Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 124

bottom to top on the ordinate, respectively. This twofold inversion on panel
(iii) in Fig. 3.13 is made to follow the arrangements of Re(ν k ) and Im(ν k )
that are both shown in descending order when proceeding from left to right on
the abscissa or from bottom to top on the ordinate on panel (vi) on the same
figure. This convenient layout reveals that all the Pade poles k = 1−25 are
aligned one after the other from right to left regarding the abscissa. The same
poles k = 1−25 are also packed together near the circumference (|z|= 1) of
the unit circle in such a way that they follow each other according to their
consecutive numbers from inside the unit circle, by being aligned upward with
respect to the ordinate.
Panel (iii) in Fig. 3.13 shows that the poles contained in the harmonic
variable z k are less scattered from each other relative to the associated dis
tributions of the complex frequencies from panel (vi) in the same figure. The
reason for this is in the exponential function of the complex frequency which
is plotted on panel (iii), whereas the frequency itself is shown on panel (vi)
in Fig. 3.13. It is seen on panel (iii) in Fig. 3.13 that all the genuine poles
retrieved by the FPT (+) are found inside the unit circle (|z|< 1), as expected.
Notice that narrow resonances k = 5−24 are shown to be near the circum
ference (|z|= 1) of the unit circle. On the other hand, the wide resonances
seen on panel (iii) in Fig. 3.13, such as k = 1−4 and k = 25, lie further from
the borderline|z|= 1.

Distributions of spectral parameters in FPT (−)

3.5.2

Figure 3.14 is, in many ways, similar to Fig. 3.13. The difference is that Fig.
3.14 displays the results of the FPT (−) . The interpretation of the results from
the FPT (+) as presented on panels (i), (ii) and (iv) - (vi) in Fig. 3.13 holds
as well with respect to the corresponding findings from the FPT (−)

shown on

panels (i), (ii) and (iv) - (vi) in Fig. 3.14.
This observation emerges from the fact that the FPT (+) and FPT (−) gen
erate indistinguishable spectral parameters for the same N/4 = 256 signal
points, as noted in subsubsection 3.1.3 . However, panel (iii) differs for Figs.
3.13 and 3.14, since the information presented here relates to the two comple
mentary regions of the initial convergence, inside and outside the unit circle,
for the FPT (+) and FPT (−) , respectively.
In order to match the configuration from panel (iii) in Fig. 3.13, the Pade
poles contained in the harmonic variable z − k , as displayed in the complex
z − −plane on panel (iii) in Fig. 3.14, are plotted with the values of Im(z − k )
in ascending order when going from bottom to top of the ordinate. This is
opposite to the ordering of Im(z k ) on panel (iii) in Fig. 3.13, as anticipated
because Im(z k ) > 0 and Im(z − k ) < 0.
Thus, in reconstructions by the FPT (+) and FPT (−) , the harmonic variables
z k and z − k are located in the first and the fourth quadrant of the complex
z + −and z − −planes, respectively. This is apparent on panel (iii) of Fig. 3.3