## 2 UWB MIMO Radar System
### MIMO Array Topology Design  
The design of the MIMO array is related to the complexity and cost of the radar system and directly affects the imaging quality. In designing MIMO array configurations, sparse array designs are often used to reduce system complexity and the number of antenna elements. In order to simplify the design of MIMO arrays, we first discuss the design factors from the following aspects. Firstly, the azimuth resolution is determined by the aperture length of the array. Therefore, the azimuth resolution can be improved by properly increasing the aperture length of the array. Secondly, after increasing the length of the array aperture, in order to satisfy the ideal imaging performance, the spacing and total number of antenna elements will also change. Specifically, due to the sparsity of the MIMO array, grating lobes and sidelobes may be generated in the imaging results, which seriously affect the quality of radar imaging. According to Nyquist sampling theory, the element spacing should be less than half of the radar wavelength in order to prevent unwanted grating lobes. On the other hand, with the decrease of the element spacing and the increase of the number of antennas, the direct coupling among the antennas will become stronger and high-level sidelobes will occur [29; 30]. Therefore, considering the above factors, the antennas should keep a certain distance and the number of antennas should not be too large. Finally, the imaging performance of UWB MIMO radar is related not only to the array configuration and number of elements, but also to the form of transmitted signals and application scenes. In this paper, a UWB SFCW signal is used to detect the short-range targets behind the wall. Therefore, it is necessary to consider the applicability of near-field conditions in through-wall detection. Generally, the longitudinal distance from the target to the MIMO array is much larger than the lateral distance between the target and the array elements. According to the verification results of [20; 29; 30; 31; 32], the focused SAR imaging and PSF results of MIMO sparse arrays and monostatic uniform arrays have almost the same performance in near-field conditions. Therefore, as for the MIMO through-wall radar system in this paper, the virtual array theory is applied to simplify the design and analysis of the MIMO sparse array. For a bistatic MIMO sparse array which contains \(M\) transmitters and \(N\) receivers, according to the theory of displaced phase center (DPC) approximation [18; 33], two paths of the \(m\)-th transmitter at the \(x_{m}\)-th position and the \(n\)-th receiver at \(x_{n}\)-th position are switched on, while all the other paths are switched off. This is equivalent (in far field) to transmitting and receiving with one single 'virtual' antenna in the median position (\(x_{m}+x_{n}\))/2 of the axis. The midpoints of each transmit and receive element are regarded as virtual elements for a linear equivalent monostatic array. Hence, for each combination of the \(m\)-th and \(n\)-th antennas, a specific pattern along the median axis is defined. As a result, we can first obtain an equivalent uniform array with \(MN\) (\(MN=M\times N\)) monostatic transceivers, which has expected resolution and low-level grating/side lobes. Then, corresponding to the reverse thinking process mentioned above, the MIMO sparse array can be designed by factorization. In this case, a synthetic aperture imaging algorithm, such as the BP-based imaging algorithm, can be directly used to image the targets. Based on the above analysis, the design steps of the MIMO array, which are illustrated in Figure 3, are as follows: Firstly, we begin with designing a linear monostatic uniform equivalent array. The effective aperture size \(L\) relative to the wavelength of the center frequency is determined by the required cross-range resolution \(\delta_{a}=\lambda R_{0}/2L\), where \(\lambda\) is wavelength, \(L\) is equivalent uniform array length, and \(R_{0}\) represents the potential distance of the target.