# Co-Channel Interference Suppression for LTE Passive Radar Based on Spatial Feature Cognition
## 3 Algorithm Description
#### 3.1.2 Spatial Spectrum Reconstruction of the Interference Signal based on the Sparse Estimation  
The CCC result at \(k=2048\) described by Equation (8) can be rewritten as:  
\[\mathbf{Y}=\mathbf{PB+Z} \tag{9}\]  
where:  
\[\mathbf{Y}=\left[y_{1}(2048),y_{2}(2048),\ldots,y_{M}(2048)\right]^{\text{T}}\]  
\[\mathbf{P}=\left[ \begin{array}{cccccc}\text{P}_{1,1,1}&\text{P}_{1,2,1}& \cdots&\text{P}_{1,N_{1}^{1},1}&\text{P}_{2,1,1}&\cdots&\text{P}_{S,N_{l}^{5}, 1}\\ \text{P}_{1,1,2}&\text{P}_{1,2,2}&\cdots&\text{P}_{1,N_{1}^{1},2}&\text{P}_{2,1,2}&\cdots&\text{P}_{S,N_{l}^{5},2}\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \text{P}_{1,1,M}&\text{P}_{1,2,M}&\cdots&\text{P}_{1,N_{l}^{1},M}&\text{P}_{2,1,M}&\cdots&\text{P}_{S,N_{l}^{5},M}\end{array} \right] \tag{10}\]  
is a \(M\times N_{I}\) matrix with each column representing the steering vector of the interference signal, \(N_{I}=\sum\limits_{s=1}^{S}N_{I}^{s}\) is the total number of the interference signal. \[\mathbf{B}=\left[ \begin{array}{cccccc}B_{1,1}&B_{1,2}&\ldots&B_{1,N_{l}^{1} }&B_{2,1}&\ldots&B_{S,N_{l}^{5}}\end{array} \right]^{\text{T}}\]  
Equation (9) describes a typical linear data generating model, with Y representing the measurement data, P the data generating matrix, and B the variable to estimate. The compressed sensing technique can be used to estimate the signal DOA according to model Equation (9). In this paper, the sparse reconstruction method proposed in literature [27] is used to estimate the DOAs of the interference. The basic idea is to solve the \(l_{1}\)-norm minimum optimization problem as follows:  
\[\min_{\mathbf{x}}\lVert\mathbf{Ax}-\mathbf{y}\rVert_{2}^{2}+\lambda\lVert\mathbf{x} \rVert_{1} \tag{11}\]  
where \(\mathbf{x}\) is an M-dimensional vector comprising the amplitude of each interference signal; \(\lVert\mathbf{x}\rVert_{1}\) is the l1-norm of the vector; \(\lambda\) is the penalty coefficient; \(\mathbf{y}\) is the measurement vector; shown previously; and \(A\) is the dictionary matrix, of which each column is a steering vector. corresponding to a specific DOA. For a uniform linear array, the dictionary matrix \(\mathbf{A}\) can be formulated as follows:  
\[\mathbf{A}=\left[ \begin{array}{cccc}\mathbf{e}^{j\pi\sin\left(\frac{-N_{\mathbf{A}}}{2N_{ \mathbf{A}}}\pi\right)}&\mathbf{e}^{j\pi\sin\left(\frac{-N_{\mathbf{A}}\bullet 1}{2N_{\mathbf{A}}}\pi \right)}&\dots&\mathbf{e}^{j\pi\sin\left(\frac{-N_{\mathbf{A}}\bullet 2N_{\mathbf{A}}}{2N_{\mathbf{A}}}\pi \right)}\\ \mathbf{e}^{j2\pi\pi\sin\left(\frac{-N_{\mathbf{A}}}{2N_{\mathbf{A}}}\pi\right)}&\mathbf{e}^{j 2\pi\pi\sin\left(\frac{-N_{\mathbf{A}}\bullet 1}{2N_{\mathbf{A}}}\pi \right)}&\dots&\mathbf{e}^{j2\pi\pi\sin\left(\frac{-N_{\mathbf{A}}\bullet 2N_{\mathbf{A}}}{2N_{\mathbf{A}}}\pi \right)}\\ \dots&\dots&\dots&\dots\\ \mathbf{e}^{j\mathbf{M}\pi\sin\left(\frac{-N_{\mathbf{A}}}{2N_{\mathbf{A}}}\pi\right)}&\mathbf{e} ^{j\mathbf{M}\pi\sin\left(\frac{-N_{\mathbf{A}}\bullet 1}{2N_{\mathbf{A}}}\pi \right)}&\dots&\mathbf{e}^{j\mathbf{M}\pi\sin\left(\frac{-N_{\mathbf{A}}\bullet 2N_{\mathbf{A}}}{2N_{\mathbf{A}}} \pi\right)}\end{array} \right] \tag{12}\]  
where \(2N_{A}\) is the number of grids into which the azimuth space\(-90\sim 90^{\circ}\) is divided. The accuracy of the DOA estimation is related to the grid length. The optimization problem shown in Equation (10) is to obtain an estimate of \(\mathbf{x}\) as sparse as possible and make the mean square error lowest at the same time. Because the co-channel interference signals are widely distributed and the number of the interference signals is greater than the antenna size, Equation (10) cannot get an accurate estimate of the amplitudes for all the interference signals. However, the amplitudes as well as the DOAs of interference signals with relatively great energy can be well estimated. These are the characteristics of the LMSE criteria. Each vector in Equation (10) is a complex vector. In the following we recast the estimation problem in Equation (10) into an equivalent one in the real-valued domain:  
\[