# Joint Design of Transmit Waveforms for Object Tracking in Coexisting Multimodal Sensing Systems
## 4 Waveform-Dependent Performance Optimization Methods
### MAI Mitigation in MU Communications Systems  
Following the proposed CoWS design, the \(m\)th communications user is assigned the OLFM waveform \(s_{m}(t)=s_{\mathrm{c,}m}(t)\) in Equation (4) with duration \(T=T_{\mathrm{c}}\) and a unique FM rate \(b_{m}\) that must satisfy:  
\[b_{m}=\frac{B}{2T_{\mathrm{c}}}-\left(\frac{M-m}{T_{\mathrm{c}}^{2}}\right)\,\quad m=1,\ldots,M\,, \tag{25}\]  
for minimum MAI, following Equation (6) [21; 40], for a given bandwidth \(B\). The maximum number of users transmitting at the same time with an assigned unique FM rate is \(M=\lfloor T_{\mathrm{c}}B/2\rfloor\) (see Section 2). This number increases with the time-bandwidth product \(T_{\mathrm{c}}B\), as shown in Figure 1a. Figure 1b also demonstrates that the number increases with both the pulse duration and the allocated bandwidth. The gross bit rate \(R\) affects the user's BER performance since, for a fixed PSK modulation order \(P\), the symbol duration is affected following the relation:  
\[R=\frac{1}{T_{\mathrm{c}}}\log_{2}(P)\,. \tag{26}\]  
This is demonstrated for an OLFM waveform in Equations (4) and (20) in Figure 2a for varying \(P\) and symbol duration \(T_{\mathrm{c}}\). As can be seen, for a given modulation order, the gross bit rate increases as the duration decreases. However, for a fixed bandwidth, when the duration decreases, the maximum number of users that can transmit over the communication channel also decreases. The BER performance for a varying signal-to-noise ratio (SNR) is shown in Figure 2b for an MU AWGN and Rayleigh fading channels. Three users are transmitting using the designed CoWS with fixed bandwidth, fixed duration, 16-PSK, and FM rates as in Equation (25). Note that for the AWGN channel, the estimated symbols needed for computing BER can be obtained as in Equations (22) and (24) but without the presence of the radar interference. The SNR is computed as \(E_{b}/N_{0}\), where \(E_{b}\) is the energy per bit and \(N_{0}\) is the variance of the AWGN samples. Note that the BER is computed theoretically and also obtained using Monte Carlo simulations in Figure 2a,b. For our proposed coexisting systems, we assumed that the MU communications system has two main objectives. The first is to maintain a desirable level of gross bit rate \(R\) for a given multiple PSK modulation order \(P\) in Equation (26); this constrains the symbol duration \(T_{\mathrm{c}}\). The second one is a minimum level of MAI; for a given bandwidth \(B\) and using OLFM waveforms, this constrains the selection of the FM rates to satisfy Equation (25). Thus, the optimal communications parameter set for a desirable level of gross bit rate and minimum MAI results in the OLFM waveform parameter set \(\Psi_{\mathrm{c}}\) given by Equation (7). Figure 2: (a) Feasible gross bit rate \(R\) as a function of symbol duration \(T_{\mathrm{c}}=T\) for phase shift keying (PSK) modulation orders \(P=2,4,8,16,32\). (b) Bit-error rate (BER) performance for varying signal-to-noise ratio (SNR) for three users transmitting over an additive white Gaussian noise (AWGN) channel and a Rayleigh fading channel using the designed OLFM waveform with 16-PSK modulation. If we assume that the actual number of users that are transmitting is \(N_{\rm c}\leq M\), then the FM rate assignment in Equation (25) can be revised to further minimize MAI. In particular, assuming that the symbol transmission between users is synchronized in time, we can select the \(N_{\rm c}\) FM rate values that correspond to the OLFM waveforms that yield the minimum correlation. In particular, the optimal communications parameter set is:  
\[\Psi_{\rm c}=\left\{T_{\rm c},\ \hat{b}_{1},\ \ldots,\ \hat{b}_{N_{\rm c}}\ \right\}= \arg\min_{ \begin{subarray}{c}N_{\rm c}\ {\rm values}\\ b\in\Theta_{\rm c}\end{subarray}