# A Novel Cross-Correlation Algorithm Based on the Differential for Target Detection of Passive Radar
## 4 Discussion
### The Influence of the Reference Channel SNR on the Performance of the Algorithms  
When considering the SNR of the reference channel, from Equation (25), the output SNR of R-D algorithm is as follows:  
\[G_{r}=\frac{\left|A_{ref}\right|^{2}\left|a_{1}\right|^{2}N^{2}}{\left(\left|A _{ref}\right|^{2}+\left|a_{1}\right|^{2}+1\right)N}=\frac{\left|a_{1}\right|^{2 }}{1+\left|a_{1}\right|^{2}/\left|A_{ref}\right|^{2}+1/\left|A_{ref}\right|^{2 }}N \tag{50}\]  
For differential operations in this paper, after compensating the time delay and frequency offset in Equation (13):  
\[ \begin{array}{l}W[\lambda,f]=\sum\limits_{l=1}^{N-1}\sum\limits_{k=1}^{N}R( l,k)\exp\left[-j2\pi(f\frac{l}{f_{s}}+\lambda k)\right]\\ \\ \left|A_{ref}\right|^{2}\left|a_{1}\right|^{2}e^{j2\pi(f_{d1}-f)\frac{l}{f_{s} }}P_{l}(k)P_{l}^{*}(k)e^{-j2\pi(\lambda_{1}+\lambda)k}\\ \left.+Z_{r,r}^{*}(l,k)\cdot\left|a_{1}\right|^{2}e^{j2\pi(f_{d1}-f)\frac{l}{f_ {s}}}P_{l}(k)e^{-j2\pi(\lambda_{1}+\lambda)k}\right.\\ \left.+Z_{r,r}^{*}(l,k)\cdot FFT\left(\begin{array}{c}a_{1}^{*}z_{sur}[n]d^{ *}[n-\lambda_{1}-l]e^{-2\pi f_{d1}(\frac{n}{f_{s}}-\frac{l}{f_{s}})}\\ +a_{1}z_{sur}^{*}[n-l]d[n-\lambda_{1}]e^{j2\pi f_{d1}(\frac{n}{f_{s}})}\\ \left.+z_{sur}^{*}[n-l]z_{sur}[n]\end{array}\right)e^{j2\pi f\frac{l}{f_{s}}}e ^{-j2\pi\lambda k}\\ +\left|A_{ref}\right|^{2}P_{l}^{*}(k)\cdot FFT\left(\begin{array}{c}a_{1}^{ *}z_{sur}[n]d^{*}[n-\lambda_{1}-l]e^{-2\pi f_{d1}(\frac{n}{f_{s}}-\frac{l}{f_ {s}})}\\ +a_{1}z_{sur}^{*}[n-l]d[n-\lambda_{1}]e^{j2\pi f_{d1}(\frac{n}{f_{s}})}\\ +z_{sur}^{*}[n-l]z_{sur}[n]\end{array}\right)e^{j2\pi f\frac{l}{f_{s}}}e^{-j2 \pi\lambda k}\\ \end{array} \right\}\end{array} \tag{51}\]  
when \(\lambda=-\lambda_{1}\) and \(f=f_{d1}\), the peak appears at the location of target 1:  
Figure 22: Comparison of the output SNR between the two algorithms: (**a**) comparison under different signal lengths: \(SNR_{sur}=-40\) dB; (**b**) comparison under different \(SNR_{sur}\): \(N=10^{4}\). \[\left|\eta[\lambda_{1},f_{d1}]\right|^{2}\approx \begin{array}{c}\eta_{sr}[ \lambda_{1},f_{d1}]^{2}+\eta_{2,1}[\lambda_{1},f_{d1}]^{2}+\eta_{2,2}[\lambda_{ 1},f_{d1}]^{2}+\eta_{2,3}[\lambda_{1},f_{d1}]^{2}+\eta_{3,1}[\lambda_{1},f_{d1} ]^{2}\\ +\eta_{3,2}[\lambda_{1},f_{d1}]^{2}+\eta_{3,3}[\lambda_{1},f_{d1}]^{2}+\eta_{3,4}[\lambda_{1},f_{d1}]^{2}+\eta_{3,5}[\lambda_{1},f_{d1}]^{2}+\eta_{3,6}[ \lambda_{1},f_{d1}]\\ +\eta_{3,7}[\lambda_{1},f_{d1}]^{2}+\eta_{3,8}[\lambda_{1},f_{d1}]^{2}+\eta_{3,9}[\lambda_{1},f_{d1}]^{2}+\eta_{3,10}[\lambda_{1},f_{d1}]^{2}+\eta_{3,11}[ \lambda_{1},f_{d1}]^{2}\\ +\eta_{3,12}[\lambda_{1},f_{d1}]^{2}+\eta_{4,1}[\lambda_{1},f_{d1}]^{2}+\eta_ {4,2}[\lambda_{1},f_{d1}]^{2}+\eta_{4,3}[\lambda_{1},f_{d1}]^{2}+\eta_{4,4}[ \lambda_{1},f_{d1}]^{2}\end{array}  \tag{52}\]  
\[\left|\eta_{sr}[\lambda_{1},f_{d1}]\right|^{2}=\sum_{l=1}^{N-1}\sum_{k=1}^{N} \biggl{\{}\left|A_{ref}\right|^{2}\left|a_{1}\right|^{2}P_{l}(R)_{l}^{*}(k) \biggr{\}}=\left|A_{ref}\right|^{4}\left|a_{1}\right|^{4}\left(\sum_{l=1}^{N- 1}\left(N-l\right)\right)^{2} \tag{53}\]  
The signal shown in Equation (53) has no phase difference after compensation and is a sequence of real numbers. Therefore, its coherent accumulation gain can be seen as a coherent accumulation of signal with data length \(\sum\limits_{l=1}^{N-1}\left(N-l\right)\).