# Target Detection of Passive Bistatic Radar under the Condition of Impure Reference Signal
## 3 Proposed Target Detection Method under the Condition of Impure Reference Signal
### Reference Signal Purification Method  
Based on the method proposed in the previous subsection, the identification of the impure component within the reference signal can be achieved. The subsequent focus of this subsection revolves around the reconstruction of the signal and the subsequent purification of the reference signal through subtraction from the original reference signal. This purification process aims to mitigate the adverse impact of impurities on target detection. Comparing (9) and (14), it can be seen that the only difference between the two formulas is \(\boldsymbol{\varphi}_{T_{1}}[l]\) in (9). For analysis, here, \(\boldsymbol{\varphi}_{T_{1}}[l]\) is rewritten as:  
\[ \begin{array}{l}\boldsymbol{\varphi}_{T_{1}}[l]=(\overline{b}_{1}-\mathbf{W }[1]b_{1})\mathbf{d}[l-\tau_{1}f_{s}]e^{j2\pi f_{\mathrm{d}}1/f_{s}}\\ -\sum\limits_{k=2}^{K}\mathbf{W}[k]\Big{\{}b_{1}\mathbf{d}[l-\tau_{1}f_{s}-k +1]e^{j2\pi f_{\mathrm{d}}1(l-k+1)/f_{s}}\Big{\}}\end{array}  \tag{34}\]  
Due to the fact that a target is received by both the reference channel and the surveillance channel, the two channels should be very close to each other. As a result, it can be considered that the energy of the target received by the two channels is approximately equal, i.e., \(\overline{b}_{1}\approx b_{1}\). In the practical scenarios of the PBR, in order to achieve full coverage of the airspace, multiple digital wave beams are generally employed for target detection. Among them, one wave beam serves as the reference channel, used to receive the reference signal, while the rest are surveillance channels, used to detect signals from the targets of interest. These digital wave beams overlap and cover each other. Therefore, this approximation is reasonable for the surveillance channels adjacent to the reference channel, but it does not hold true for other surveillance channels that are far away from the reference channel. Based on this approximation, (34) is further expressed as:  
\[\boldsymbol{\varphi}_{T_{1}}[l]=(1-\mathbf{W}[1])b_{1}\mathbf{d}[l-\tau_{1}f_{ s}]e^{2\pi f_{\text{d1}}l/f_{s}}-\sum_{k=2}^{K}\mathbf{W}[k]\Big{\{}b_{1} \mathbf{d}[l-\tau_{1}f_{s}-k+1]e^{2\pi f_{\text{d1}}(l-k+1)/f_{s}}\Big{\}} \tag{35}\]  
where \((1-\mathbf{W}[1])b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\) is the real target term after clutter suppression, while \(b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\) is the original term of this target, and the other term is the set of false targets originated from the real target. When the target in the reference signal is eliminated, the false targets will also disappear. In other words, if \(b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\) can be reconstructed, and then the impure reference signal with the existence of \(b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\) can be purified. Although it is almost impossible to reconstruct \(b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\) directly, it can be reconstructed indirectly from \((1-\mathbf{W}[1])b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\). Among \((1-\mathbf{W}[1])b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\), \(\tau_{1}f_{s}\) and \(f_{\text{d1}}\) can be obtained by the proposed localization method, while \(\mathbf{W}[1]\) is known. Therefore, as long as \((1-\mathbf{W}[1])b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\) is estimated, \(b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\) can be reconstructed. To achieve this purpose, inspired by the ECA, \((1-\mathbf{W}[1])b_{1}\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\) in \(\boldsymbol{\varphi}_{T_{1}}[l]\) can be estimated by utilizing the subspace projection method similar to the ECA.