantities
are computed by the FPT (−) with the signal lengths N/4 = 256,N/2 = 512
and N = 1024, is anticipated from panels (iv) - (vi) in Table 3.2 .

3.5.4 Distributions of fundamental frequencies in FPT (±) near
full convergence
Figure 3.16 displays the distributions of the complex fundamental frequencies
reconstructed by the FPT (+) (left column) and FPT (−) (right column) near
full convergence at 3 partial signal lengths N P = 180, 220, 260. On panels (i)
and (iv) at N P = 180, the frequency for peak k = 11 is missing from the
reconstructed parameters in the FPT (+) and FPT (−) . However, at N P = 220