## 3 Through-Wall Imaging Method with the Improved BP-Based Algorithm  
The traditional cross-correlation based TDBP algorithm is simple and practical in free space. However, when considering the existence of a concrete wall with 22.5 cm thickness, the imaging results focused directly by the BP imaging algorithm will cause errors between the imaged positions and the true positions.  
Generally speaking, the influence of the wall on the location of target imaging is mainly due to the following two reasons: firstly, the slow propagation of EM waves in the wall produce additional  
Figure 11: Photograph of designed MIMO radar prototype.  
Figure 10: Total gain as a function of frequency of the designed Vivaldi antenna.  
time delay. This causes the targets' positions in the imaging results to be farther from than their actual positions. Secondly, the refraction and scattering of EM waves inside and outside the wall mean that the original propagation path is not a straight line. These errors are mainly affected by the relative dielectric constant of the wall. The above two points are briefly illustrated in Figure 12. In this paper, we propose an improved TDBP algorithm to reconstruct target locations behind the wall from MIMO sparse array radar data.  
In the TDBP algorithm, the travel time for the wave field to propagate from the transmitter to the target and scatter back to the receiver must be known. In [37], we used ray tracing methods to efficiently calculate travel time in the TDBP algorithm for SIMO radar data. Assume that there are M paths passing through the wall from one antenna. The point \(p\) in the imaging area is a pixel on the \(m\)-th path. The incident angle is \(\alpha_{1}\). The refraction angle is \(\alpha_{2}\). The average propagation velocity of an EM wave inside the wall is \(v_{w}\). The relative dielectric constant of the wall is \(\varepsilon_{w}\). The thickness of the wall is \(d_{2}\). The distance from the equivalent phase center of MIMO antenna array to the surface of the wall is \(d_{3}\). The distance from the point \(p\) to the wall is \(d_{1}\). According to [PERSON]'s law, we can obtain  
\[\frac{\sin\alpha_{1}}{\sin\alpha_{2}}=\frac{c}{v_{w}} \tag{7}\]  
where \(v_{w}=c\sqrt{1/\varepsilon_{w}}\).  
The travel time from the one antenna to the point \(p\) is  
\[T_{p}=\left(\frac{d_{1}+d_{3}}{\cos\alpha_{1}}+\frac{d_{2}\sqrt{ \varepsilon_{w}}}{\cos\alpha_{2}}\right)\!\!\frac{1}{c} \tag{8}\]  
Then, the one-way travel time of all \(M\) propagation paths can be calculated in the same way, and the travel time of any point behind the wall can be obtained by interpolation from the M paths. The one-way travel time of any antenna can be computed similarly. Considering that there may be a large number of antennas in a MIMO array (16 in this paper), we designed a parallel algorithm on the MATLAB platform to improve the computational efficiency of one-way travel time (Figure 13, Part I). Assume the one-way travel time in the imaging area from the transmitter \(T_{x1}\) is \(T_{t1}\) and the receiver \(R_{x1}\) is \(T_{r1}\). For a pair of transmitting and receiving antennas, \(T_{x1}\) and \(R_{x1}\), the whole travel time behind the wall can be obtained by \(T_{1}=T_{r1}+T_{t1}\). According to the designed sparse MIMO array, we can  
Figure 12: Imaging geometry for through-wall radar (TWR) with the designed MIMO array.  
obtain the whole travel time from all transmitters and receivers (at the pair of \(T_{x1}\)-\(R_{x2}\), \(R_{x3}\),... \(R_{x8}\) to \(T_{x8}-R_{x8}\)) to the imaging area behind the wall (Figure 13, Part II).  
Finally, the TDBP algorithm can be directly applied to focus the data received from the MIMO antennas and reconstruct the target location (Figure 13, Part III). In this paper, \(M=501\) ray paths for every antenna are constructed, and the imaging range does not include the wall.