# 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
The accumulated gain is shown in the following Equations (58)-(61):  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}\approx\sum_{l=1}^{N-1}\left(N -l\right) \tag{58}\]  
\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}\approx\left|a_{1}\right|^{4} \sum_{l=1}^{N-1}\left(N-l\right) \tag{59}\]  
\[\left|\eta_{3,8}[\lambda_{1},f_{d1}]\right|^{2}\approx\left|a_{1}\right|^{2} \sum_{l=1}^{N-1}\left(N-l\right) \tag{60}\]  
\[\left|\eta_{4,3}[\lambda_{1},f_{d1}]\right|^{2}\approx\left|A_{ref}\right|^{4 }\sum_{l=1}^{N-1}\left(N-l\right) \tag{61}\]  
\[\left|\eta_{2,1}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}A_{ref}^{*}z_{ref}[n]d^{*}[n-l]|a_{1}|^{2}P_{l}(k)\biggr{\}} \right\}^{2} \tag{62}\]  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]0\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]1  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]2  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]3  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]4  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]5  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]6  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]7  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]8  
\[\left|\eta_{3,9}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\biggl{\{}z_{ref}[n]z_{ref}^{*}[n-l]FFT(z_{sur}^{*}[n-l]z_{sur}[n])e^{j2 \pi f_{d1}\frac{l}{l}}e^{-j2\pi\lambda_{1}k}\biggr{\}}\right\}^{2} \tag{54}\]9  
The signal in Formulas (62)-(72) includes both completely related items and incompletely related items. In the same differential order, due to the phase difference of the signal itself, only incoherent accumulation can be achieved. Meanwhile, it is possible that the correlation of each signal changes from approximately uncorrelated to completely correlated under different differential orders.