# A Novel Cross-Correlation Algorithm Based on the Differential for Target Detection of Passive Radar
## 4 Discussion
### Reference Signal without Noise
Its accumulated gain is approximately the incoherent accumulated gain of the signal with length \(\sum\limits_{l=1}^{N-1}(N-l)\), as shown in Equation (40),  
\[|\eta_{zz}[\tau,f]|^{2}\approx\left|A_{ref}\right|^{4N-1}(N-l) \tag{40}\]  
\[|\eta_{rs_{2}}[\tau_{1},f_{d1}]|^{2}=\left\{\sum\limits_{l=1}^{N-1}\sum \limits_{k=1}^{N}\left\{\left|A_{ref}\right|^{2}P_{l}^{*}(k)\cdot FFT \left(a_{1}^{*}z_{sur}[n]d^{*}[n-\lambda_{1}-l]e^{-2\pi f_{d1}(\frac{n}{\pi}- \frac{l}{f_{s}})}\right)e^{j2\pi f_{d1}\frac{l}{f_{s}}}e^{-j2\pi\lambda_{1}k} \right\}\right\}^{2} \tag{41}\]  
\[|\eta_{rs_{2}}[\tau_{1},f_{d1}]|^{2}=\left\{\sum\limits_{l=1}^{N-1}\sum \limits_{k=1}^{N}\left\{\left|A_{ref}\right|^{2}P_{l}^{*}(k)\cdot FFT \left(a_{1}^{*}z_{sur}[n]d^{*}[n-\lambda_{1}-l]e^{-2\pi f_{d1}(\frac{n}{\pi}- \frac{l}{f_{s}})}\right)e^{j2\pi f_{d1}\frac{l}{f_{s}}}e^{-j2\pi\lambda_{1}k} \right\}\right\}^{2} \tag{42}\]  
The signal shown in Equations (41) and (42) has phase difference between different sampling points under the same differential order \(l\). Therefore, the incoherent accumulation is realized under the same differential order \(l\). There is a correlation under different differential orders. As shown in Table 1, when the signals of Equation (41) and (42) are completely correlated under different differential orders, the output SNR of the algorithm in this paper can reach the theoretical lower bound. Its accumulation gain is the sum of incoherent accumulation under the same differential order \(l\) and coherent accumulation gain under different differential orders, as shown in Equation (43) and (44) below:  
\[ \begin{split}\left|\eta_{rs_{2}}[\tau,f]\right|^{2}=\left|A_{ref} \right|^{4}&|a_{1}|^{2}\bigg{\{}(N-1)\sum\limits_{l=1}^{N-1}(N-l)- \left[1+2+3+\cdots+(N-1)\right]\bigg{\}}\\ &\approx\left|A_{ref}\right|^{4}&|a_{1}|^{2}(N-2) \sum\limits_{l=1}^{N-1}(N-l)\end{split}  \tag{43}\]  
\[ \begin{split}\left|\eta_{sr_{2}}[\tau,f]\right|^{2}\approx\left|A_{ ref}\right|^{4}&|a_{1}|^{2}\bigg{(}(N-1)\sum\limits_{l=1}^{N-1}(N-l)- \left[1+2+3+\cdots+(N-1)\right]\bigg{)}\\ &\approx\left|A_{ref}\right|^{4}&|a_{1}|^{2}(N-2) \sum\limits_{l=1}^{N-1}(N-l)\end{split}  \tag{44}\]  
In the above Formulas (43) and (44), it is assumed that the signals are completely correlated under different cancellation orders and the accumulation is equivalent to N-1 times of signal amplification. Therefore, the accumulation result is the incoherent accumulation gain of the signal with length \(\sum\limits_{l=1}^{N-1}(N-l)\) multiplied by N-1. However, due to the difference method, the signal lengths under different difference orders are different, and the influence of the signal length needs to be subtracted, so the result in the formula is obtained. When the signal part in Equations (41) and (42) is completely uncorrelated under different differential orders, the output SNR of the algorithm in this paper can reach the theoretical upper bound. Its accumulated gain is the sum of incoherent accumulation at the same differential order \(l\) and incoherent accumulation gain at different differential orders, and its accumulated gain is approximately the incoherent accumulated gain of the signal with length \(\sum\limits_{l=1}^{N-1}(N-l)\), as shown in Formulas (45) and (46) below:  
\[ \begin{split}\left|\eta_{rs_{2}}[\tau,f]\right|^{2}\approx\left|A _{ref}\right|^{4}&|a_{1}|^{2}\sum\limits_{l=1}^{N-1}(N-l)\\ &\left|\eta_{sr_{2}}[\tau,f]\right|^{2}\approx\left|A_{ref} \right|^{4}&|a_{1}|^{2}\sum\limits_{l=1}^{N-1}(N-l)\end{split}