# A Novel Cross-Correlation Algorithm Based on the Differential for Target Detection of Passive Radar
## 2 Materials and Methods
### Algorithm Description and Analysis
#### 2.2.2 Novel Cross Correlation Algorithm  
In this paper, a novel cross correlation processing method is proposed. First, the received echo signal and reference signal are converted into multiple signals by differential operations. At this time, there is a phase difference between the target part and the noise part of the multi-channel signals, so coherent integration between the multi-channel signals cannot be achieved. Therefore, through frequency domain processing, this paper transforms the target part of each channel signal into an in-phase signal and the interference and noise part into a non-in-phase signal, thus realizing the twice coherent processing of multiple target signals and greatly improving the target output SNR. The specific treatment scheme is as follows:  
The reference signal and echo signal will be differentiated as follows:  
\[ \begin{array}{l}r_{ref}[l,n]=s_{ref}[n]s_{ref}^{*}[n-l]\\ =\left|A_{ref}\right|^{2}d[n]d^{*}[n-l]+A_{ref}^{*}z_{ref}[n]d^{*}[n-l]+A_{ref} [n]\cdot z_{ref}^{*}[n-l]+z_{ref}[n]z_{ref}^{*}[n-l]\\ n=1,\ldots,N,l=1,\ldots,N-1\end{array}  \tag{5}\]  
\[ \begin{array}{l}r_{sur}[l,n]=s_{sur}[n]s_{sur}^{*}[n-l]\\ =\sum\limits_{m=1}^{N_{M}}|a_{m}|^{2}d[n-\lambda_{m}]d^{*}[n-\lambda_{m}-l]e^{ j2\pi f_{dm}\frac{l}{\beta_{s}}}+\sum\limits_{m=1}^{N_{M}}\sum\limits_{c=1,c \
eq m}^{N_{M}}a_{m}a_{c}^{*}d[n-\lambda_{m}]d^{*}[n-\lambda_{c}-l]e^{j2\pi f_{ dm}\frac{\eta}{\beta_{s}}}e^{-j2\pi f_{dc}(\frac{\eta}{\beta_{s}}-\frac{l}{\beta_{s}})}\\ +\sum\limits_{m=1}^{N_{M}}a_{m}^{*}z_{sur}[n]d^{*}[n-\lambda_{m}-l]e^{-j2\pi f _{dm}(\frac{\eta}{\beta_{s}}-\frac{l}{\beta_{s}})}+\sum\limits_{m=1}^{N_{M}}a _{m}z_{sur}^{*}[n-l]d[n-\lambda_{m}]e^{j2\pi f_{dm}(\frac{\eta}{\beta_{s}})}+ z_{sur}^{*}[n-l]z_{sur}[n]\\ n=1,\ldots,N,l=1,\ldots,N-1\end{array}  \tag{6}\]  
where \(l=\eta_{s}\). \(r_{ref}[l,n]\), \(r_{sur}[l,n]\), respectively, represent the \(l\)-order difference of the reference signal and the echo signal (that is, to multiply the corresponding point between the signal and the signal after time delay) and \(l\) is an integer less than \(N\). * represents the conjugation of the signal. In (6), the first term represents the conjugate point multiplication of the same target signal and its signal after \(l\)-order delay. The second term represents the conjugate point multiplication of the \(m\)-th target signal and the \(c\)-th target signal after \(l\)-order delay. It can be seen from (6) that after the differential operation, in the first term, the Doppler frequency offset of the target echo becomes a constant term with \(l\). Then, we assume  
\[p_{l}(n)=d[n]d^{*}[n-l] \tag{7}\]  
Equations (5) and (6) can be written as follows:  
\[ \begin{array}{l}r_{ref}[l,n]=s_{ref}[n]\cdot s_{ref}^{*}[n-l]=\left|A_{ref} \right|^{2}p_{l}[n]+z_{r,r}(l,n)\\ n=1,\ldots,N,l=1,\ldots,N-1\end{array}  \tag{8}\]  
\[ \begin{array}{l}\text{where }z_{r,r}[l,n]=A_{ref}^{*}z_{ref}[n]d^{*}[n-l]+A_{ ref}d[n]\cdot z_{ref}^{*}[n-l]+z_{ref}[n]z_{ref}^{*}[n-l].\end{array}  \tag{9}\]  
\[ \begin{array}{l}r_{sur}[l,n]=s_{sur}[n]\cdot s_{sur}^{*}[n-l]=\sum\limits_{ m=1}^{N_{M}}|a_{m}|^{2}e^{j2\pi f_{dm}\frac{l}{\beta_{s}}}p_{l}[n-\lambda_{m}]+ \zeta[l,n]\\ n=1,\ldots,N,l=1,\ldots,N-1\end{array}  \tag{10}\]  
where  
\[ \begin{array}{l}\zeta[l,n]=\sum\limits_{m=1}^{N_{M}}\sum\limits_{c=1\
eq m }^{N_{M}}a_{m}a_{c}^{*}d[n-\lambda_{m}]d^{*}[n-\lambda_{c}-l]e^{j2\pi f_{dm} \frac{\eta}{\beta_{s}}}e^{-j2\pi f_{dc}(\frac{\eta}{\beta_{s}}-\frac{l}{\beta_ {s}})}\\ +\sum\limits_{m=1}^{N_{M}}a_{m}^{*}z_{sur}[n]d^{*}[n-\lambda_{m}-l]e^{-j2\pi f _{dm}(\frac{\eta}{\beta_{s}}-\frac{l}{\beta_{s}})}+\sum\limits_{m=1}^{N_{M}}a _{m}z_{sur}^{*}[n-l]d[n-\lambda_{m}]e^{j2\pi f_{dm}(\frac{\eta}{\beta_{s}})}+ z_{sur}^{*}[n-l]z_{sur}[n]\\ n=1,\ldots,N,l=1,\ldots,N-1\end{array}  \tag{11}\]The Fourier transformation of (8) and (9) results in (11) and (12):  
\[ \begin{array}{l}R_{ref}[l,k]=FFT\Big{(}r_{ref}(l,n)\Big{)}=\Big{|}A_{ref}\Big{|} ^{2}P_{l}(k)+Z_{r,r}(l,k)\\ k=1,\ldots,N,l=1,\ldots,N-1\end{array}