# A Novel Cross-Correlation Algorithm Based on the Differential for Target Detection of Passive Radar
## 4 Discussion  
In the previous section, the simulation results of the proposed algorithm with the simulation data and the measured data were compared to the traditional R-D algorithm. In this section, we will theoretically deduce the output SNR of the two algorithms and give the final results to prove the effectiveness and the advantages of the proposed algorithm.  
According to classical radar theory, when the sampling rate of the signal is \(f_{s}=B\), the accumulation gain \(G_{c}\) of the conventional CW radar can reach the maximum value, which is the product of the effective bandwidth \(B\) of the signal and the accumulation time \(T\), \(G_{c}=BT=N\).  
The algorithm presented in this paper uses signal differential to transform a single channel signal into a multi-channel signal. Comparing Equations (3) with (6), the cross term is introduced into the differential signals, which results in a reduction in the SNR. The cross terms introduced are as follows:  
\[\begin{array}{rl}r_{cross}[l,n]&=\sum\limits_{m=1}^{N_{M}}a_{m}z_{sur}^{*}[n -l]d[n-\lambda_{m}]e^{j2\pi f_{dm}(\frac{n}{f_{s}})}+\sum\limits_{m=1}^{N_{M}} a_{m}^{*}z_{sur}[n]d^{*}[n-\lambda_{m}-l]e^{-j2\pi f_{dm}(\frac{n}{f_{s}}-\frac{l}{f_{s}})}  
For the convenience of analysis, set \(d[n]\) is the power to the normalized signal, the modulus of which is 1, and set \(z_{ref}\) and \(z_{sur}\) are the power to the normalized Gaussian white noise. Since the calculation of the accumulated gain is independent of the number of targets and the target echo energy is much lower than the noise energy, we assume that there is only one target in the echo signal (i.e., \(N_{m}=1\) in (3)). Thus, the SNR of the reference channel before accumulation is \(SNR_{ref\_RD}=\left|A_{ref}\right|^{2}\). The SNR of the echo channel before accumulation is \(SNR_{sur}=\left|a_{1}\right|^{2}\).  
Then, (4) is expanded as follows:  
\[\begin{array}{l}\chi[\lambda_{1},f_{d1}]=\frac{N}{n}s_{sur}[n]s_{ref}^{*}[n -\lambda]e^{-2\pi/\frac{\pi}{\lambda}}=\frac{N}{n}\Lambda_{ref}a_{1}d^{*}[n- \lambda]d[n-\lambda_{1}]e^{2\pi f_{m}\frac{\pi}{\lambda}}e^{-2\pi/\frac{\pi}{ \lambda}}+\sum\limits_{n=1}^{N}A_{nf}d^{*}[n-\lambda]e^{-2\pi f/\frac{\pi}{ \lambda}}z_{sur}[n]\\ +\sum\limits_{n=1}^{N}a_{1}d[n-\lambda_{1}]e^{2\pi f_{m}\frac{\pi}{\lambda}} z_{ref}^{*}[n-\lambda]e^{-2\pi f/\frac{\pi}{\lambda}}+\sum\limits_{n=1}^{N}z_{sur}[n ]z_{ref}^{*}[n-\lambda]e^{-2\pi f/\frac{\pi}{\lambda}}\end{array} \tag{24}\]  
when \(\tau=\tau_{1}\), \(f=f_{d1}\), the peak appears at the target location,  
\[\begin{array}{rl}\left|\chi[\lambda_{1},f_{d1}]\right|^{2}&=\left|\chi_{sr}[ \lambda_{1},f_{d1}]+\chi_{sr_{2}}[\lambda_{1},f_{d1}]+\chi_{sz}[\lambda_{1},f _{d1}]+\chi_{zz}[\lambda_{1},f_{d1}]\right|^{2}\\ &\approx\left|\chi_{sr}[\lambda_{1},f_{d1}]^{2}+\chi_{sr_{2}}[\lambda_{1},f_{ d1}]^{2}+\chi_{sz}[\lambda_{1},f_{d1}]^{2}+\chi_{zz}[\lambda_{1},f_{d1}]^{2} \right|^{2}\end{array} \tag{25}\]  
where  
\[\chi_{sr}[\lambda_{1},f_{d1}]=\sum\limits_{n=1}^{N}A_{ref}a_{1}|d[n-\tau_{1}f _{s}]|^{2}=A_{ref}a_{1}N \tag{26}\]  
\[\chi_{rs_{2}}[\lambda_{1},f_{d1}]=\sum\limits_{n=1}^{N}A_{ref}d^{*}[n-\tau_{1 }f_{s}]z_{sur}[n]e^{-2\pi f_{d1}\frac{\pi}{\lambda}} \tag{27}\]  
\[\chi_{sr_{2}}[\lambda_{1},f_{d1}]=\sum\limits_{n=1}^{N}a_{1}d[n-\tau f_{s}]z_{ ref}^{*}[n-\tau_{1}f_{s}] \tag{28}\]  
\[\chi_{zz}[\lambda_{1},f_{d1}]=\sum\limits_{n=1}^{N}z_{sur}[n]z_{ref}^{*}[n- \tau_{1}f_{s}]e^{-2\pi f_{d1}\frac{\pi}{\lambda}} \tag{29}\]  
Since \(\chi_{rz}\), \(\chi_{sz}\), and \(\chi_{zz}\) in the integral term can be regarded as stationary noise [30],  
\[\chi_{rs_{2}}[\lambda_{1},f_{d1}]\approx A_{ref}\sqrt{N} \tag{30}\]  
\[\chi_{sr_{2}}[\lambda_{1},f_{d1}]\approx a_{1}\sqrt{N} \tag{31}\]  
\[\chi_{zz}[\lambda_{1},f_{d1}]\approx\sqrt{N} \tag{32}\]