# Clutter Cancellation and Long Time Integration for GNSS-Based Passive Bistatic Radar
## 3 The Clutter Cancellation and Proposed Long Time Integration Algorithm  
### The Principle of Extensive Cancellation Algorithm  
Unlike the traditional active radar, a reference antenna is required in passive radar for collecting the direct path signal that is transmitted by illuminators as the reference signal to accomplish the cross-correction. The surveillance antenna is mainly applied to capture the target echo; however, it is inevitably polluted by the direct-path and multipath signals. The strong clutter will degrade the target detection performance. Hence, it is critical to remove the clutter from the captured data before cross-correlation.  
The sampled signal in surveillance channel is  
\[\mathbf{s}_{surv}=\left[\begin{array}{cccccc}s_{surv}(0)&s_{surv}(1)&\ldots&s_{ surv}(n)&\ldots&s_{surv}(N-1)\end{array}\right]^{T} \tag{10}\]  
where \(N\) is the number of samples within the CPI.  
Assuming that the clutter echoes are backscattered from the first \(R\) range cells, it is needed to extract the \(N+R-1\) samples of the signal in reference channel:  
\[\mathbf{s}_{ref}=\left[\begin{array}{cccccc}s_{ref}(-R+1)&\ldots&s_{ref}(0)&s_{ ref}(1)&\ldots&s_{ref}(n)&\ldots&s_{ref}(N-1)\end{array}\right]^{T} \tag{11}\]  
The signal model of ECA is for exploring by minimizing the residual signal power after clutter cancellation, thus  
\[\min_{\mathbf{\alpha}}\left\{\left\|\mathbf{s}_{surv}-\mathbf{\Lambda}_{\mathbf{\alpha}} \right\|^{2}\right\} \tag{12}\]  
with  
\[\mathbf{\Lambda}=\textbf{B}\left[\mathbf{\Lambda}_{-P}\textbf{S}_{ref}\ldots\mathbf{ \Lambda}_{-1}\textbf{S}_{ref}\textbf{S}_{ref}\mathbf{\Lambda}_{1}\textbf{S}_{ref} \ldots\mathbf{\Lambda}_{P}\textbf{S}_{ref}\right] \tag{13}\]  
where **B** is an incidence matrix that selects the last \(N\) rows of the matrix,  
\[\textbf{B}=\left\{b_{ij}\right\}_{i=1,\ldots,N\ j=1,\ldots,N+R-1^{\prime}},\ b_{ij}=\begin{cases}1&i=j-R-1\\ 0&\text{otherwise}\end{cases} \tag{14}\]  
\(\mathbf{\Lambda}_{P}\) is the diagonal matrix corresponding to the \(p\)-th Doppler cell.  
\[\textbf{S}_{ref}=\left[\begin{array}{cccccc}\mathbf{s}_{ref}&\textbf{D}\textbf{ s}_{ref}&\textbf{D}^{2}\textbf{s}_{ref}&\ldots&\textbf{D}^{R-1}\textbf{s}_{ ref}\end{array}\right] \tag{15}\]  
where **D** is a \(0/1\) permutation matrix that applies a delay of a single sample that is defined as:  
\[\textbf{D}=\left\{d_{ij}\right\}_{i,j=1,\ldots,N+R-1^{\prime}},\ d_{ij}= \begin{cases}1&i=j+1\\ 0&\text{otherwise}\end{cases} \tag{16}\]  
The columns of the matrix **X** define a \(M\)-dimensional clutter subspace, where \(M=(2P+1)*R\). The filter coefficients \(\mathbf{\alpha}=\left[\alpha_{0}\ \alpha_{1}\ \ldots\ \alpha_{R-1}\right]^{T}\) are evaluated based on the least square approach and it can be expressed as  
\[\mathbf{\alpha}=\left(\textbf{X}^{H}\textbf{X}\right)^{-1}\textbf{X}^{H}\mathbf{s}_{surv} \tag{17}\]The surveillance signal after clutter cancellation can be expressed as  
\[\mathbf{s}_{ECA}=\mathbf{s}_{surv}-\mathbf{\chi}\alpha=\Big{[}\mathbf{I}_{N}-\mathbf{\chi}\Big{(} \mathbf{X}^{H}\mathbf{X}\Big{)}^{-1}\mathbf{X}^{H}\Big{]}\mathbf{s}_{surv}=\mathbf{P} \mathbf{s}_{surv} \tag{18}\]  
where \(\mathbf{P}\) is the projection matrix projecting the surveillance channel signal in the subspace orthogonal to the clutter subspace.  
After clutter cancellation, the received signals in surveillance channel are target echoes and thermal noise. Subsequently, cross-correlation and KT are applied to accomplish the range compression and range cell migrations, respectively.