# Efficient Multipath Clutter Cancellation for UAV Monitoring Using DAB Satellite-Based PBR
## 4 Performance Analysis
### Suppression Capability  
Notice that ECA-C/LF can be regarded as a linear, time-invariant system once the reference signal is determined, ensuring that the contribution from different clutters/targets to the cancellation vector could be calculated independently. Without loss of generality, only the on-grid echoes are discussed in this section, and the suppression capability of ECA-CF could be demonstrated based on the following simplified hypotheses similar to the following [34]:  
1. The surveillance channel includes a stationary clutter and a moving target/clutter at the same range bin with a Doppler frequency of \(f_{d}\), in which the sampled vector becomes the following: \[\mathbf{s}_{\text{sur}}=a_{c}\mathbf{d}_{0}+a_{t}\mathbf{d}_{f_{d}}+\mathbf{w},\] (25) where \(a_{c}\) and \(a_{t}\) are the complex amplitudes. 2. The correlation between the reference signal's delayed replicas is negligible. Thus, the clutter base turns into a column vector: \[\mathbf{c}=a_{\text{ref}}\mathbf{d}_{0}\] (26)
3. The segmentation used by the 1D-CCF calculation is consistent with that of ECA-C/LF's evaluation. With the above assumptions, the estimation of each batch, or \(\hat{\mathbf{\theta}}_{0,b}\), is reduced to a single complex coefficient:  
\[\hat{\theta}_{0,b} =\frac{a_{\text{ref}}^{H}\mathbf{d}_{0,b}^{H}\left[a_{c}\mathbf{d}_{0,b}+a _{t}\mathbf{d}_{f_{d},b}+\mathbf{w}_{b}\right]}{\left|a_{\text{ref}}\right|^{2}\mathbf{d} _{0,b}^{H}\mathbf{d}_{0,b}}\, \tag{27}\] \[=\frac{a_{\text{ref}}^{H}}{\left|a_{\text{ref}}\right|^{2}}[a_{c} +a_{t}\chi_{b}(f_{d})+w_{b}]\]  
where \(w_{b}\) indicates the noise component and \(\chi_{b}(f_{d})\) represents the Doppler slice of the \(b\)th batch's normalized ambiguity function without relative delay. For a constant modulus signal with normalized amplitude, it can be expressed as follows:  
\[\chi_{b}(f_{d}) =\frac{B\sin\left(\pi\frac{Nf_{d}}{Bf_{s}}\right)}{N\sin\left(\pi \frac{f_{d}}{f_{s}}\right)}e^{j\pi\frac{\left(2 Nb+N-B\right)f_{d}}{Bf_{s}}}. \tag{28}\]  
Figure 3: Total and average Flops of different ECA versions. Equation (27) needs to be smoothed with a circular filter before clutter suppression. Assuming that \(Nf_{d}/f_{s}\in Z\) and \(\sum\limits_{l=0}^{B}\tilde{h}[l]=1\), the cancellation vector for each batch can be reconstructed as follows:  
\[ \begin{split}\tilde{e}_{b}^{\prime}&=a_{\text{ref}} \mathbf{d}_{0,b}\sum\limits_{l=0}^{B}\hat{\theta}_{0,l}\tilde{h}[b-l]\\ &=\mathbf{d}_{0,b}\bigg{[}a_{c}+w_{b}^{\prime}+a_{t}\chi_{0}(f_{d})e^ {\jmath\pi\frac{Nf_{d}}{2\beta f_{s}}b}H(f_{d})\bigg{]}^{\prime}\end{split}  \tag{29}\]  
where \(w_{b}^{\prime}\) is the filtered noise component and \(H(f_{d})\) denotes the frequency response of \(\tilde{h}[b]\), the definitions of which are as follows:  
\[ \begin{split} w_{b}^{\prime}&=\sum\limits_{l=0}^{B- 1}\tilde{h}[b-l]w_{l}\\ H(f_{d})&=\sum\limits_{b=1}^{B}h[b]e^{-\jmath \pi\frac{Nf_{d}}{2\beta f_{s}}b}\end{split}  \tag{30}\]  
With the above formulation, the 1D-CCF of each batch's canceled signal is  
\[\mathbf{c}_{b}^{H}\big{(}\mathbf{s}_{\text{sur},b}-\tilde{e}_{b}^{\prime}\big{)}=\frac {Na_{\text{ref}}^{H}}{B}\bigg{\{}a_{t}\chi_{0}(f_{d})e^{\jmath\pi\frac{Nf_{d}} {2\beta f_{s}}b}[1-H(f_{d})]+\big{(}w_{b}-w_{b}^{\prime}\big{)}\bigg{\}}, \tag{31}\]  
and the output of inter-batch accumulation compensated with a Doppler of \(f_{d}\) can be expressed as  
\[\sum\limits_{b=1}^{B}e^{-\jmath\pi\frac{Nf_{d}}{2\beta f_{s}}b}\mathbf{c}_{b}^{H} \big{(}\mathbf{s}_{\text{sur},b}-\tilde{e}_{b}^{\prime}\big{)}=Na_{\text{ref}}^{ H}a_{t}\chi_{0}(f_{d})[1-H(f_{d})]+w^{\prime\prime}. \tag{32}\]  
Compared with the ideal 2D-CCF output \(Na_{\text{ref}}^{H}a_{t}\), the first term in (32) has a amplitude attenuation of  
\[L(f_{d})=|\chi_{0}(f_{d})[1-H(f_{d})]|. \tag{33}\]  
The above derivation applies to any \(f_{d}\) for ECA-LF as the filter in (29) is replaced with a linear one.