# Efficient Multipath Clutter Cancellation for UAV Monitoring Using DAB Satellite-Based PBR
## 3 Suppression Algorithm
, \tag{10}\]  
where \(\hat{\mathbf{\theta}}_{f_{\text{flu}}^{\text{du}},b}\) represents the solution of each decomposed problem, for which the expression is  
\[\hat{\mathbf{\theta}}_{f_{\text{flu}}^{\text{du}},b}=\left(\mathbf{C}_{f_{\text{flu}}^{ \text{du}},b}^{H}\mathbf{C}_{f_{\text{flu}}^{\text{du}},b}^{\text{glu}}\right)^{- 1}\mathbf{C}_{f_{\text{flu}}^{\text{du}},b}^{H}\mathbf{s}_{\text{sur},b}. \tag{11}\]  
The above derivation is only tenable when \(\mathbf{C}_{f_{\text{flu}}^{\text{du}},b}^{H}\mathbf{C}_{f_{\text{flu}}^{\text{du}},b}\) is non-singular for all \(b=1,\cdots,B\). Otherwise, the matrix inversion in (11) does not exist. When the oversampling rate is moderately low, such a premise is well guaranteed because \(\mathbf{C}_{f_{\text{flu}}^{\text{du}}}\) contains only the components with the same Doppler frequency, which prevents the rank reduction caused by the decreasing Doppler resolution. Apart from that, with a stationary reference signal, \(\mathbf{C}_{f_{\text{flu}}^{\text{du}},b}^{H}\mathbf{C}_{f_{\text{flu}}^{\text{du}},b}\), and \(\mathbf{C}_{f_{\text{flu}}^{\text{du}}}^{H}\mathbf{C}_{f_{\text{flu}}^{\text{du}}}\) can be approximated by their expectations that differ only in amplitude. Hence, the matrix calculation in (10) can be substituted with a scalar weighting for simplicity at the cost of negligible accuracy loss, of which the expression is as follows:  
\[ \begin{split}\hat{\mathbf{\theta}}_{f_{\text{flu}}^{\text{du}}}& =\sum_{b=1}^{B}\left\|\mathbf{s}_{\text{ref,b}}\odot\mathbf{\phi}_{f_{\text {flu}}^{\text{du}},b}\right\|_{2}^{2}\hat{\mathbf{\theta}}_{f_{\text{flu}}^{\text{du }},b}\\ &=\sum_{b=1}^{B}\left\|\mathbf{s}_{\text{ref,b}}\right\|_{2}^{2}\hat{ \mathbf{\theta}}_{f_{\text{flu}}^{\text{du}},b}\end{split} , \tag{12}\]  
where \(\left\|\ \right\|_{2}\) and \(\odot\) represent the \(l_{2}\) norm and the Hadamard product, respectively. \(\mathbf{s}_{\text{ref}}\odot\mathbf{\phi}_{f_{\text{flu}}^{\text{du}}}\) indicates the vector of a reference signal modulated by the clutter Doppler phase, of which the column partitions are denoted with subscript \(b\) similar to that in (10). Due to its constant modulus, \(\mathbf{\phi}_{f_{\text{flu}}^{\text{clu}}}\) can be ignored during the calculation. When the reference channel has stable received power and (8) is segmented evenly, the weighting coefficients \(\left\|\mathbf{s}_{\text{ref,b}}\right\|_{2}^{2}/\left\|\mathbf{s}_{\text{ref}}\right\| _{2}^{2}\equiv 1/B\) and (12) can be further simplified as follows:  
\[\hat{\mathbf{\theta}}_{f_{\text{flu}}^{\text{clu}}}=\frac{1}{B}\sum_{b=1}^{B}\hat{ \mathbf{\theta}}_{f_{\text{flu}}^{\text{clu}},b}. \tag{13}\]  
The above derivation extends the evaluation stage in ECA-E (S) from stationary clutters to moving ones based on the orthogonality between different sub-clutter bases, with which the optimal output is achievable by independent cancellation of different Doppler frequencies. However, the processing faces the dimension inflation caused by Doppler extension, as \(\hat{\mathbf{\theta}}_{f_{\text{m}}^{\text{du}},b}\) has to be calculated separately for different \(f_{m}^{\text{clu}}\). Further approximations are needed to reduce this computational burden. Notice that the decreasing Doppler resolution drastically enhances the correlation between different \(\hat{\mathbf{\theta}}_{f_{\text{flu}}^{\text{du}},b}\) within the same batch. When \(\left|f_{m}^{\text{clu}}\right|\ll f_{s}B/N\), the following expressions can be taken into (11)  
\[ \begin{split}\mathbf{C}_{f_{\text{flu}}^{\text{du}},b}^{H}\mathbf{C}_{f_{ \text{flu}}^{\text{du}},b}&=\mathbf{C}_{0,b}^{H}\mathbf{C}_{0,b}\\ \mathbf{C}_{f_{\text{flu}}^{\text{du}},b}^{H}\mathbf{s}_{\text{sur},b}& \approx e^{-j2\pi\frac{N\text{flu}}{\beta_{f_{\text{flu}}}}b}\mathbf{C}_{0,b}^{H} \mathbf{s}_{\text{sur},b}.\end{split}