## 3 Signal Processing
### Signal Model  
Two types of signal models are formulated in two different domains: the fast time-slow time domain and the fast time-azimuth angle domain. First, we derive the signal model in the fast time-slow time domain. We then translate this model into the fast time-azimuth angle domain. As the fast time-azimuth angle domain is based on the MIMO radar signal model, it can be used more conveniently in subsequent signal processing steps. Figure 8 shows the generic MIMO video SAR model [25]:  
In the system model, a broadside, uniform spatial sampling condition is assumed. Therefore, the PRF is selected such that the following relationship is satisfied [6]:  
\[f_{p}=\frac{v}{MN\cdot\Delta x} \tag{8}\]  
where \(f_{p}\) is the sweep repetition frequency, and \(\Delta x\) is the distance between antenna elements of the virtual array. Figure 8: Signal model of the generic multiple-input multiple-output (MIMO) video SAR. Figure 7: Antenna geometry: (**a**) The actual array; (**b**) The equivalent virtual array. Tx: transmitting; Rx: receiving. * Signal model in the fast time-slow time domain. In this sub-section, we define the transmitted waveform of the proposed MIMO ViSAR, and derive the received signal, the reference signal for dechirping in fast time, and the dechirped signal in the fast time-slow time domain. The transmitted waveform vector of the proposed MIMO ViSAR system can be expressed in vector form as [3]:  
\[\mathbf{s}_{T}\left(t,t_{a}\right)\equiv\left[ \begin{array}{c}ss_{0}\left(t,t _{a}\right)\\ \vdots\\ ss_{m}\left(t,t_{a}\right)\\ \vdots\\ ss_{M-1}\left(t,t_{a}\right)\end{array} \right] \tag{9}\]  
where the transmitted signal of the \(m\)-th Tx antenna, \(ss_{m}\left(t,t_{a}\right)\), is the BFD FMCW waveform [28] and can be written in complex form as follows:  
\[ss_{m}\left(t,t_{a}\right)=\text{rect}\left(\frac{t-t_{a}}{T_{d}}\right)\exp \left[j2\pi\left(f_{c}+m\Delta f_{b}\right)\left(t-t_{a}\right)+j\pi k_{r}\left( t-t_{a}\right)^{2}\right] \tag{10}\]  
where \(t_{a}=n_{s}T_{d}\) is slow time, \(n_{s}\) is the sweep number, \(t\) is the time variable, \(B_{r}\) and \(T_{d}\) are the bandwidth and sweep duration of the BFD FMCW waveform, respectively, and \(k_{r}=\frac{B_{r}}{T_{d}}\), which is the chirp rate of the waveform. The beat frequency division offset, which means the frequency offset between the transmitted signals, should be decided such that the orthogonality within the range swath is maintained as follows:  
\[\Delta f_{b}>\frac{B_{r}}{(MN-1)T_{d}}\frac{2R_{sw}}{c} \tag{11}\]  
where \(R_{sw}\) is the range swath and \(c\) is the speed of light. In the proposed MIMO ViSAR system, the signal from an ideal point scatterer, received at the th Rx antenna, is a delayed version of the transmitted signal and can be derived using convolution as [25],  
\[ \begin{array}{l}rr_{n}\left(t,t_{a}\right)=\sum_{m=0}^{M-1}ss_{m}\left(t,t_ {a}\right)*h_{nm}\left(t,t_{a}\right)=\sum_{m=0}^{M-1}a_{nm}ss_{m}\left(t- \tau_{nm}\left(x_{t},y_{t},z_{t}\right),t_{a}\right)\\ =\sum_{m=0}^{M-1}a_{nm}\text{rect}\left(\frac{t-t_{a}-\tau_{nm}\left(x_{t},y_ {t},z_{t}\right)}{T_{d}}\right)\\ \quad\cdot\exp\left[j2\pi\left(f_{c}+m\Delta f_{b}\right)\left(t-\tau_{nm} \left(x_{t},y_{t},z_{t}\right)\right)+j\pi k_{r}\left(t-t_{a}-\tau_{nm}\left( x_{t},y_{t},z_{t}\right)\right)^{2}\right]\end{array}  \tag{12}\]  
where, \(a_{nm}\) denotes the complex coefficient representing the scattering and the path loss, \(h_{nm}\left(t,t_{a}\right)\) is the ideal channel impulse response between the \(n\)-th Rx antenna and the \(m\)-th Tx antenna, and \(\tau_{nm}\left(x_{t},y_{t},z_{t}\right)\) is the round trip time delay from the \(m\)-th Tx antenna to the point target at position \(\left(x_{t},y_{t},z_{t}\right)\) and from the point target to the \(n\)-th Rx antenna. The reference signal for dechirping in fast time, which is a replica of the transmitted waveform delayed by the time to the center of the swath, can be written as,  
\[