# MIMO Radar Imaging Method with Non-Orthogonal Waveforms Based on Deep Learning
## 2 Imaging Model of Non-Orthogonal Waveform MIMO Radar
### Coupling Interference Induced by Non-Orthogonal Waveforms  
To realize MIMO radar imaging, it is necessary to separate the waveform of the MIMO radar echo. Usually, the matched filter method is used. A group of matched filters are used to separate the signals with different waveforms from the mixed echo. The complex range profile of the target is reconstructed at the same time. Ignoring the influence of noise, the separation result of the echo after the \(m\)th matched filter can be expressed as  
\[\boldsymbol{y}_{mn}=\left(\boldsymbol{T}\odot\boldsymbol{\varphi}_{m}\right) ^{*}\cdot\boldsymbol{y}_{n}=\left(\boldsymbol{T}\odot\boldsymbol{\varphi}_{m }\right)^{*}\cdot\boldsymbol{T}\odot\boldsymbol{S}\cdot\boldsymbol{\theta}_{n} \tag{4}\]  
\(\boldsymbol{y}_{mn}\) represents the waveform component in the echo corresponding to the \(m\)th transmitted waveform \(\boldsymbol{\varphi}_{m}\), where  
\[\begin{array}  
\[\begin{array}{ll}\boldsymbol{y}_{mn}&=\left[\boldsymbol{T}^{0}\boldsymbol{\varphi}_{ m},\boldsymbol{T}^{1}\boldsymbol{\varphi}_{m},\cdots,\boldsymbol{T}^{K-1} \boldsymbol{\varphi}_{m}\right]^{*}\cdot\left[\boldsymbol{T}^{0}\boldsymbol{S},\boldsymbol{T}^{1}\boldsymbol{S},\cdots,\boldsymbol{T}^{K-1}\boldsymbol{S} \right]\cdot\left[\boldsymbol{0},\cdots,\left(\boldsymbol{\theta}_{n}^{k} \right)^{T},\cdots,\boldsymbol{0}\right]^{T}\\ &=\left[\boldsymbol{T}^{0}\boldsymbol{\varphi}_{m},\boldsymbol{T}^{1} \boldsymbol{\varphi}_{m},\cdots,\boldsymbol{T}^{K-1}\boldsymbol{\varphi}_{m} \right]^{*}\cdot\boldsymbol{T}^{k}\boldsymbol{S}\cdot\boldsymbol{\theta}_{n}^{k} \end{array} \tag{7}\]  
The result of the \(k\)th distance grid, where the target is located, of \(\boldsymbol{y}_{mn}\) is  
\[\begin{array}{ll}\boldsymbol{y}_{mn}(k)&=\left(\boldsymbol{T}^{k} \boldsymbol{\varphi}_{m}\right)^{*}\cdot\boldsymbol{T}^{k}\boldsymbol{S}\cdot \boldsymbol{\theta}_{n}^{k}\\ &=\boldsymbol{\varphi}_{m}{}^{*}\left(\boldsymbol{T}^{k}\right)^{T}\cdot \boldsymbol{T}^{k}\boldsymbol{S}\cdot\boldsymbol{\theta}_{n}^{k}\\ &=\boldsymbol{\varphi}_{m}{}^{*}\boldsymbol{S}\cdot\boldsymbol{\theta}_{n}^{k} \\ &=\left[\boldsymbol{\varphi}_{m}^{*}\boldsymbol{\varphi}_{0},\boldsymbol{ \varphi}_{m}^{*}\boldsymbol{\varphi}_{1},\cdots,\boldsymbol{\varphi}_{m}^{*} \boldsymbol{\varphi}_{M-1}\right]\cdot\left[\widetilde{\sigma}_{q,mn}, \widetilde{\sigma}_{q,1n},\cdots,\widetilde{\sigma}_{q,(M-1)n}\right]^{T} \end{array} \tag{8}\]  
When the transmission waveform meets the ideal orthogonal condition (\(m=m^{\prime}\) when \(\boldsymbol{\varphi}_{m}^{*}\boldsymbol{\varphi}_{m^{\prime}}=1\); \(m\
eq m^{\prime}\) when \(\boldsymbol{\varphi}_{m}^{*}\boldsymbol{\varphi}_{m^{\prime}}=0\)), it can be obtained \(\boldsymbol{y}_{mn}(k)=\widetilde{\sigma}_{q,mn}\). This means that the separation result only contains the target signal corresponding to the \(m\)th transmission waveform, and there is no component of other waveforms. MIMO radar imaging can be realized by integrating the complex scattering coefficients \(\widetilde{\sigma}_{q,mn}\) obtained from different receiving and transmitting channels, and then compressing them in the azimuth direction.  
The above considers that the transmission waveform is ideally orthogonal, but in practice, waveforms with the same frequency cannot meet the above orthogonal conditions. As can be seen from (8), the separation result includes not only the signal component \(\widetilde{\sigma}_{q,mn}\) corresponding to the \(m\)th transmission waveform, but also a large amount of coupling interference from other transmission waveforms, which will lead to a serious decline in the reconstruction quality of the target image.