# Co-Channel Interference Suppression for LTE Passive Radar Based on Spatial Feature Cognition
## 3 Algorithm Description
### Co-Channel Interference Suppression based on Spatial Feature Cognition  
On getting the DOA of the interference, we use the beamforming technique to obtain the reference signal of each interference as:  
\[S_{ref_{i}}(n)=WS_{\text{echo}}(n)\;r=1,2,\cdots,N_{R}n=1,\dots,N \tag{15}\]where \(S_{\text{echo}}(n)=[S_{1}(n),S_{2}(n),\cdots,S_{M}(n)]^{\text{T}}\) is the surveillance signal vector received by the antenna array; and W is the weight vector obtained by using the worst-case optimization beamforming algorithm [25], which can extract the reference signal of an interference with a specific DOA and can also suppress the other interferences, thus can get a good INR.  
The signal extracted above can be further purified by re-generating the signal with the signal bits demodulated from \(S_{ref_{t}}(n)\). We denote by \(\widetilde{S}_{ref_{t}}(n)\), \(r=1,2,\cdots,\widetilde{N}_{R}\) the purified signal, where \(\widetilde{N}_{R}\) is the signal sample number of the re-generating signal with the incorrect demodulated bits removed.  
On getting the reference signal \(\widetilde{S}_{ref_{t}}(n)\), ECA-B can be used to suppress the interference signal from a specific CCBS, i.e.,  
\[\mathbf{S}_{\text{c}n}\mathbf{=}\mathbf{S}_{\mathbf{m}}\mathbf{-}\mathbf{S}_ {\mathbf{I}}\left(\mathbf{S}_{\mathbf{I}}^{\text{H}}\mathbf{S}_{\mathbf{I}} \right)^{-1}\mathbf{S}_{\mathbf{I}}^{\text{H}}\mathbf{S}_{\mathbf{m}}\ m=1, \cdots M \tag{16}\]  
where \(\mathbf{S}_{\mathbf{m}}=[S_{m}(0),S_{m}(1),\cdots,S_{m}(N)]^{\text{T}}\) is the surveillance signal received by the antenna element m.  
The expression \(\mathbf{S}_{\mathbf{I}}=\left[\widetilde{\mathbf{S}}_{\mathbf{I}_{1}}, \widetilde{\mathbf{S}}_{\mathbf{I}_{2}},\cdots\widetilde{\mathbf{S}}_{ \widetilde{I}_{\widetilde{N}_{R}}}\right]\) is the subspace matrix of the interference signal constructed by the reference signal; and \(\widetilde{\mathbf{S}}_{\mathbf{I}_{k}}\) is subspace matrix generated by the \(r\)-th reference signal, i.e.,  
\[\widetilde{\mathbf{S}}_{\mathbf{I}_{r}}=\left[\begin{array}{cccc}\widetilde {S}_{ref_{t}}(0-L)&\widetilde{S}_{ref_{t}}(0-L+1)&\cdots&\widetilde{S}_{ref_{ t}}(0-L+2L)\\ \widetilde{S}_{ref_{t}}(1-L)&\widetilde{S}_{ref_{t}}(2-L+1)&\cdots&\widetilde{S }_{ref_{t}}(2-L+2L)\\ \cdots&\cdots&\cdots&\cdots\\ \widetilde{S}_{ref_{t}}(N-L)&\widetilde{S}_{ref_{t}}(N-L+1)&\cdots&\widetilde{ S}_{ref_{t}}(N-L+2L)\\ \end{array}\right] \tag{17}\]  
where \(L\) is the maximum delay bin. In Equation (16), each column of the interference subspace matrix is a delayed replica of the reference signal. The delay can be positive and negative, because the reference signal obtained may corresponding to a multipath signal, not the direct signal.  
The worst-case optimization beamforming technique is exploited again to further suppress the interference residual, i.e.,  
\[S_{\text{echo\_b}}(n)=W\mathbf{S}_{\text{c}}(n)n=1,\ldots,N \tag{18}\]  
where \(\mathbf{S}_{\text{c}}(n)\mathbf{=}\left[\mathbf{S}_{\text{c}_{1}}(n),\mathbf{ S}_{\text{c}_{2}}(n),\cdots\mathbf{S}_{\text{c}_{M}}(n)\right]^{\text{T}}\) is the interference residual after time domain cancellation; \(W\) is the beamforming weight.  
After the temporal and spatial suppression, the cascaded cancellation for one loop is finished. At this step, we estimate the INR of the interference residual remaining in the surveillance signal to assess the interference level. If the INR is still high, i.e., it is larger than a predefined threshold \(\eta\), then the cancellation loop is repeated. According to [10], the INR of the co-channel interference can be estimated as:  
\[INR\approx\frac{L_{sym}y_{\text{xor}}[2048]}{L_{cp}y_{\text{xor}}[0]-L_{sym}y _{\text{xor}}[2048]} \tag{19}\]  
where \(L_{sym}\) is the length of an OFDM symbol and it is \(2048+144\) according to the LTE protocol; \(L_{cp}\) is the length of CP in one OFDM symbol and it is \(144\) in LTE; and \(y_{\text{xor}}[k]\) is expressed as:  
\[y_{\text{xor}}[k]=\sum_{n=1}^{N}S_{\text{echo\_b}}(n)S^{*}_{\text{echo\_b}}(n+k) \tag{20}\]