# Efficient Multipath Clutter Cancellation for UAV Monitoring Using DAB Satellite-Based PBR
## 3 Suppression Algorithm
, \tag{14}\]and \(\hat{\theta}_{f^{\text{clu}}_{\text{fl}},b}\) becomes  
\[\hat{\theta}_{f^{\text{clu}}_{\text{fl}},b} \approx e^{-j2\pi\frac{N\ell^{\text{clu}}_{\text{fl}}}{B_{\text{fl}} }b}\Big{(}\mathbf{C}^{H}_{0,b}\mathbf{C}_{0,b}\Big{)}^{-1}\mathbf{C}^{H}_{0,b}\mathbf{s}_{\text{sur },b} \tag{15}\] \[=e^{-j2\pi\frac{N\ell^{\text{clu}}_{\text{fl}}}{B_{\text{fl}}}b} \hat{\theta}_{0,b}\]  
Thus, only the zero-Doppler estimations need to be solved at each batch, and the coefficients of moving clutters can be obtained by  
\[\hat{\theta}_{f^{\text{clu}}_{\text{fl}}}=\frac{1}{B}\sum_{b=1}^{B}e^{-j2\pi \frac{N\ell^{\text{clu}}_{\text{fl}}}{B_{\text{fl}}}b}\hat{\theta}_{0,b}. \tag{16}\]  
With (16), the redundant calculation in the evaluation stage is well eliminated. The same approximation can be applied to the cancellation stage as well, where the following vector is reconstructed to achieve a suppression factor of \(H\Big{(}f^{\text{clu}}_{\text{m}}\Big{)}\) at specific Doppler frequencies:  
\[\hat{\epsilon}^{\prime}_{b}=\mathbf{C}_{0,b}\sum_{m=1}^{M}e^{j2\pi\frac{N\ell^{ \text{clu}}_{\text{fl}}}{B_{\text{fl}}}b}H\Big{(}f^{\text{clu}}_{\text{m}} \Big{)}\hat{\theta}_{f^{\text{clu}}_{\text{fl}}}. \tag{17}\]  
The concatenated vector of (17), \(\hat{\epsilon}^{\prime}=\big{[} \begin{array}{cccc}\hat{\epsilon}^{\prime \intercal}_{1}&\dots&\hat{\epsilon}^{\prime\intercal}_{B}\end{array} \big{]}^{T}\), can be regarded as a substitution of \(\hat{\epsilon}\) with a stepped Doppler phase instead of a continuous one. Since \(\left|f^{\text{clu}}_{\text{m}}\right|\ll f_{\text{s}}B/N\), \(\hat{\epsilon}^{\prime}\), and \(\hat{\epsilon}\) have similar outputs after 2D-CCF, its impact on detection is negligible. In fact, \(\mathbf{C}_{0,b}\) can be viewed as a low rank approximation of the corresponding column partition of \(\mathbf{C}\), and (17) is the result of projection. Considering that \(f^{\text{clu}}_{\text{m}}/f_{\text{s}}\) is an \(N\)-dimensional Fourier basis, (17) can be reformulated with a circular filter:  
\[\hat{\epsilon}^{\prime}_{b} =\mathbf{C}_{0,b}\overline{\theta}_{b} \tag{18}\] \[=\mathbf{C}_{0,b}\sum_{l=0}^{B}\tilde{h}[b-l]\hat{\theta}_{0,l}^{ \intercal}\]  
where \(\overline{\theta}_{b}\) is the output filtered by the following weights:  
\[\tilde{h}[b]=\sum_{m=1}^{M}e^{j2\pi\frac{N\ell^{\text{clu}}_{\text{fl}}}{B_{ \text{fl}}}b}H\Big{(}f^{\text{clu}}_{\text{m}}\Big{)}. \tag{19}\]  
According to the former analysis, an ECA version based on batch segmentation and circular filter is derived. As discussed in [34], reducing the sampling length decreases the Doppler resolution, bringing about a higher correlation between the low-Doppler clutter bases and the zero-Doppler one. As a result, each batch's evaluation contains more nonzero-frequency components, some of which are eliminated by the proposed algorithm, named ECA-Circular Filter (ECA-CF), to control the notch width and to retain the target's energy. Unlike ECA-B, ECA-CF tends to segment the data thoroughly to enlarge the Doppler tolerance range, for which the suppression area is mainly determined by the filter weights instead of the batches' number. ECA-ES in [35] can be regarded as a particular case of ECA-CF with a constant filter weight of \(\tilde{h}[b]\equiv 1/B\), of which the passband determines that ECA-ES cannot be applied to situations requiring Doppler extension. In contrast, ECA-CF makes clutter cancellation more flexible as the filter weights in (18) can be adjusted separately for different range bins if varying rejection ranges in the Doppler domain are required in practical processing. Although the nonconstant filter weights increase a certain amount of calculation burden, it can be neglected compared with coefficients evaluation as the cost of matrix factorization is much higher. The above derivation is based on the hypothesis that the extended clutter base only contains the components satisfying \(N/f^{\text{m}}_{\text{clu}}/fs\in Z\).