Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 129

the absent 11th peak is recovered by both variants of the FPT, as observed
on panels (ii) and (v). Specifically, in the FPT (+) full convergence to all the
exact digits is achieved at N P = 260 in panel (iii). On the other hand, the
FPT (−) completely converges at N P = 220 and N P = 260 in panels (v) and
(vi). We have confirmed that stable estimation of the fundamental frequencies
in the FPT (±) continues at N P > 260.

Convergence of fundamental amplitudes in FPT (−)

3.5.5

Figure 3.17 displays the distributions of the absolute values of amplitudes (re
lated to the corresponding fundamental frequencies) that were reconstructed
with the FPT (−) . These distributions are computed as a function of chemi
cal shifts by using the same sequence of signal lengths as in Fig. 3.15 , i.e.,
N/32 = 32,N/16 = 64,N/8 = 128,N/4 = 256,N/2 = 512 and N = 1024.
On panels (i), (ii) and (iii) in Fig. 3.17 we show the absolute values of the
amplitude distributions of the 10, 14 and 20 resonances reconstructed for
N/32 = 32,N/16 = 64 and N/8 = 128, respectively. The ordinate axes on
the entire left column via panels (i) (iii) in Fig. 3.17 are enlarged compared
to panels (iv) (vi) on the same figure. A similar enlargement was needed
as well for Fig. 3.15 for the distribution of the retrieved complex frequencies,
but only on panel (i) for the shortest partial signal length N/32 = 32. We
then see that the reconstructed|d − k

|'s in Fig. 3.17 are more scattered from
the associated exact values than in the case of the corresponding complex
frequencies on the left column in Fig. 3.15.
Thus, prior to convergence, there appears to be a greater sensitivity of
the reconstructed absolute values of the amplitudes on panels (i) (iii) in
Fig. 3.17, relative to the corresponding complex frequencies from panels (i)
(iii) in Fig. 3.15, when both sets of spectral parameters are computed with
the same partial signal length in the interval N/M < N/4 = 256. This is
anticipated because the complex frequencies{ν − k

}extracted by the FPT (−)
are based only upon the roots of the denominator polynomial Q − K . However,
the absolute values of the amplitudes{|d − k |}rely upon both the numerator

P − K and denominator Q − K polynomials, as per (2.183).
Clearly, any extra numerical inaccuracies due to the additional parame
ters to be obtained for the expansion coe cients{p − r

}in P − K yield more pro
nounced departures from the exact results for the amplitudes than for frequen
cies, prior to reaching full convergence. Also, the reconstructed amplitudes
converge more slowly than the corresponding frequencies for the considered
FID, because it is di cult to recover exactly zero values for all the phases
{φ − k }to match the associated exact set{φ k }. This is reflected on panels (i)-

(iii) in Fig. 3.17, because here the nonconverged|d − k |'s contain an admixture

of Im(d − k ) = 0, i.e., φ − k |= d − k .

In contrast, on panels (iv) - (vi) in Fig. 3.17, all the 25 absolute values of
the amplitudes|d − k

= 0. Thus, in these cases we have|d − k

|show the feature Im(d − k ) = 0, or equivalently, φ − k

= 0.