# A Novel Cross-Correlation Algorithm Based on the Differential for Target Detection of Passive Radar
## 4 Discussion
### Reference Signal without Noise  
When the SNR of the reference channel is high or the reference channel is pure, the noise of reference channel can be ignored. The gain in coherent accumulation comes from the in-phase addition of \(N\) signal samples. The maximum gain in signal component energy is \(N^{2}\), whereas out-phase signals can only achieve incoherent accumulation and the maximum accumulation gain is \(N\). From Formula (24), ignoring the reference channel noise, (24) can be rewritten as follows:  
\[|\chi[\lambda_{1},f_{d1}]|^{2}\approx\left|\chi_{sr}[\lambda_{1},f_{d1}]^{2}+ \chi_{rs_{2}}[\lambda_{1},f_{d1}]^{2}\right| \tag{33}\]  
Then, the output SNR of the traditional R-D method is shown in (34):  
\[G_{RD}=10\log_{10}(\frac{\left|A_{ref}\right|^{2}\left|a_{1}\right|^{2}N^{2}}{ \left|A_{ref}\right|^{2}N})=\left|a_{1}\right|^{2}N \tag{34}\]For the algorithm in this paper, when the reference channel noise is ignored and there is only one target, Formula (13) can be written as,  
\[R[l,k]=|A_{ref}|^{2}|a_{1}|^{2}e^{2\pi f_{1}\frac{l}{\pi}B_{l}(k)P_{l}^{*}(k)e^{-2 \pi\tau_{1}\frac{l}{\pi}k}} \tag{35}\] \[+|A_{ref}|^{2}P_{l}^{*}(k)\cdot FFT\left(a_{1}^{2}z_{sur}[n]d^{*} [n-\lambda_{1}-l]e^{-2\pi f_{1}(\frac{n}{\pi}-\frac{l}{\pi})}+a_{1}z_{sur}^{*}[n -l]d[n-\lambda_{1}]e^{2\pi f_{1}(\frac{n}{\pi})}+z_{sur}^{*}[n-l]z_{sur}[n]\right)\]  
After compensating above time delay and frequency offset, the accumulate results can be obtained as follows:  
\[W[\lambda,f]=\sum\limits_{l=1}^{N-1}\sum\limits_{k=1}^{N}R(l,k) \exp\left[-j2\pi(f\frac{l}{f_{s}}+\lambda k)\right] \tag{36}\] \[=\sum\limits_{l=1}^{N-1}\sum\limits_{k=1}^{N}\left\{ \begin{array}[] {c}\left|A_{ref}\right|^{2}|a_{1}|^{2}P_{l}(k)P_{l}^{*}(k)e^{2\pi(f-f_{d1}) \frac{l}{\pi}e^{-j2\pi(\lambda_{1}+\lambda)k}}\\ +\left|A_{ref}\right|^{2}P_{l}^{*}(k)\cdot FFT\left(\begin{array}{c}a_{1}^{ *}z_{sur}[n]d^{*}[n-\lambda_{1}-l]e^{-2\pi f_{d1}(\frac{n}{\pi}-\frac{l}{\pi}) }\\ +a_{1}z_{sur}^{*}[n-l]d[n-\lambda_{1}]e^{2\pi f_{d1}(\frac{n}{\pi})}\\ +z_{sur}^{*}[n-l]z_{sur}[n]\end{array}\right)e^{j2\pi f\frac{l}{\pi}e^{-j2\pi \lambda k}}\end{array} \right\}\]  
when \(\lambda=-\lambda_{1},f=f_{d1}\), the peak appears at location of target 1:  
\[|\eta[\tau_{1},f_{d1}]|^{2}\approx\left|\eta_{sr}[\tau_{1},f_{d1}]^{2}+\eta_{ sr_{2}}[\tau_{1},f_{d1}]^{2}+\eta_{s_{2}\tau}[\tau_{1},f_{d1}]^{2}+\eta_{zz}[ \tau_{1},f_{d1}]^{2}\right| \tag{37}\]  
\[|\eta_{sr}[\tau_{1},f_{d1}]|^{2}=\sum\limits_{l=1}^{N-1}\sum\limits_{k=1}^{N} \left\{\left|A_{ref}\right|^{2}|a_{1}|^{2}P_{l}(k)P_{l}^{*}(k)\right\}=\left|A _{ref}\right|^{4}|a_{1}|^{4}\left(\sum\limits_{l=1}^{N-1}(N-l)\right)^{2} \tag{38}\]  
The signal shown in Equation (38) 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 \(\sum\limits_{l=1}^{N-1}(N-l)\). \[|\eta_{zz}[\tau,f]|^{2}=\left\{\sum\limits_{l=1}^{N-1}\sum\limits_{k=1}^{N} \left\{\left|A_{ref}\right|^{2}P_{l}^{*}(k)\cdot FFT(z_{sur}^{*}[n-l]z_{sur}[ n])e^{j2\pi f\frac{l}{f_{s}}}e^{-j2\pi\tau f_{s}k}\right\}\right\}^{2} \tag{39}\]  
Under the same differential order \(l\), the signal of Equation (39) has phase difference between different sampling points, so its accumulation is incoherent. However, under different differential orders \(l\), it can be seen from Table 2 that the correlation of the signal in Equation (39) is weak and it can be approximated as non-correlation. At this time, the signal in Equation (39) realizes two incoherent accumulations.