# High-Resolution Through-the-Wall Radar Imaging with Exploitation of Target Structure
## 2 Signal Model
### Interior Wall Multipath  
While the original signals reflect once or more times at the walls, the target returns at different propagation paths will be received with different delays. As shown in Figure 1, in the scenario of radar detection, we consider the dominant multipath component generated by the reflections at the internal surface. As indicated by the black and blue arrows, the signal is directly transmitted from the transceiver to the target, denoted by \(T_{p}\), and then the scattered echo reflects back to the transceiver and vice versa. The direct return path is indicated by the red arrow. Assuming the geometry information is accessible, the time delays corresponding to the different multipath returns can be accurately acquired according to geometric relationships. Figure 1: Direct path (red arrows) and multipath propagation (blue arrows) model. When there exists one direct path and \(K-1\) indirect paths, the signal propagation model can be considered as the superposition of all propagation paths and formulated as [20]  
\[\mathbf{\hat{y}}=\mathbf{\Phi}_{1}\mathbf{w}_{1}+\mathbf{\Phi}_{2}\mathbf{w}_{2}+ \cdots+\mathbf{\Phi}_{K}\mathbf{w}_{K}, \tag{4}\]  
where \(\mathbf{\Phi}_{k}\in\mathcal{C}^{MN\times N_{k}N_{y}}\) and \(\mathbf{w}_{k}\in\mathcal{C}^{N_{k}N_{y}}\) with \(k\in\{1,\cdots K\}\) represent the sensing matrix and the complex reflectivities mentioned in Section 2.1 of the \(k\)th propagation path, respectively. When \(k=1\), \(\mathbf{\Phi}_{k}\) and \(\mathbf{w}_{k}\) correspond to the direct path. In addition, the remaining \(\mathbf{\Phi}_{k}\) and \(\mathbf{w}_{k}\) correspond to the indirect path, where the elements of \(\mathbf{\Phi}_{k}\) are phase terms, similar to (3), containing the corresponding time delay. It is worth mentioning that the elements of target reflectivity vectors \(\mathbf{w}_{k},k\in\{1,\cdots K\}\) have the same nonzero support on all propagation paths regardless of the different reflectivity coefficients. Since the measurements of different propagation paths generally cannot be solved in TWRI, to overcome this difficulty, we rewrite (4) as  
\[\mathbf{\hat{y}} =\mathbf{\Psi}\mathbf{\hat{w}}+\mathbf{e}, \tag{5}\] \[\mathbf{\hat{w}} =[\mathbf{w}_{1},\cdots,\mathbf{w}_{K}]^{T}\in\mathcal{C}^{N_{k} N_{y}K},\] (6) \[\mathbf{w}_{k} =[w_{k1},\cdots,w_{k,NxN_{y}}]\in\mathcal{C}^{1\times N_{k}N_{y}}, \tag{7}\]  
where \(\mathbf{\hat{y}}\in\mathcal{C}^{MN}\) denotes the measurement data vector, \(\mathbf{\Psi}=[\mathbf{\Phi}_{1},\cdots,\mathbf{\Phi}_{K}]\in\mathcal{C}^{MN \times N_{k}N_{y}K}\) denotes the joint sensing matrix, and \(\mathbf{\hat{w}}\) denotes the combined complex reflectivity vector. For generalization, (2) plus a measurement noise vector \(\mathbf{e}\in\mathcal{C}^{MN}\) is considered as the measurements. Within the CS framework, only a portion of the full measurements is used for high-quality sparse reconstruction. Therefore, we first decrease the amount of data collected. Mathematically, the data reduction operation is equivalent to implementing a measurement or downsampling matrix \(\mathbf{D}\in\mathcal{R}^{J\times MN}\) on the total measurements, where \(J\ll MN\) is the number of reduced measurements. A binary measurement matrix \(\mathbf{D}\in\{0,1\}^{J\times MN}\) is a proper option for the stepped-frequency radar system in this article [20]. Therefore, the measurement vector after undersampling \(\mathbf{\hat{y}}\in\mathcal{C}^{I}\) can be given by  
\[\mathbf{\hat{y}}=\mathbf{D}\mathbf{\hat{y}}=\mathbf{D}\mathbf{ \hat{w}}+\mathbf{D}\mathbf{e} =\mathbf{A}\mathbf{\hat{w}}+\mathbf{\hat{e}}, \tag{8}\] \[\mathbf{A}=\mathbf{D}\mathbf{\hat{\tau}} \in\mathcal{C}^{I\times N_{k}N_{y}},\] (9) \[\mathbf{\hat{\epsilon}} =\mathbf{D}\mathbf{e}\in\mathcal{C}^{I}, \tag{10}\]  
where \(\mathbf{A}  
is the precision (inverse variance) of complex Gaussian distribution, and \(\delta(\cdot)\) is the Dirac delta function.