# 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
(2) Influence on the Cross-Ambiguity Function (CAF): Followed by the RDCC, in the PBR, the SNR of target detection is improved by utilizing the RDCC based on CAF [29], which is defined as:  
\[\xi[r,p]=\sum\limits_{l=1}^{L}\mathbf{S}_{\text{err}}[l]\mathbf{S}_{\text{ref }}^{*}[l-r]e^{-2\pi pl/L} \tag{16}\]  
where (\(\bullet\))* means the conjugate operation, \  
\[\xi_{T_{3}}^{2}[r,p]=\sum_{l=1}^{L}\Big{(}\overline{b}_{2}\mathbf{d}[l-\tau_{2}f_{ s}]e^{j2\pi f_{\text{d}2}l/f_{s}}\Big{)}b_{1}^{*}\mathbf{d}^{*}[l-\tau_{1}f_{s}-r]e^{-j2 \pi f_{\text{d}1}l/f_{s}}e^{-j2\pi pl/L} \tag{28}\]  
\[\xi_{T_{3}}^{3}[r,p]=\sum_{l=1}^{L}\Big{(}\overline{b}_{2}\mathbf{d}[l-\tau_{2} f_{s}]e^{j2\pi f_{\text{d}2}l/f_{s}}\Big{)}\mathbf{z}_{\text{ref}}^{*}[l-r]e^{-j2 \pi pl/L} \tag{29}\]  
\[\xi_{z}[r,p]=\sum_{l=1}^{L}\mathbf{Z}[l]\left\{ \begin{array}{l}a_{ref} \mathbf{d}[l-r]\\ +b_{1}\mathbf{d}[l-\tau_{1}f_{s}-r]e^{j2\pi f_{\text{d}1}l/f_{s}}\\ +\mathbf{z}_{\text{ref}}[l-r]\end{array} \right\}^{*}e^{-j2\pi pl/L} \tag{30}\]  
where \(\xi_{c}^{1}\), \(\xi_{c}^{2}\), and \(\xi_{c}^{3}\) represent the matching results of the residual clutter with the direct path clutter, target echo, and noise in the reference signal, respectively. \(\xi_{T_{1}}^{1}\), \(\xi_{T_{1}}^{2}\), and \(\xi_{T_{1}}^{3}\) denote the matching results of the target existing in both the surveillance channel and reference channel, respectively. \(\xi_{T_{2}}^{1}\), \(\xi_{T_{2}}^{2}\), and \(\xi_{T_{2}}^{3}\) are the matching results of the false targets, respectively. \(\xi_{T_{3}}^{1}\), \(\xi_{T_{3}}^{2}\), and \(\xi_{T_{3}}^{3}\) are the matching results of the target only received by the surveillance channel, respectively. \(\xi_{z}\) refers to the matching results of the noise. After the ECA, theoretically, the residual clutter will not affect the CAF of the target. In addition, due to the random nature of noise, the CAF of the noise does not matter. Furthermore, although the target signal in residual surveillance signal and the one in the reference signal can be matched, the energy intensity is negligible. Therefore, the matching results involved in these cases will be ignored in the following analysis, and (17) is further simplified as:  
\[\xi[r,p]=\xi_{T_{1}}^{1}[r,p]-\xi_{T_{2}}^{1}[r,p]+\xi_{T_{3}}^{1}[r,p] \tag{31}\] \[=\sum_{l=1}^{L}\Big{(}(\overline{b}_{1}-\mathbf{W}[1]b_{1}) \mathbf{d}[l-\tau_{1}f_{s}]e^{j2\pi f_{\text{d}1}l/f_{s}}\Big{)}a_{ref}^{*} \mathbf{d}^{*}[l-r]e^{-j2\pi pl/L}\] \[-\sum_{l=1}^{L}\Big{(}\sum_{k=2}^{K}\mathbf{W}[k]\Big{\{}b_{1} \mathbf{d}[l-\tau_{1}f_{s}-k+1]e^{j2\pi f_{\text{d}1}(l-k+1)/f_{s}}\Big{\}} \Big{)}\] \[+\sum_{l=1}^{L}\Big{(}\overline{b}_{2}\mathbf{d}[l-\tau_{2}f_{s}] e^{j2\pi f_{\text{d}2}l/f_{s}}\Big{)}a_{ref}^{*}\mathbf{d}^{*}[l-r]e^{-j2\pi pl/L}\]  
Following a similar derivation, substituting (5) and (14) into (16) produces:  
\[\overline{\xi}[r,p]=\overline{\xi}_{T_{1}}^{1}[r,p]+\xi_{T_{3}}^{1}[r,p] \tag{32}\] \[=\sum_{l=1}^{L}\Big{(}\overline{b}_{1}\mathbf{d}[l-\tau_{1}f_{s}] e^{j2\pi f_{\text{d}1}l/f_{s}}\Big{)}a_{\text{ref}}^{*}\mathbf{d}^{*}[l-r]e^{-j2 \pi pl/L}\] \[+\sum_{l=1}^{L}\Big{(}\overline{b}_{1}\mathbf{d}[l-\tau_{1}f_{s}] e^{j2\pi f_{\text{d}1}l/f_{s}}\Big{)}a_{\text{ref}}^{*}\mathbf{d}^{*}[l-r]e^{-j2 \pi pl/L}\]  
Comparing (31) and (32), it can be found that the amplitude of target \(T_{1}\) in (31) is lower than that of target \(T_{1}\) in (32); that is, the impure reference signal will negatively benefit the detection of the target received by both the surveillance channel and reference channel. On the other hand, compared to (32), (31) has one more matching term, which has the same Doppler shift as target \(T_{1}\), but the range cells demonstrate a pseudo-random distribution related to the clutter suppression coefficients, thereby interfering with the normal detection of target \(T_{1}\).