# Joint Design of Transmit Waveforms for Object Tracking in Coexisting Multimodal Sensing Systems
## 4 Waveform-Dependent Performance Optimization Methods
### Approach I: Coexistence Waveform Design Approach by Minimizing System Interference
#### 4.2.1 Approach I-A: Radar Has Knowledge of Symbol Duration of Communications Users  
We assumed that the only prior knowledge the radar has is the fixed symbol duration \(T_{\rm c}\) used by all the communications users. This is demonstrated with the block diagram in Figure 3. The radar system also assumes that all possible communications users are transmitting at any given time; this would correspond to the maximum possible communications interference. Based on this information, we first set the radar OLFM waveform duration equal to the symbol duration, set \(T_{\rm r}=T_{\rm c}\). Then, since both systems use OLFM waveforms, and the maximum frequency used in the system is \(B\), the radar can easily determine that the maximum number of possible communications users is \(M=\lfloor T_{\rm r}B/2\rfloor\). Use of the CoWS design by both systems and Equation (25) also provide the information that the users' FM rates are \(b_{1},\ldots,b_{M}\); also, by design, \(b_{1}<b_{2}<\ldots<b_{M}\). In order to design the FM rate of the radar OLFM waveform, we can consider increasing the range resolution for tracking the position of a target or minimizing the interference between the two systems. As the range resolution is given by:  
\[\sigma_{\rm r}=\frac{c_{0}}{4\,b_{\rm r}T_{\rm r}}\,, \tag{28}\]  
where \(c_{0}\) is the wave velocity for electromagnetic propagation in free space, then since \(T_{\rm r}=T_{\rm c}\) is fixed, the only way to maximize \(\sigma_{\rm r}\) is to maximize the FM rate \(b_{\rm r}\). The maximum possible value is \(b_{M}\), which is already taken by a communications user. However, we can select \(b_{\rm r}=-b_{M}\) to also improve the range resolution. As we show next, this also reduces the interference between the two systems. The correlation between a shifted radar LFM waveform with FM rate \(b_{\rm r}\) and the OLFM waveform used by the \(m\)th communication user is given by:  
\[\Gamma(b_{\rm r},b_{m},\tau)=\bigg{|}\int_{0}^{T_{\rm r}}s_{\rm r}(t-\tau;b_{ \rm r})\,s_{\rm c,m}^{*}(t;b_{m})\,dt\bigg{|}^{2}=\bigg{|}\int_{0}^{T_{\rm r}} 2\sqrt{t(t-\tau)}\,\,e^{j2\pi b_{\rm r}(t-\tau)^{2}}\,e^{-j2\pi b_{\rm r}t^{2} }\,dt\bigg{|}^{2}\,, \tag{29}\]  
where \(b_{m}\) is given in Equation (25), \(m=1,\ldots,M\), and \(T_{\rm r}=T_{\rm c}\). This correlation term must be minimized to reduce the interference between the two systems. As the minimization cannot be computed in closed form, we evaluated it numerically. Figure 4a shows the correlation as a function of the radar FM rate \(b_{\rm r}\) using \(B=10\) MHz, \(T_{\rm r}=4\)\(\mu\)s, \(\tau=0\), and the highest FM rate, \(b_{M}=B/(2T_{\rm r})=1.25\) GHz, that corresponds to user \(m=M\). As can be seen, the radar FM rate that minimizes the correlation corresponds to the negative of the highest FM rate [40]; that is, the optimal radar FM rate is \(b_{\rm r}=-b_{M}\). A similar result is obtained when the time-delay \(\tau\) is allowed to vary, as shown in Figure 4b. A depiction of the resulting coexisting scheme in the time-frequency plane is shown in Figure 5a. Figure 3: Block diagram overview of coexistence waveform design Approach I-A: The radar receiver has knowledge of the OLFM waveform duration, \(T_{c}\), that is common to all \(M\) communications users. 2.2 Approach I-B: Radar Has Knowledge of Communications User Parameter Set \(\mathbf{\Psi_{c}}\)  
If we assume that the radar system has knowledge of the optimal communications signal parameter set \(\Psi_{c}\) in Equation (27) for \(N_{c}{\leq}M\) active communications users, then the radar waveform design can be further improved. This is demonstrated with the block diagram in Figure 6. Using this knowledge, the radar OLFM waveform parameter set \(\mathbf{\Psi_{r}}=\{T_{r},\ b_{r}\}\) can be designed to minimize the communications interference \(\tilde{\mathbf{X}}_{c}(\Psi_{c},\Psi_{r})\) in Equation (19) over the radar CPI.