## 2 UWB MIMO Radar System
### SFCW Signal Model and Conventional BP Algorithm  
For the SFCW based radar system, the range resolution is determined by the frequency bandwidth. The UWB signal can be used to obtain higher range resolution with a low hardware requirement. Therefore, we utilize the UWB SFCW signal model in this paper.  
Figure 2 shows a linear MIMO sparse array with an SFCW signal, which contains M transmitters and N receivers. Assuming the \(m\)-th transmitter and \(n\)-th receiver are located at (\(x_{m}\),0) and (\(x_{n}\),0), where \(m=1,2,\ldots,M\) and \(n=1,2,\ldots,N.\) Given a point target located at (\(x_{0}\),\(y_{0}\)), the average distance between the target and the \(l\)-th pair of transmitter and receiver (\(l\)-th sampling point) is given by  
\[R_{l}=(R_{m}+R_{n})/2=\bigg{[}\sqrt{(x_{m}-x_{0})^{2}+y_{0}^{2}}+\sqrt{(x_{n} -x_{0})^{2}+y_{0}^{2}}\bigg{]}\bigg{/}2 \tag{1}\]  
where \(l=(m-1)N+n=1,2,\ldots,MN.\)  
Then, we assume that the whole imaging scene is discretized by \(l\) grids in the \(x\) axis and \(J\) grids in the \(y\) axis. There are a set of targets and each one is located at a given grid in the \(x\)-\(y\) coordinate. For the \(l\)-th sampling point and the \(q\)-th frequency \(f_{q}=f_{0}+(q-1)\Delta f\) the demodulated received signal can be expressed as  
\[S(l,q)=\sum_{i=1}^{l}\sum_{j=1}^{l}\alpha(x_{i},y_{j})\exp(-j2k_{q}R_{l}(x_{i},y_{j}))+n(l,q) \tag{2}\]  
where \(k_{q}=2\pi f_{q}/c\) is the wavenumber, \(R_{l}(x_{i}\),\(y_{i}\)) is the average distance between the grid at (\(x_{i}\),\(y_{j}\)) and the \(l\)-th sampling point;\(f_{0}\) is the start frequency, \(\Delta f\) is the frequency step, \(\alpha(x_{i}\),\(y_{j})\) is the reflection coefficient of target, \(c\) is the velocity of light, and \(n(l,q)\) is the noise.  
The traditional BP algorithm is applied in the case of free space or uniform medium, and the target echo delay directly corresponds to the linear distance between the target and the antenna. One of the typical BP methods is the frequency-domain back projection (FDBP) method. It coherently sums the received signals from all the frequencies and sampling points to estimate the reflection coefficients of targets. With the received signal \(S(l,q)\), the reflection coefficient of a point (\(x_{i}\),\(y_{j}\)) is estimated by  
Figure 2: The signal model of the UWB SFCW radar with a MIMO array.  
\[\widetilde{\alpha}(x_{i},y_{j})=\frac{1}{QL}\sum_{q=1}^{Q}\sum_{l=1}^{L}S(l,q)\exp( +\mathrm{j}2k_{q}R_{l}(x_{i},y_{j})) \tag{3}\]  
In practice, the time-domain implementation of BP method (TDBP) [9; 20] is most commonly used in the highly suboptimal aperture length case for its simplicity. It can significantly save computing time. The formulation of TDBP can be written as  
\[\widetilde{\alpha}(x_{i},y_{j})=\frac{1}{L}\sum_{l=1}^{L}S_{t}(l,2R_{l}(x_{i}, y_{j})/c) \tag{4}\]  
where  
\[S_{t}(l,2R_{l}(x_{i},y_{j})/c)=\frac{1}{Q}\sum_{q=1}^{Q}S(l,q)\exp(+\mathrm{j }2k_{q}R_{l}(x_{i},y_{j})) \tag{5}\]  
Equations (4) and (5) are the range compressed signal of the \(l\)-th sampling point, which can be easily obtained by the inverse fast Fourier transformation (IFFT) and interpolation process.  
Although TDBP imaging methods are simple and convenient to implement, their imaging results may have high-level artifacts. Strong artifacts make weak targets undetectable and artifacts of many targets produce spurious peaks, leading to negative effects on 2D imaging. In order to effectively suppress artifacts, the cross-correlation-based TDBP (CC-TDBP) algorithm is applied [27; 28], which can be expressed as  
\[\widetilde{\alpha}(x_{i},y_{j})=\sum_{l_{1}=1}^{MN-1}\sum_{l_{2}=l_{1}+1}^{MN }S_{t}(l_{1},\tau(l_{1},x_{i},y_{j}))S_{t}(l_{2},\tau(l_{2},x_{i},y_{j})) \tag{6}\]  
where \(\tau(l_{1},x_{i},y_{j})=2R_{l_{1}}(x_{i},y_{j})/c\) and \(\tau(l_{2},x_{i},y_{j})=2R_{l_{2}}(x_{i},y_{j})/c\).