# 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
 \tag{11}\]  
where \(FFT(\cdot)\) represent the Fourier transform and \(P_{l}(k)\), \(Z_{r,r}(l,k)\) represent the Fourier transform of \(p_{l}(n)\), \(z_{r,r}(l,n)\), respectively. \[ \begin{array}{l}R_{sur}[l,k]=FFT[r_{sur}(l,n)]=\sum\limits_{m=1}^{N_{M}}|a_{ m}|^{2}e^{2\pi f_{dm}\frac{l}{\gamma_{s}}}P_{l}(k)e^{-2\pi\lambda_{m}k}+FFT( \zeta[l,n])\end{array}  \tag{12}\]  
Multiplying the conjugate points of (11) and (12) gives  
\[ \begin{array}{l}R[l,k]=R_{sur}[l,k]R_{ref}^{*}[l,k]=\Big{|}A_{ref}\Big{|}^{2 }\sum\limits_{m=1}^{N_{M}}|a_{m}|^{2}e^{2\pi f_{dm}\frac{l}{\gamma_{s}}}P_{l} (k)P_{l}^{*}(k)e^{-2\pi\lambda_{m}k}+H(l,k)\end{array}  \tag{13}\]  
where \(H(l,k)\) is shown as follows:  
\[ \begin{array}{l}H[l,k]=Z_{r,r}^{*}(l,k)\cdot\sum\limits_{m=1}^{N_{M}}|a_{m}| ^{2}e^{j2\pi f_{dm}\frac{l}{\gamma_{s}}}P_{l}(k)e^{-j2\pi\lambda_{m}k}+Z_{r,r} ^{*}(l,k)\cdot FFT(\zeta[l,n])+\Big{|}A_{ref}\Big{|}^{2}P_{l}^{*}(k)\cdot FFT (\zeta[l,n])\end{array}  \tag{14}\]  
As can be seen from Equation (13), the phase of the transmitted signal contained in the first item is eliminated and transformed into a constant sequence \(P_{l}(k)P_{l}^{*}(k)\), and there are only complex phases caused by the target delay and Doppler. Thus, in \(\Big{|}A_{ref}\Big{|}\sum\limits_{m=1}^{N_{M}}|a_{m}|^{2}e^{j2\pi f_{dm}\frac {l}{\gamma_{s}}}P_{l}(k)P_{l}^{*}(k)e^{-j2\pi\lambda_{m}k}\), there are phase differences \(e^{-j2\pi\lambda_{m}k}\) that are only related to the time delay for various differential orders \(l\) and there are phase differences \(e^{j2\pi f_{dm}\frac{l}{\gamma_{s}}}\) only  
When \(\lambda=\lambda_{m}\) and \(f=f_{dm}\), the maximum peak in Equation (17) is obtained, as shown in (18),  
\[V_{m}=\Big{|}A_{ref}\Big{|}^{2}|a_{m}|^{2}\sum\limits_{l=1}^{N-1}D_{sum,l}+\sum \limits_{l=1}^{N-1}\sum\limits_{k=1}^{N}H(l,k)e^{-2\pi f\frac{l}{f_{s}}}e^{j2\pi \lambda k} \tag{18}\]  
where \(D_{sum,l}=\sum\limits_{k=1}^{N}D_{l}\). The target echo becomes constant, allowing twice coherent accumulation to be achieved, whereas the noise and interference part achieves twice incoherent accumulation (the specific analysis can be seen in Part IV Algorithm Performance Analysis). Thus, the target time delay and Doppler can be accurately estimated. In order to realize the above delay Doppler compensation, the two-dimensional Fourier transform is adopted in this paper. For different orders \(l\), the row vector takes the order \(l\) as the variable with a constant phase difference of \(e^{-\beta\pi\lambda_{m}k}\) and the column vector takes the discrete frequency \(k\) as the variable with a constant phase difference of \(e^{j2\pi f_{dm}\frac{l}{f_{s}}}\). A two-dimensional matrix with \(l\) and \(k\) as the variables can be obtained through Equation (13), and the maximum dimensions of this matrix are given by the \((N-1)\times N\) dimensional matrix, as shown in (19),  
\[ \begin{bmatrix}R[1,1]&\cdots&R[1,N-1]\\ R[2,1]&\cdots&R[2,N-1]\\ \vdots&\ddots&\vdots\\ R[N-1,1]&\cdots&R[N-1,N-1]\\ R[N,1]&\cdots&R[N,N-1]\\ \end{bmatrix}  \tag{19}\]  
After a two-dimensional Fourier transform is performed in (19), we can obtain (20),  
\[ \begin{array}{l}Q[\lambda,f]=\sum\limits_{l=1}^{N-1}\sum\limits_{k=1}^{N}R( l,k)\exp\left[-j2\pi(f\frac{l}{f_{s}}+\lambda k)\right]\\ =\sum\limits_{l=1}^{N-1}\sum\limits_{k=1}^{N}\Bigg{\{}A_{ref}^{2}\sum\limits_{ m=1}^{N_{M}}\bigg{\{}a_{m}^{2}D_{l}e^{j2\pi(f-f_{dm})\frac{l}{f_{s}}}e^{-j2\pi( \lambda_{m}+\lambda)k}\bigg{\}}+H(l,k)\exp\left[-j2\pi(f\frac{l}{f_{s}}+ \lambda k)\right]\bigg{\}}\end{array}  \tag{20}\]  
where \(\lambda\) and \(f\) are the time delay and frequency offset search compensation factors, respectively, and (20) can be used to obtain the peak value when \(f=f_{dm}\) and \(\lambda=\lambda_{m}\) so as to obtain an accurate time delay and the Doppler estimation of the target.