Digital Signal Processing Reference
In-Depth Information
•
Radar parameters
:
- Carrier frequency
f
c
=
1 GHz;
- Available bandwidth
B
10 MHz;
- Number of OFDM subcarriers
L
=
=
3;
- Subcarrier spacing of
Δf
=
B/(L
+
1
)
=
2
.
5MHz;
- Pulse width
T
P
=
1
/Δf
=
400 ns;
- Pulse repetition interval
T
=
4ms;
- Number of coherent pulses
N
20;
- All the transmit OFDM weights were equal, i.e.,
a
l
=
=
1
/
√
L
∀
l
.
•
Simulation parameters
:
To apply a sparse estimation, we partitioned the viable relative speeds from
24
.
5 to 25 m/s with steps of 0
.
05 m/s. We generated the noise samples from a
C
N
LN
(
0
,
I
N
⊗
.
Hence, for all the results presented in this section we ensured a constant noise-
power distribution among all the subchannels.
To solve the MOO problem (
3.40
), we employed the NSGA-II with the follow-
ing parameters: population size
)
distribution with
=[
1
,
0
.
1
,
0
.
01
;
0
.
1
,
1
,
0
.
1
;
0
.
01
,
0
.
1
,
1
]
=
1000, number of generations
=
100, crossover
=
=
probability
0
.
1. The initial population of 1000
different values of
a
were generated randomly, but ensuring that the total-energy
constraint
a
H
a
0
.
9, and mutation probability
1 was satisfied. Furthermore, at each generation of the NSGA-
II, we imposed the total-energy constraint on the children-chromosomes by intro-
ducing an 'if-statement' in the 'genetic-operator' portion of the NSGA-II code.
However, satisfying the hard-equality constraint
a
H
a
=
1 was difficult to simu-
late due to the numerical precision errors. That is why we relaxed it with a softer
constraint by considering 0
.
999
=
a
H
a
≤
≤
1
.
001.
3.5.1 Results of the MOO Problem
The results of the MOO problem are depicted in Figs.
3.4
,
3.5
and Figs.
3.6
,
3.7
for
the target Scenarios I and II, respectively. We maintained a fixed SNR of 0 dB for
these simulations. The initial population of 1000 different values of
a
were gener-
ated randomly. Considering a Cartesian coordinate system with
|
a
1
opt
|
,
|
a
2
opt
|
, and
|
as the axes, the initial population is represented on the surface of a sphere
restricted to the first octant, as shown by circles in Figs.
3.4
(a) and
3.6
(a) for the
two different target scenarios. The values of the associated objective functions are
also indicated by circles respectively in Figs.
3.4
(b) and
3.6
(b), whose coordinate
systems are constructed with the three objective functions representing the axes on
the logarithmic scales.
We represent the Pareto-optimal solutions by squares in Figs.
3.4
(a) and
3.6
(a)
and the associated Pareto-optimal objective values by squares in Figs.
3.4
(b)
and
3.6
(b), for the Scenarios I and II, respectively. In Scenario I, when the target
had the strongest reflectivity along the second subcarrier, we got the optimal so-
lutions varying from
a
3
opt
|
T
to
T
on an
|
a
opt
|=[
0
.
6189
,
0
.
6691
,
0
.
4119
]
|
a
opt
|=[
0
,
1
,
0
]
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