Digital Signal Processing Reference
In-Depth Information
and (vi) should be reviewed together, as they are complementary.
For this reason, we chose to use the descending order for the values of
Im(ν
k
) when proceeding from bottom to top of the ordinate axis. Thereby,
the ordinate is also inverted similarly to the abscissa on panels (iii) and (vi)
for the configuration of the poles of complex frequencies according to the kind
of resonances displayed on panel (v).
This is especially elucidating considering that panel (v) is placed immedi
ately above panel (vi) on the same Fig. 3.13. Here, as seen panel (v), most
resonances, e.g., k = 5−24 are rather narrow as implied by the relatively
small values of Im(ν
k
). Thus, these imaginary frequencies are quite close to
the real axis. As a result, these resonances are seen on panel (vi) in a group
in the middle part of this subplot. In contrast, panel (v) in Fig. 3.13 shows
wider resonances, e.g., k = 1−4 and k = 25 with larger values of Im(ν
k
).
Thus, such imaginary frequencies are deeper in the complex plane and these
resonances are quite distant from the real axis, as observed on the far left and
the far right parts of panel (vi).
Besides panel (vi) in Fig. 3.13, graphic presentations of the reconstructed
and the input data for the spectral parameters are also presented on panels
(i) - (iii). Panel (i) in Fig. 3.13 depicts the distribution of the absolute
values of the amplitudes at different chemical shifts. It follows from panel (i)
that the quantities|d
k
|do not represent the heights of the absorption peaks
from panels (iv) and (v). Instead, the absorption peak heights are directly
proportional to the quotient|d
k
|/Im(ν
k
), as per (2.186). Thus in Fig. 3.13,
panel (ii) displays the distribution of the quotients of the absolute values of
the amplitudes and the imaginary frequencies. It can be observed from panel
(ii) that all the 25 ratios|d
k
|/Im(ν
k
) are, in fact, proportional to the heights
h
k
of the corresponding peaks in the absorption component shape spectra
from panel (v) in Fig. 3.13.
Panel (iii) from Fig. 3.13 shows, in the complex z
+
−plane, the distribution
of the Pade poles using the complex harmonic variable z
k
. This is the zoomed
is in displaying only the first quadrant in Fig. 3.13, since the rest of the
complex z
+
−plane does not contain any genuine resonance. Moreover, panel
(iii) from Figs. 3.3 and 3.13 differ in arrangements of the values of Re(z
k
)
and Im(z
k
).
Unlike chemical shifts, there is no special reason in Fig. 3.3 or 3.10 to
abide by the universal convention of the ascending order of the values for
Re(z
k
) when proceeding from left to right on the abscissa, and similarly for
the ascending order of the values of Im(z
k
) when passing from bottom to top
on the ordinate. As such, this usual practice is seen on panel (iii) in Fig. 3.3
as well as on panels (i) and (iv) in Fig. 3.10. However, as to panel (iii) in
Fig. 3.13, both axes Re(z
k
) and Im(z
k
) are reversed. This implies that the
values on the abscissa and ordinate on panel (iii) in Fig. 3.13 are arranged
in descending order when going from left to right on the abscissa and from
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