# Target Detection of Passive Bistatic Radar under the Condition of Impure Reference Signal
## 2 Signal Model and Problem Analysis
### Analysis of the Influence of Signal Model on Target Detection  
In this subsection, the impact of the new signal model on the PBR target detection is analyzed from two aspects:  
(1) Influence on Clutter Suppression: In the PBR, the extensive cancellation algorithm (ECA) [27; 28] is a widely used algorithm for clutter suppression. The fundamental concept behind the ECA is to reduce clutter by projecting the surveillance signal onto the orthogonal subspace created by the reference signal and its time-delayed cells, and its projection coefficients vector is given by:  
\[\mathbf{W}=\left(\mathbf{V}^{\text{H}}\mathbf{V}\right)^{-1}\mathbf{V}^{ \text{H}}\mathbf{S}_{\text{sur}} \tag{7}\]  
where \((\bullet)^{\text{H}}\) represents the Hermitian transpose operation and \(\mathbf{V}\) denotes the orthogonal subspace created by the reference signal \(\mathbf{S}_{\text{ref}}\) and its time-delayed cells. Furthermore, the key expression of the ECA can be given as:  
\[\mathbf{S}_{\text{err}}[l]=\mathbf{S}_{\text{sur}}[l]-\sum\limits_{k=1}^{K} \mathbf{W}[k]\mathbf{S}_{\text{ref}}[l-k+1] \tag{8}\]  
where \(\mathbf{S}_{\text{err}}\) and \(\mathbf{W}\) denote the residual signal and the corresponding projection matrix of the ECA, respectively. In addition, \(K\) represents the maximum time-delayed cells of the reference signal. Substituting (5) and (6) into the above formula, (8) is further expressed as:  
\[\mathbf{S}_{\text{err}}[l]=\boldsymbol{\varphi}_{c}[l]+\boldsymbol{\varphi}_{ T_{1}}[l]+\boldsymbol{\varphi}_{T_{2}}[l]+\boldsymbol{\varphi}_{z}[l] \tag{9}\]  
where  
\[ \begin{array}{ll}\boldsymbol{\varphi}_{c}[l]=\sum\limits_{c=1}^{N_{c}}a_{c} \mathbf{d}[l-\tau_{c}f_{s}]-\sum\limits_{k=1}^{K}\mathbf{W}[k]a_{\text{ref}} \mathbf{d}[l-k+1]\end{array}  \tag{10}\]  
\[ \begin{array}{ll}\boldsymbol{\varphi}_{T_{1}}[l]&=(\overline{b}_{1}-\mathbf{ W}[1]b_{1})\mathbf{d}[l-\tau_{1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}}\\ &-\sum\limits_{k=2}^{K}\mathbf{W}[k]\Big{\{}b_{1}\mathbf{d}[l-\tau_{1}f_{s}-k+ 1]e^{2\pi f_{\text{d1}}(l-k+1)/f_{s}}\Big{\}}\\ &\boldsymbol{\varphi}_{T_{2}}[l]=\overline{b}_{2}\mathbf{d}[l-\tau_{2}f_{s}]e^ {2\pi f_{\text{d2}}l/f_{s}}\end{array}  \tag{12}\]  
\[ \begin{array}{ll}\boldsymbol{\varphi}_{z}[l]=\mathbf{z}_{\text{sur}}[l]-\sum \limits_{k=1}^{K}\mathbf{W}[k]\mathbf{z}_{\text{ref}}[l-k+1]\end{array}  \tag{13}\]  
where \(\boldsymbol{\varphi}_{c}\), \(\boldsymbol{\varphi}_{T_{1}}\), \(\boldsymbol{\varphi}_{T_{2}}\), and \(\boldsymbol{\varphi}_{z}\) denote residual clutter, the echo around target 1, the echo around target 2, and the noise, respectively. For comparison, in the same way, substituting (1) and (6) into (8) produces:  
\[\overline{\mathbf{S}}_{\text{err}}[l]=\boldsymbol{\varphi}_{c}[l]+\overline{ \boldsymbol{\varphi}}_{T_{1}}[l]+\boldsymbol{\varphi}_{T_{2}}[l]+\boldsymbol{ \varphi}_{z}[l] \tag{14}\]  
where \(\boldsymbol{\varphi}_{c}\), \(\boldsymbol{\varphi}_{T_{2}}\), and \(\boldsymbol{\varphi}_{z}\) are the same as the three terms in (9), and the only difference between (9) and (14) is \(\overline{\boldsymbol{\varphi}}_{T_{1}}\), which is given by:  
\[\overline{\boldsymbol{\varphi}}_{T_{1}}[l]=\overline{b}_{1}\mathbf{d}[l-\tau_ {1}f_{s}]e^{2\pi f_{\text{d1}}l/f_{s}} \tag{15}\]  
From (11) and (15), when the reference signal is impure, it can be inferred that in the presence of an impure reference signal, the target echo within the surveillance signal after the ECA will inevitably be influenced by the same target signal received by the reference channel. The potential impact on target detection can be categorized as follows: firstly, the amplitude of the real target will experience attenuation, thereby reducing the signal-to-noise ratio (SNR) associated with the target. Then, some false targets attached to the real target may appear.