# MIMO Radar Imaging Method with Non-Orthogonal Waveforms Based on Deep Learning
## 2 Imaging Model of Non-Orthogonal Waveform MIMO Radar  
### MIMO Radar Echo Signal Model  
Consider a \(M\)-transmitting and \(N\)-receiving linear array MIMO radar. The spacings between transmitting and receiving array elements are \(Nd\) and \(d\), respectively. In this case, the MIMO array can be equivalent to a uniform transceiver array. Let \(\varphi_{m}(t)\) represent the \(m\)th baseband transmitting signal. Consider a far-field target containing \(U\) scattering points, and \(Q\) is the \(q\)th scattering point on the target. \(T_{m}Q\) and \(R_{n}Q\) represent the distances between the scattering point and the transmitting array element and the receiving array element, respectively. The baseband echo signal received by the \(n\)th receiving array element can be expressed as  
\[y_{n}(t)=\sum_{q=0}^{U-1}\sum_{m=0}^{M-1}\widetilde{\sigma}_{q,mn}\varphi_{m} \left(t-\tau_{q,mn}\right) \tag{1}\]  
where \(n=0,1,\ \cdots,\ N-1\), \(\tau_{q,mm}=(T_{m}Q+R_{n}Q)/c\), \(c\) is the speed of light, \(\widetilde{\sigma}_{q,mn}=\sigma_{q}\exp[-j2\pi(T_{m}Q+R_{n}Q)/\lambda], \sigma_{q}\) represents the scattering coefficient of the scattering point, and \(\lambda\) is the working wavelength of the radar. When the line of sight angle of the target \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Pros** & **Cons** \\ \hline Waveform design methods [6; 7; 8; 9; 10] & Wide research and various alternative waveforms. & There are no completely orthogonal waveforms with the same frequency. Mutual coupling of the applied waveforms always exists. \\ \hline Signal processing methods [11; 12; 13] & Suitable for existing non-orthogonal waveforms. & The actual effect is limited. \\ \hline Sparse recovery methods [14; 15; 16; 17] & Suitable for existing non-orthogonal waveforms, good imaging effect. & Sensitive to model errors, basis mismatch, high computational complexity, and large storage amount. \\ \hline Deep learning method (the proposal) & Suitable for existing non-orthogonal waveforms, good imaging effect, high computation efficiency, and small storage amount. & Specially designed imaging network is necessary. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of methods of suppressing the mutual coupling among different waveforms. is known, the echo envelope of different transmit and receive channels can be aligned by pre-compensation of the envelope. At this time, \(\tau_{q,mn}\) in (1) does not change with \(m\),\(n\), so it can be re-expressed as \(\tau_{q}\). Assuming that the range imaging scope is \(\Delta R\), the number of sampling points in \(\Delta R\) will be \(K=2\Delta R\cdot f_{s}/c\) with the sampling frequency \(f_{s}\). Similarly, by sampling \(\varphi_{m}(t)\) with the frequency \(f_{s}\), we can obtain a transmission waveform vector \(\boldsymbol{\varphi}_{m}\) with the length of \(L\). Assuming that the scattering point \(Q\) is located in the \(k\)th range grid, \(k=0,1,\ \cdots,\ K-1\), the range kernel matrix of the \(k\)th range grid is defined as \(\boldsymbol{T}_{k}= \begin{pmatrix}\boldsymbol{0}_{k\times L}\\ \boldsymbol{I}_{L\times L}\\ \boldsymbol{0}_{(K-k)\times L}\end{pmatrix} \in\mathbb{C}^{(K+L)\times L}\), and then the echo signal of the scattering point \(Q\) received by the \(n\)th receiving array element can be expressed in vector form  
\[\boldsymbol{y}_{n}^{q}=\boldsymbol{T}_{k}\cdot\boldsymbol{S}\cdot\boldsymbol {\theta}_{n}^{k} \tag{2}\]  
where \(\boldsymbol{\theta}_{n}^{k}=\left[\widetilde{\sigma}_{q,0n},\widetilde{\sigma }_{q,1n},\cdots,\widetilde{\sigma}_{q,(M-1)n}\right]^{T}\in\mathbb{C}^{M \times 1}\), \(\boldsymbol{S}=[\boldsymbol{\varphi}_{0},\boldsymbol{\varphi}_{1},\cdots, \boldsymbol{\varphi}_{M-1}]\in\mathbb{C}^{L\times M}\). The target echo is the superposition of the echoes of all scattering points on the target.