Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 131

Thus, in the cases of panels (iv) - (vi) in Fig. 3.17 , it follows that|d − k |= d − k

as in the corresponding exact values for|d k |from Table 3.1 , where φ k = 0,

|= d k (1≤k≤K). In the FPT (−) , any inaccuracies in the inter
mediate stage of computations should disappear for all the 25 reconstructed
amplitudes once convergence has occurred. This expectation, based on the
oretical grounds, is actually fulfilled already at N/M≥N/4 = 256 as seen
on panels (iv), (v) and (vi) on Fig. 3.17 where a quarter, one half and full
signal length is used, respectively. This graphic visualization stems from the
exact agreement between all the absolute values of the amplitudes in the input
data (Table 3.1) and the corresponding values obtained by the FPT (−)

so that|d k

with

N/4 = 256,N/2 = 512 and N = 1024 ( Table 3.2 ) .

Distribution of fundamental amplitudes in FPT (±) near
full convergence

3.5.6

Figure 3.18 shows the absolute values of the amplitudes reconstructed by
the FPT (+) (left column) and FPT (−) (right column) near full convergence
at three partial signal lengths N P = 180, 220, 260. In the FPT (+) complete
convergence to all the exact digits is attained at N P = 260 in panel (iii). In
contrast, the FPT (−) fully converges at N P = 220 and N P = 260 in panels
(v) and (vi). Our computations show that stable estimation of the absolute
values of the amplitudes in the FPT (±) continues at N P > 260.
We emphasize that the most interesting features are actually shown prior
to full convergence in the FPT (−) . This can be seen on, e.g., panel (iv) from
Figs. 3.16 and 3.18, particularly by referring to panels (i) and (iv) in Fig.
3.11 for the component and total shape spectra. It should be recalled that
Fig. 3.11 shows that the converged total shape spectrum from panel (iv) is
obtained without achieving convergence of the component spectra. This was
due to a compensation effect through which the 11th peak was absent, and
the 12th peak was overestimated.
Panel (iv) in Figs. 3.16 and Fig. 3.18 provides the explanation for this
compensation. Peak k = 12 on panels (iv) in Fig. 3.16 has a smaller imaginary
frequency than the corresponding exact value, Im(ν − 12 ) < Im(ν 12 ). In contrast,
panel (iv) in Fig. 3.18 it is seen that|d − 12 |>|d 12 |so that |d − 12 |/Im(ν − 12 ) >

}and{ν − k ,d − k |d 12 }are the pairs of the input and

FPT (−) −reconstructed parameters, respectively.
According to (2.186), the ratio|d − k

|/Im(ν 12 ). Here,{ν k ,d k

|/Im(ν − k ) is proportional to the height

h − k |/Im(2πν − k ) and h k = |d k |/Im(2πν k ).

The height h k itself is proportional to the k th peak area which is, in turn,
proportional to the concentration of the k th resonance. Consequently, the
overestimation of peak k = 12 stems from the observed relation h − 12 > h 12 .
Moreover, a review of the numerical values of these spectral parameters from
Table 3.3 demonstrates that the area of the 12th peak is overestimated by the
corresponding amount of the absent 11th peak. Thus, this compensation effect

of the k th peak, i.e., h − k = |d − k