# Joint Design of Transmit Waveforms for Object Tracking in Coexisting Multimodal Sensing Systems
## 4 Waveform-Dependent Performance Optimization Methods
### Approach II: Coexistence Waveform Design Approach by Multiobjective Optimization  
In Approach I, the radar transmit waveform was designed only based on minimizing the interference between the coexisting systems. The design did not incorporate the minimization of the MSE for estimating range and velocity of a target, even though this is a very important performance metric for radar systems. Whereas minimizing the interference is a joint system performance metric, the MSE is only a radar specific performance metric. However, both of these metrics affect the transmit radar ULFM waveform parameter set \(\Psi_{\rm r}\). Although not necessarily conflicting objectives, the two performance metrics may need to be traded off, depending on the expected system outcome. For example, the bandwidth of the radar waveform can be chosen to be large for a higher range resolution, as then the MSE of the time delay estimate is decreased. However, it is not known how increasing the radar waveform bandwidth can affect the interference between the two systems. The coexisting waveform design optimization is thus formulated as a multiobjective or Pareto optimization in order to obtain a set of acceptable tradeoff optimal solutions [42; 43]. Figure 7 provides an overview of this joint waveform system design. In particular, we considered an optimization problem over a waveform parameter vector \(\mathbf{g}\) with \(L\geq 2\) objective functions denoted by \(\zeta_{l}(\mathbf{g})\), \(l\!=\!1,\ldots L\). We also considered a set of waveforms \(\mathbf{G}\) whose parameter vectors \(\mathbf{g}\) can be used by both systems. The multiobjective optimization can then be written as [42]:  
\[\tilde{\mathbf{g}}=\min_{\mathbf{g}\in\mathbf{G}}\ \ [\zeta_{1}(\mathbf{g})\ \ \ldots\ \ \zeta_{L}(\mathbf{g})]^{\rm T}\.\]  
Figure 6: Block diagram overview of the coexistence waveform design Approach I-B: The radar receiver has knowledge of the OLFM waveform parameters, \(\{T_{\rm c},\ b_{1},\ldots,b_{M}\}\), used by the \(M\) communications users. For the coexistence radar and communications problem, the waveform parameter vector is given by \(\mathbf{g}=[\Psi_{\mathrm{c}}\ b_{\mathrm{r}}\ T_{\mathrm{r}}]^{\mathrm{T}}\). It consists of the radar OLFM waveform parameters \(b_{\mathrm{r}}\) and \(T_{\mathrm{r}}\), constrained using Equation (31), and the communications user OLFM waveform parameters \(\Psi_{\mathrm{c}}=\left\{\hat{T}_{\mathrm{c}},\hat{b}_{1},\ldots,\hat{b}_{N_{ \mathrm{c}}}\right\}\) in Equation (27). Thus, for the communication system, the OLFM waveform parameters are already optimized to reduce MAI between users using Equation (6) for \(N_{\mathrm{c}}\leq M\). The overall waveform set \(\mathbf{G}\) thus consists of OLFM waveforms that satisfy Equations (6) and (31) for communications and radar, respectively. Although different objective functions for each system can be selected, we concentrated on using the radar time-delay estimation MSE and the system coexistence interference. In particular, we selected the radar time-delay estimation MSE, denoted by \(\zeta_{1}([\Psi_{\mathrm{c}}\ b_{\mathrm{r}}\ T_{\mathrm{r}}]^{\mathrm{T}})\), and the interference between the radar and communications waveforms, denoted by \(\zeta_{2}([\Psi_{\mathrm{c}}\ b_{\mathrm{r}}\ T_{\mathrm{r}}]^{\mathrm{T}})\)). The computation of the joint optimization approach was performed by sampling the boundaries associated with the feasible regions of the parameters of the radar waveform and the communications users waveforms. The complexity of computing the correlation between system waveforms is largely dominated by a dense matrix product whose order of magnitude is \(O(N_{\mathrm{r}}N_{\mathrm{r}}^{2})\); here, \(N_{\mathrm{r}}\) and \(N_{\mathrm{r}}\) are the total number of range and Doppler bins, respectively, used for processing at the radar receiver in Section 3.2.2.