# Efficient Multipath Clutter Cancellation for UAV Monitoring Using DAB Satellite-Based PBR
## 3 Suppression Algorithm  
An advanced ECA version is proposed to realize efficient and flexible multipath clutter suppression in the focused scenario. Let \(f_{s}\) denote the sampling frequency of the PBR system; the matrix form of (2) after discrete sampling becomes  
\[\mathbf{s}_{\text{sur}}=\mathbf{C}\mathbf{\theta}+\mathbf{T}\mathbf{\alpha}+\mathbf{w}, \tag{4}\]  
where \(\mathbf{s}_{\text{sur}}\) is a \(N\times 1\) vector containing the surveillance signal. \(\mathbf{C}\) and \(\mathbf{T}\) represent the clutter base and target base, for which the column vectors are built with replicas of the reference signal with different delays and Doppler frequencies. \(\mathbf{\theta}\) and \(\mathbf{\alpha}\) are the corresponding coefficient vectors. \(\mathbf{w}\) denotes the noise vector. To achieve effective clutter suppression, the standard ECA evaluates the cancellation weights by solving the following LS problem:  
\[\mathbf{\hat{\theta}}=\left(\mathbf{C}^{H}\mathbf{C}\right)^{-1}\mathbf{C}^{H}\mathbf{s}_{\text{ sur}}, \tag{5}\]  
with which the restored clutter component can be obtained by \(\mathbf{\hat{\epsilon}}=\mathbf{C}\mathbf{\hat{\theta}}\) and the signal after suppression turns into \(\mathbf{s}_{\text{ECA}}=\mathbf{s}_{\text{sur}}-\mathbf{\hat{\epsilon}}\). When the base matrices \(\mathbf{C}\) and \(\mathbf{T}\) are orthogonal to each other, it can be proven that the clutter is perfectly canceled, and \(\mathbf{s}_{\text{ECA}}\) only consists of the target echo and the noise within the orthogonal space of \(\mathbf{C}\), the expression of which is as follows:  
\[\mathbf{s}_{\text{ECA}}=\mathbf{T}\mathbf{\alpha}+\left(\mathbf{I}-\mathbf{C}\Big{(}\mathbf{C}^{H}\bm {C}\Big{)}^{-1}\mathbf{C}^{H}\right)\mathbf{w}. \tag{6}\]  
In order to reduce the matrix dimension, the extended clutter base \(\mathbf{C}\) and estimated coefficient vector \(\mathbf{\hat{\theta}}\) are divided according to Doppler frequency, i.e.,  
\[\mathbf{C} =\left[ \begin{array}{ccc}\mathbf{C}_{f_{1}^{\text{clu}}}&\dots&\bm {C}_{f_{\text{clu}}}\\ \mathbf{\hat{\theta}}&=\left[\begin{array}{ccc}\mathbf{\hat{\theta}}_{f_{1}^{\text{ clu}}}&\dots&\mathbf{\hat{\theta}}_{f_{\text{clu}}}^{\text{clu}}\\ \end{array}\right]^{T},\end{array} \right. \tag{7}\]  
where the subscripts denote the Doppler of each block. Without losing generality, the size of each sub-clutter base, or \(\mathbf{C}_{f_{\text{m}}^{\text{clu}}}\), is assumed to be \(N\times K\), fro which the column vectors correspond to the \(k\)th range bin in order. When \(Nf_{m}^{\text{clu}}/f_{s}\) is an integer for all \(m\), \(\mathbf{C}^{H}\mathbf{C}\) could be approximated with a block diagonal matrix. Thus, \(\mathbf{\hat{\theta}}_{f_{\text{m}}^{\text{clu}}}\) can be estimated separately:  
\[\mathbf{\hat{\theta}}_{f_{\text{m}}^{\text{clu}}}=\Big{(}\mathbf{C}_{f_{\text{m}}^{ \text{clu}}}^{H}\mathbf{C}_{f_{\text{m}}^{\text{clu}}}\Big{)}^{-1}\mathbf{C}_{f_{\text {m}}^{\text{clu}}}^{H}\mathbf{s}_{\text{sur}}. \tag{8}\]  
Similar to [35], the surveillance vector and the sub-clutter base are partitioned into \(B\) batches along the column, i.e.,  
\[\mathbf{s}_{\text{sur}} =\left[ \begin{array}{ccc}\mathbf{s}_{\text{sur},1}^{T}&\cdots&\mathbf{ s}_{\text{sur},B}^{T}\\ \end{array} \right]^{T} \tag{9}\] \[\mathbf{C}_{f_{\text{m}}^{\text{clu}}} =\left[ \begin{array}{ccc}\mathbf{C}_{f_{\text{m}}^{\text{clu}},1}^{ T}&\cdots&\mathbf{C}_{f_{\text{m}}^{\text{clu}},B}^{T}\\ \end{array} \right]^{T},\]and the decomposed form of (8) is given by  
\[ \begin{split}\hat{\mathbf{\theta}}_{f_{\text{flu}}^{\text{du}}}& =\sum_{b=1}^{B}\left(\mathbf{C}_{f_{\text{flu}}}^{H}\mathbf{C}_{f_{\text{flu}}^{ \text{du}}}\right)^{-1}\mathbf{C}_{f_{\text{flu}}^{\text{du}},b}^{H}\mathbf{s}_{\text{sur },b}\\ &=\sum_{b=1}^{B}\left(\mathbf{C}_{f_{\text{flu}}}^{H}\mathbf{C}_{f_{\text{flu }}^{\text{du}}}\right)^{-1}\mathbf{C}_{f_{\text{flu}}^{\text{du}},b}^{H}\mathbf{C}_{f_{ \text{flu}}^{\text{du}},b}^{H}\mathbf{\hat{\theta}}_{f_{\text{flu}}^{\text{du}},b} \end{split}