Digital Signal Processing Reference
In-Depth Information
Fig. 3.7
Convergence of the
objective functions to the
Pareto-optimal values in
Scenario II
sideration. If
o
j
(k
2
,
3
,...,
100, denote two
vectors of objective-functions respectively computed at the
(k
−
1
)
and
o
j
(k)
,for
j
=
1
,
2
,
3
,k
=
1
)
th and
k
th gen-
erations over the entire population, then their relative changes were calculated as
−
. It is quiet evident from these plots that the Pareto-
optimal solutions were reached very quickly even within the tenth generation, par-
ticularly for the first two objective functions (
3.21
) and (
3.31
).
o
j
(k)
−
o
j
(k
−
1
)
/
o
j
(k)
3.5.2 Improvement in Detection and Estimation Performance
We demonstrate the performance improvement due to the adaptive waveform de-
sign at several SNR values in terms of the squared Mahalanobis distance, weighted
trace of CRB matrix, and squared upper bound on sparse-error. These results are
shown in Figs.
3.8
and
3.9
for the target Scenarios I and II, respectively. As we
expect, the Mahalanobis-distance measure improved as we increased the SNR val-
ues, but the trace of CRB matrix decreased and the upper bound on the sparse-
estimation error remained unchanged. In each figure, the red-colored lines (in to-
tal 1000 of them) represent the variations of the objective-functions, associated
with the entire population of 1000 solutions; whereas the blue-colored line sh
o
ws
their counterparts corresponding to the fixed (nonadaptive) waveform
a
1
/
√
L
=
=
T
. In both the target scenarios, we found that all the Pareto-
optimal solutions produced better performances, in terms of the Mahalanobis dis-
tance and trace of CRB matrix, when compared to those with the fixed waveform.
However, with respect to the squared upper bound on sparse-error, only a subset
of the Pareto-optimal solutions was found to show improved performance than that
with the fixed waveform.
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