**4. Simulation Results**

In this section, some numerical results are provided to illustrate the performance of the proposed LPI-based joint dwell time and bandwidth optimization strategy for multi-target tracking in a radar network. A multi-target tracking scenario with six radars and two targets is considered. In order to simplify the problem, we assume that all the radars in the network systems have the same system parameters. Then we can utilize the default values for the system parameters, as given in Table 2.


**Table 2.** Radar network system parameters.

The velocities of target 1 and target 2 are: (1300, 530)m/s and (−1300,−530)m/s, respectively. It is also assumed that the tracking process lasts 150 s.

Figure 2 depicts the distribution of the radar network, the true trajectories of the two targets and the estimated trajectories of the targets according to the proposed strategy.

**Figure 2.** Target trajectory and radar network deployment.

This part first gives the simulation results under the non-undulating RCS model. Assuming that the reflection coefficients of all targets is 1 at any observation time, we define this situation as RCS case 1. In this case, the radar selection and dwell time allocation are only related to the distance and relative position of the target to the radar.

Figure 3 shows the radar selection and bandwidth allocation results of the two targets, while Figure 4 gives the dwell time allocation results. In each figure, on the left side is the radar index, on the right side is the different intensity of the bandwidth and dwell time, which is depicted in different colors. Moreover, the blue areas in each figure indicate that the radar selection variable *u q <sup>i</sup>*,*<sup>k</sup>* = 0, while the lines in different colors mean that *u q <sup>i</sup>*,*<sup>k</sup>* = 1, with different colors representing the intensity of the transmitted bandwidth and dwell time. We can conclude that the radar network tends to assign the two radars closest to a specified target for tracking tasks, and more dwell time and bandwidth resources will be allocated to the selected radar, which is farther from the target.

**Figure 3.** Radar selection and bandwidth allocation in radar cross section (RCS) case 1.

**Figure 4.** Radar selection and dwell time allocation in RCS case 1.

To show the superiority of the proposed joint dwell time and bandwidth optimization strategy, the optimization algorithm without considering the bandwidth allocation is compared to a benchmark. Figure 5 shows the comparison of total dwell time for two different algorithms. From the result we can see that the proposed strategy can reduce the total dwell time of the radar network compared with the benchmark.

**Figure 5.** Comparison of total dwell time for different algorithms in RCS case 1.

Define the root mean square error (RMSE) for the tracking accuracy of all targets at time index *k* as:

$$\text{RMSE}(k) = \sum\_{q=1}^{Q} \sqrt{\frac{1}{N\_{\text{MC}}} \sum\_{n=1}^{N\_{\text{MC}}} \left\{ \left[ \mathbf{x}\_{k}^{q} - \mathbf{x}\_{n,\text{k}\natural}^{q} \right]^{2} + \left[ y\_{k}^{q} - \mathbf{y}\_{n,\text{k}\natural}^{q} \right]^{2} \right\}} \tag{48}$$

where *<sup>N</sup>*MC <sup>=</sup> 100 represents the Monte Carlo experiment number, and , *x*ˆ *q n*,*k*|*k* , *y*ˆ *q n*,*k*|*k* is the location estimate at the *n*th trial.

The RMSE of the proposed strategy and the benchmark are evaluated in Figure 6, respectively. The results prove that the tracking accuracy has not been sacrificed too much after allocating the bandwidth, which is acceptable to our tracking tasks.

**Figure 6.** Root mean squared error (RMSE) in two algorithms for target tracking in RCS case 1.

In order to further analyze the impact of the target RCS on radar selection and radar resource allocation results, a second RCS model is also considered, which can be defined as RCS case 2, where it is depicted in Figure 7. In this case, the reflection coefficient of the two targets to radar 3 and radar 4 change with time, while the RCS of the two targets to the other radars remain unchanged at any observation time. In Figure 7, the red and black lines represent the RCS values of target 1 to radar 3 and target 2 to radar 4 at each moment, respectively, which fluctuate around 10.3 m2. Similarly, the green and blue lines represent the RCS values of target 1 to radar 4 and target 2 to radar 3, respectively, which fluctuate around 2.3 m2.

**Figure 7.** RCS case 2.

Figures 8 and 9 illustrate the optimization results of target 1 and target 2 with the proposed strategy in RCS case 2 at every time index, respectively.

**Figure 8.** Radar selection and bandwidth allocation in RCS case 2.

**Figure 9.** Radar selection and bandwidth allocation in RCS case 2.

Compared with Figures 3 and 4, we can draw the following conclusions. During the whole tracking process, the number of times that radar 3 irradiated target 1 and radar 4 irradiated target 2 increase significantly. In addition, during the period in which radar 2 and radar 3 irradiate target 1 together, radar 2, which is closer to the target, but has a lower reflection coefficient, is allocated more bandwidth and dwell time resources. Similarly, this phenomenon also exists in the resource allocation of target 2. In summary, it can be concluded that the reflection coefficient of the target also affects the radar selection and radar resource allocation results. The radar network system will preferentially select the radar with higher reflection coefficient to irradiate the target. Furthermore, the system tends to allocate more resources to the radar with lower reflection coefficient to the target.

Figures 10 and 11 show the performance comparison of the two algorithms in RCS case 2. Obviously, it is consistent with the conclusions of RCS case 1, thus verifying the stability of the proposed strategy.

**Figure 10.** Comparison of total dwell time for different algorithms in RCS case 2.

**Figure 11.** RMSE in two algorithms for target tracking in RCS case 2.

Define the target tracking average root mean square error (ARMSE) as:

$$\text{ARMSE}(k) = \sum\_{q=1}^{Q} \sqrt{\frac{1}{N\_{\text{MC}}} \sum\_{n=1}^{N\_{\text{MC}}} \frac{1}{N\_{k}^{q}(n)} \sum\_{k=1}^{N\_{k}^{q}(n)} \left\{ \left[ \mathbf{x}\_{k}^{q} - \mathbf{f}\_{n, \text{k}\natural}^{q} \right]^{2} + \left[ \mathbf{y}\_{k}^{q} - \mathbf{f}\_{n, \text{k}\natural}^{q} \right]^{2} \right\}} \tag{49}$$

where *N<sup>q</sup> k* (*n*) denotes the number of times that the radar network radiated *q*th target at time index *k*. Figure 12 shows the ARMSE comparison between the proposed strategy and the benchmark in the two RCS cases. With respect to the target tracking accuracy, the latter is slightly better than the former, but the gap is not large, and is within an acceptable range. In conclusion, the proposed strategy effectively improves the LPI performance of the radar network without sacrificing too much accuracy.

**Figure 12.** Average root mean square error (ARMSE) comparison of two algorithms for target tracking.
