# Target Detection of Passive Bistatic Radar under the Condition of Impure Reference Signal
## 2 Signal Model and Problem Analysis
### Signal Model  
Figure 1 illustrates a representative detection scenario in the PBR system, considering the presence of the impure reference signal. The received signal within the surveillance channel adheres to the conventional assumptions of the PBR system, encompassing components such as direct path or multipath clutter, target echo, and noise signal. In contrast to the traditional reference signal, the reference channel in the newly proposed signal model not only comprises the direct path clutter, a limited amount of multipath clutter, and noise signal, but it also incorporates the reception of target echo signals. The presence of moving echo signals within the reference channel induces unexpected Doppler frequency shifts, posing several challenges to the conventional target detection methods employed in PBR systems.  
Following the process of analog-to-digital sampling and subsequent digital down-conversion, the ideal reference signal is characterized by the following model:  
\[\mathbf{S}_{\text{ref}}[l]=a_{\text{ref}}\mathbf{d}[l]+\mathbf{z}_{\text{ref} }[l] \tag{1}\]  
with  
\[l=1,\cdots,L \tag{2}\]  
where \(\mathbf{S}_{\text{ref}}\)\([l]\) denotes the \(n\)th sampling datum received by the reference channel, \(\mathbf{d}[l]\) represents the \(l\)th sampling data of the direct path signal, \(a_{\text{ref}}\) is the corresponding complex amplitude, the received noise signal is represented by \(\mathbf{z}_{\text{ref}}[l]\), and \(L\) is the data length. Whena target appears in the airspace in the direction the reference channel is pointed at, the reference signal represented by (1) can be rewritten as:  
\[\mathbf{S}_{\text{ref}}[l]=a_{\text{ref}}\mathbf{d}[l]+\sum_{n=1}^{N_{T}}b_{n} \mathbf{d}[l-\tau_{n}f_{\text{s}}]e^{2\pi f_{\text{dm}}l/f_{\text{s}}}+\mathbf{ z}_{\text{ref}}[l] \tag{3}\]  
where \(b_{n}\), \(\tau_{n}f_{s}\), and \(f_{\text{dm}}\) denote the complex amplitude, range unit respecting to direct path signal, and Doppler frequency shift of the \(n\)th target echo in the reference signal, respectively. Moreover, \(f_{\text{s}}\) is the baseband signal sampling rate, and \(N_{T}\) represents the number of targets in the reference signal.  
Similarly, the surveillance signal received by the surveillance channel can be represented as:  
\[\mathbf{S}_{\text{sur}}[l]=\sum_{c=1}^{N_{C}}a_{c}\mathbf{d}[l-\tau_{c}f_{ \text{s}}]+\sum_{m=1}^{N_{M}}\bar{b}_{m}\mathbf{d}[l-\tau_{m}f_{\text{s}}]e^{2 \pi f_{\text{dm}}l/f_{\text{s}}}+\mathbf{z}_{\text{sur}}[l] \tag{4}\]  
where \(a_{c}\) and \(\tau_{c}f_{\text{s}}\) are the complex amplitude and range cell of the direct path or multipath clutter in the surveillance signal, respectively; \(N_{C}\) is the number of clutters; \(\bar{b}_{m}\), \(\tau_{m}f_{s}\), and \(f_{\text{dm}}\) denote the complex amplitude, range unit, and Doppler frequency shift of the \(m\)th target echo in the surveillance signal, respectively; \(N_{M}\) is the target number; and \(\mathbf{z}_{\text{sur}}\) is the echo noise.  
Generally, the target echo received by the reference channel can also be simultaneously detected by the surveillance channel, that is, \(N_{M}\geq N_{T}\). For the sake of analytical convenience and without sacrificing generality, let \(N_{T}\) = 2 and \(N_{T}\) = 1, so the reference signal and echo signal represented by (3) and (4), respectively, can be further updated as:  
\[\mathbf{S}_{\text{ref}}[l]=a_{\text{ref}}\mathbf{d}[l]+b_{1}\mathbf{d}[l-\tau _{1}f_{\text{s}}]e^{2\pi f_{\text{dm}}l/f_{\text{s}}}+\mathbf{z}_{\text{ref}} [l] \tag{5}\]  
Figure 1: Sketch map of the PBR system with impure reference signal.  
\[\begin{array}{ll}\mathbf{S}_{\text{sur}}[l]&=\sum\limits_{c=1}^{N_{c}}a_{c} \mathbf{d}[l-\tau_{c}f_{s}]+\overline{b}_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f _{\text{d1}}l/f_{s}}\\ &+\overline{b}_{2}\mathbf{d}[l-\tau_{2}f_{s}]e^{2\pi f_{\text{d2}}l/f_{s}}+ \mathbf{z}_{\text{sur}}[l]\end{array} \tag{6}\]