# Joint Design of Transmit Waveforms for Object Tracking in Coexisting Multimodal Sensing Systems
## 5 Simulations Results
### Approach II Simulation  
In Approach I, it was demonstrated that the interference between radar and communications systems is reduced at the cost of decreasing radar system performance in terms of MSE. This approach aims to use Pareto optimization to obtain parameters that are jointly optimal for both the communications and radar system waveform parameters. The feasible parameter vectors that correspond to the objective functions of time-delay estimation MSE and interference between the two systems appear as two-dimensional points in Figure 10. The points were obtained using the conditions in Equations (6) and (31), and the same system parameters were utilized as in Approach I-B. The optimal communications FM rates in \(\Psi_{\rm c}\) were obtained by employing simulated annealing as discussed in Section 4.1 and satisfy Equations (6) and (27). For the radar parameter conditions in Equation (31), the waveform bandwidth is constrained between 3 and 10 MHz, and the radar pulse duration is between 1 to 4 \(\upmu\)s. The Pareto frontier (in green) in Figure 10 connects the efficient solutions of the Pareto maximization. Efficient solutions are those that are not dominated by other possible outcomes in the optimization problem.  
The efficient points on the Pareto frontier obtained from the simulated feasible vectors are labeled as A, B, C, D and shown in maroon in Figure 10. These points correspond to OLFM waveforms whose radar parameters \((\bar{b}_{\rm r},\bar{T}_{\rm r})\) are optimal. The coordinates of these points provide the time-delay estimation MSE and the system correlation. The minimum correlation is achieved by point A, and the miminum time delay estimate MSE is achieved by point B. The coordinates of points C and D have time-delay estimate MSEs that are close in value. We simulated both systems using the OLFM waveform radar parameters that resulted from the four Pareto efficient points. The four communications users' BER performance is shown in Figure 11a-d for each Pareto design. The figure shows that Design B yields the best BER performance for all four users; this is because point A was used in Figure 10 that corresponds to the minimum correlation value. Figure 12 demonstrates the estimation MSE for both time delay and Doppler. This figure shows that Pareto Design A provides the minimum time delay estimation MSE at high SINR. This follows because the point used from Figure 10 is point B, the point of minimum MSE. The tradeoff between joint system performance and design selection can be seen by the fact that the design that provides the best BER performance (Design B) does not provide the best MSE estimation performance; the MSE performance of Design B is slightly worse than for the other designs at high SINR. Note that the Pareto optimal points that lie on the frontier are only optimal in the sense that they provide the best tradeoff performance when used to design the transmit system waveforms.  
Figure 9: (**a**) BER and (**b**) time-delay estimation mean-squared error (MSE) performance (**red**) for coexisting systems using communications parameters \(\Psi_{\rm c}\) in Equation (27) and radar parameters \(\Psi_{\rm r}\) in Equation (32). In (**b**), also shown (**green**) is the MSE performance obtained using a radar waveform that minimizes the MSE.  
The selected waveforms are, however, suboptimal, as they do not simultaneously result in both minimum system correlation and minimum radar MSE estimation performance.  
Figure 11: Four communications users BER performance using the Pareto designs: (**a**) A; (**b**) B; (**c**) C; and (**d**) D.  
Figure 10: The possible solutions obtained by minimizing both the time-delay estimation MSE and the system correlation are shown in blue; the efficient Pareto solutions (shown in maroon) are connected (using green) with the Pareto frontier.