# 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
Therefore, when the signals are completely correlated, the algorithm in this paper obtains the theoretical lower bound of the accumulated gain, and each accumulated gain is shown in the following formulas:  
\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]0  
\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]1  
\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]2  
\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]3  
\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]4  
\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]5  
\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]6  
\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]7  
When the signals are non-correlated, the algorithm in this paper obtains the theoretical upper bound of the accumulated gain, and accumulated gains are shown in the following formulas:  
The theoretical upper bound output SNR in this paper is as follows:\[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]8 \[\left|\eta_{2,3}[\lambda_{1},f_{d1}]\right|^{2}=\left\{\sum_{l=1}^{N-1}\sum_{k =1}^{N}\left\{z_{ref}[n]z_{ref}^{*}[n-l]|a_{1}|^{2}P_{l}(k)\right\}\right\}^{2} \tag{55}\]9  
The theoretical lower bound output SNR in this paper is as follows:  
\[G_{lower} =\frac{\left|A_{ref}\right|^{4}\left|a_{1}\right|^{4}\left(\sum \limits_{l=1}^{N-1}l\right)^{2}}{\left\{ \begin{array}{c}\left(N-2\right) \left\{2\left|A_{ref}\right|^{4}\left|a_{1}\right|^{2}+2\left|A_{ref}\right|^ {2}\left|a_{1}\right|^{4}+4\left|A_{ref}\right|^{2}\left|a_{1}\right|^{2}+2 \left|A_{ref}\right|^{2}+\left|a_{1}\right|^{2}\right\}\sum\limits_{l=1}^{N-1} \left(N-l\right)\\ +\left(\left|a_{1}\right|^{4}+\left|A_{ref}\right|^{4}+\left|a_{1}\right|^{2}+ 1\right)\sum\limits_{l=1}^{N-1}\left(N-l\right)\end{array} \right\}} \tag{96}\]  
When the SNR of the reference channel tends to infinity, the derivation in Equations (95) and (96) is consistent with that in Equations (48) and (49). Figure 23 shows the output SNR under different reference SNR. It can be seen from Figure 23 that the algorithm in this paper is more greatly affected by the SNR of the reference channel. However, when the SNR is relatively high, the estimation performance is better than that attained with traditional R-D processing. In the detection of the passive radar, the reference antenna generally points directly to the direction of the emitter, so that the SNR of the reference channel is generally higher than 0 dB. Therefore, in practical applications, the estimation performance of this algorithm is better than that of the traditional R-D algorithm. From the above analysis, the performance of this algorithm is higher than that of the traditional R-D algorithm. With the increase of SNR of target echo channel and accumulation time, it increases continuously. However, the detection performance of the algorithm in this paper is greatly affected under lower reference channel SNR.