# Joint Design of Transmit Waveforms for Object Tracking in Coexisting Multimodal Sensing Systems
## 3 Spectrum Sharing Radar and Communications Systems
### Pulse-Doppler Radar Receiver
#### 3.2.2 Radar Receiver Processing  
The processing at the radar receiver requires the estimation of the target range and velocity at each time step. The overall received waveform at the receiver must be correlated with all possible time-delayed and frequency-shifted versions of the transmitted signal. Due to the pulse-Doppler processing, range is estimated from slow-time processing that involves the PRI time step \(k\), whereas velocity is estimated from fast-time processing that involves the time sample \(n\). The correlation matrix \(\mathbf{A}_{\tau}\)\(\in\)\(\mathbb{C}^{\text{K}\times N_{\tau}}\) over all possible \(N_{\tau}\) time delays (or range bins) is first formed as:  
\[\mathbf{A}_{\tau}=\mathbf{Z}_{\tau}^{\text{H}}\,\mathbb{S}_{\tau}\,,\]  
where \(\mathbf{Z}_{\tau}\) is the received signal matrix in Equation (12), \(\mathbb{S}_{\tau}\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times N_{\tau}}\) is \(\mathbb{S}_{\tau}\)\(=\)\([\mathbf{s}_{\tau}(\tau_{1};k)\ \ldots\ \mathbf{s}_{\tau}(\tau_{N_{\tau}};k)]\), \(\mathbf{s}_{\tau}(\tau_{\ell};k)\) is given in Equation (13), and \(\tau_{\ell}\), \(\ell\)\(=\)\(1,\ldots,N_{\tau}\) denotes the \(\ell\)th time-delay or range bin. The domain \([T_{\text{r}},T_{\text{PRI}}]\) of \(\tau_{\ell}\) represents the domain of unambiguous target returns, where \(T_{\text{r}}\) is the duration of the transmit radar signal \(s_{\tau}(t)\). The \((\ell,m)\)th element \(a_{\ell,k}\), \(\ell\)\(=\)\(1,\ldots,N_{\tau}\), \(k\)\(=\)\(1,\ldots,K\), of \(\mathbf{A}_{\tau}\) is given by:  
\[a_{\ell,k}=\sum_{n=1}^{N_{\text{s}}}z_{\tau,k}[n]\,s_{\tau}^{*}(nT_{s}-\tau_{ \ell}-kT_{\text{PRI}})=\mathbf{z}_{\tau,k}^{\text{H}}\,\mathbf{s}_{\tau}(\tau_{ \ell};k)\,. \tag{14}\]  
In Equation (14), the target range can be estimated when the time-shifted transmitted signal is correlated with the received signal at the \(k\)th PRI in Equation (10). In Equation (14), the vector \(\mathbf{z}_{\tau,k}\) can also be given by:  
\[\mathbf{z}_{\tau,k}=\sqrt{P_{\tau}}\,\mathbf{s}_{\tau}(\tau_{0};k)\,e^{-j2\pi\
u_ {0}\,k\,T_{\text{PRI}}}+\mathbf{x}_{\text{c},k}+\mathbf{w}_{k},\quad k=1, \ldots,K\,, \tag{15}\]  
where \(\mathbf{s}_{\tau}(\tau_{0};k)\) is defined in Equation (13). After \(KThe radar processed output, with the noise and communications interference component, is given by \(\mathbf{X}_{r,\
u,\tau}=\mathbf{F}_{\
u}\ \mathbf{D}(\
u_{0})\ \mathbf{S}_{\mathrm{r}}^{ \mathrm{H}}(\tau_{0})\ \bar{\mathbf{S}}_{\mathrm{r}}\) and has the form of the ambiguity function of the transmitted signal. This follows from its dependence on \(\mathbf{S}_{\mathrm{r}}(\tau_{0})=[\mathbf{s}_{\mathrm{r}}(\tau_{0};1)\dots \mathbf{s}_{\mathrm{r}}(\tau_{0};K)]\) and \(\bar{\mathbf{S}}_{\mathrm{r}}=[\mathbf{s}_{\mathrm{r}}(\tau_{1};k)\ \dots\ \mathbf{s}_{ \mathrm{r}}(\tau_{N_{r}};k)]\), where \(\mathbf{s}_{\mathrm{r}}(\tau_{0};k)\), \(\ell=0,1,\dots,N_{\tau}\), is defined in terms of the radar transmit waveform in Equation (13). When this transmit waveform is selected to follow the CoWS, the radar processed output directly depends on the radar parameter set \(\Psi_{\mathrm{r}}=\{T_{\mathrm{r}},\ b_{\mathrm{r}}\}\) in Equation (7), and it is given by:  
\[\mathbf{X}_{r,\
u,\tau}(\Psi_{\mathrm{r}})=\mathbf{F}_{\
u}\ \mathbf{D}(\
u_{0})\ \mathbf{S}_{\mathrm{r}}^{ \mathrm{H}}(\tau_{0};\Psi_{\mathrm{r}})\ \bar{\mathbf{S}}_{\mathrm{r}}(\Psi_{ \mathrm{r}}). \tag{18}\]  
From Equation (17), the interference from the communications system is given by:  
\[\tilde{\mathbf{X}}_{\mathrm{c}}(\Psi_{\mathrm{c}},\Psi_{\mathrm{r}})=\mathbf{ F}_{\
u}\ \mathbf{X}_{\mathrm{c}}^{\mathrm{H}}(\Psi_{\mathrm{c}})\ \bar{\mathbf{S}}_{\mathrm{r}}(\Psi_{ \mathrm{r}}). \tag{19}\]  
Note that it depends on both the radar parameter set, due to the term \(\bar{\mathbf{S}}_{\mathrm{r}}(\Psi_{\mathrm{r}})\) and the parameters \(\bar{\mathbf{V}}_{\mathrm{c}}\) in Equation (8) of all the users transmitting, due to the term \(\mathbf{X}_{\mathrm{c}}\) that depends on \(x_{\mathrm{c},k}[n]\) in Equation (9).