# Joint Design of Transmit Waveforms for Object Tracking in Coexisting Multimodal Sensing Systems
## 3 Spectrum Sharing Radar and Communications Systems
### Pulse-Doppler Radar Receiver  
#### 3.2.1 Radar Received Waveform  
The pulse-Doppler radar was assumed to transmit \(K\) pulses over the CPI. The received baseband signal from the \(k\)th transmitted pulse, \(k=1,\ldots,K\), using a sampling period \(T_{s}\) and a PRI \(T_{\rm PRI}\), is:  
\[z_{r,k}[n]=x_{r,k}[n]+x_{\rm c,k}[n]+w_{k}[n],\ \ \ \ n=1,\ldots,N_{s}\,, \tag{9}\]  
where \(x_{\rm c,k}[n]\) is the observed communications signal on the collocated receiver, \(w_{k}[n]\) is additive white Gaussian noise (AWGN), \(N_{\rm s}=\left\lfloor T_{\rm PRI}/T_{s}\right\rfloor\) is the number of samples, and the radar return is:  
\[x_{r,k}[n]=\sqrt{P_{\rm r}}\ s_{\rm r}(nT_{\rm s}-\tau_{0}-kT_{\rm PRI})\,e^{ -2\pi\tau_{0}\,k\,T_{\rm PRI}},\ \ \ \ n=1,\ldots,N_{s}\,. \tag{10}\]  
Here, \(P_{\rm r}\) is the power of the radar return signal and \(s_{\rm r}(t)\) is the radar transmitted waveform. Over the CPI, both the time shift \(\tau_{0}\) and the frequency shift \(\
u_{0}\) are assumed to be constant. Observing all \(K\) transmitted pulses over the CPI, the overall radar received waveform is given by:  
\[z_{\rm r,CPI}[n]=\sum_{k=1}^{K}z_{r,k}[n],\ \ \ n=1,\ldots,N_{s}\,. \tag{11}\]The overall waveform over the CPI in Equation (11) can also be expressed in vector form \(\mathbf{z}_{\tau,\text{CPI}}\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times 1}\) as \(\mathbf{z}_{\tau,\text{CPI}}\)\(=\)\(\text{vect}(\mathbf{Z}_{\tau})\), where \(\text{vect}(\mathbf{Z}_{\tau})\) denotes vectorization of matrix \(\mathbf{Z}_{\tau}\) by stacking the matrix columns into a single column vector. The matrix \(\mathbf{Z}_{\tau}\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times K}\) is given by:  
\[\mathbf{Z}_{\tau}=\left[\mathbf{z}_{\tau,1}\ \ \ldots\ \ \mathbf{z}_{\tau,K} \right]=\sqrt{P_{\tau}}\ \mathbf{S}_{\tau}(\tau_{0})\ \mathbf{D}^{\text{H}}(\
u_{0})+\mathbf{X}_{ \text{c}}+\mathbf{W}, \tag{12}\]  
where \(\mathbf{z}_{\tau,k}\)\(=\)\([z_{\tau,k}[1]\ldots z_{\tau,k}[N_{s}]]^{\mathsf{T}}\), \(\mathbf{z}_{\tau,k}\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times 1}\), \(\mathsf{T}\) and \(\mathsf{H}\) denote vector transpose and Hermitian transpose, respectively, \(\mathbf{S}_{\tau}(\tau_{0})\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times K}\) is \(\mathbf{S}_{\tau}(\tau_{0})\)\(=\)\([\mathbf{s}_{\tau}(\tau_{0};1)\ldots\mathbf{s}_{\tau}(\tau_{0};K)]\), and \(\mathbf{s}_{\tau}(\tau_{0};k)\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times 1}\) is:  
\[\mathbf{s}_{\tau}(\tau_{0};k)=\left[s_{\tau}(T_{\text{s}}-\tau_{0}-kT_{\text{ PRI}})\ \ \ldots\ \ s_{\tau}(N_{\text{s}}T_{\text{s}}-\tau_{0}-kT_{\text{PRI}})\right]^{ \mathsf{T}}. \tag{13}\]  
The other matrices in Equation (12) include the diagonal Doppler matrix \(\mathbf{D}(\
u_{0})\)\(\in\)\(\mathbb{C}^{\text{K}\times K}\) given by \(\mathbf{D}(\
u_{0})\)\(=\)\(\text{diag}(\mathbf{d}(\
u_{0}))\), with diagonal entries \(\mathbf{d}(\
u_{0})\)\(=\)\([e^{j2\pi\
u_{0}\text{T}_{\text{PRI}}}\ \ldots\ e^{j2\pi K\
u_{0}\text{T}_{\text{PRI}}}]\), \(\mathbf{d}(\
u_{0})\)\(\in\)\(\mathbb{C}^{\text{1}\times K}\); matrix \(\mathbf{X}_{\text{c}}\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times K}\) given by \(\mathbf{X}_{\text{c}}\)\(=\)\([\mathbf{x}_{\text{c},1}\ \ldots\ \mathbf{x}_{\text{c},K}]\) with \(\mathbf{x}_{\text{c},k}\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times 1}\), \(\mathbf{x}_{\text{c},k}\)\(=\)\([\mathbf{x}_{\text{c},k}[1]\ \ldots\ \mathbf{x}_{\text{c},k}[N_{\text{s}}]]^{\mathsf{T}}\); and matrix \(\mathbf{W}\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times K}\) given by \(\mathbf{W}\)\(=\)\([\mathbf{w}_{1}\ \ldots\ \mathbf{w}_{K}]\) with \(\mathbf{w}_{k}\)\(\in\)\(\mathbb{C}^{N_{\text{s}}\times 1}\), \(\mathbf{w}_{k}\)\(=\)\([w_{k}[1]\ \ldots\ w_{k}[N_{\text{s}}]]^{\mathsf{T}}\). Note that the columns of matrix \(\mathbf{X}_{\text{c}}\) correspond to the communications symbols of all the users over each PRI.