## 3 Signal Processing
### Signal Processing Procedure
\begin{split}&\mathbf{M}_{BFD}\left(t_{r},\theta\right)\\ &\equiv\left[\begin{array}{ccc}rR_{0,BFD2}\left(t_{r},\theta \right)&0&0\\ 0&\ddots&0\\ 0&0&rR_{M-1,BFD2}\left(t_{r},\theta\right)\end{array}\right]\\ &=\left[\begin{array}{ccc}r_{0,BFD2}\left(t_{r}\right)&0&0\\ 0&\ddots&0\\ 0&0&r_{M-1,BFD2}\left(t_{r}\right)\end{array}\right]R_{az}\left(\theta\right) \end{split}  \tag{39}\]  
Note that each element of the matrix, \(\mathbf{rR}_{BFD}\left(t_{r},\theta\right)\), represents the signal after BFD FMCW demodulation of the corresponding elements of the virtual array. Following a range Fourier transform of Equation (38), the \(N\times M\) range compression output matrix can be obtained as follows:  
\[\mathbf{R}\mathbf{R}_{RC}\left(f_{r},\theta\right)=\int_{-\infty}^{\infty} \mathbf{r}\mathbf{R}_{BFD}\left(t_{r},\theta\right)e^{-j2\pi f_{r}t_{r}}dt_{r} \tag{40}\]  
where the \(nm\)-th element of this matrix is the Fourier transform of the dechirped transmitted signal vector, with respect to fast time, as shown in Equation (38) as follows:  
\[ \begin{split}& RR_{RC,nm}\left(f_{r},\theta\right)\\ &=\mathrm{F}_{\mathrm{r}}\left[rR_{m,BFD2}\left(t_{r},\theta\right)b_{ n}\left(\theta\right)a_{m}\left(\theta\right)\right]\\ &=\mathrm{F}_{\mathrm{r}}\left[\exp\left[-j2\pi k_{r}\left(\tau_{0}- \tau_{ref}\right)\left(t_{r}-\tau_{0}\right)\right]\right]\cdot\exp\left[-j \frac{4\pi}{\lambda}r_{0}\right]\\ &\quad\cdot\exp\left(j2\pi f_{0}\tau_{ref}\right)\cdot R_{az} \left(\theta\right)\cdot b_{n}\left(\theta\right)a_{m}\left(\theta\right)\\ &=T_{d}\mathrm{sinc}\left(\pi T_{d}\left(f_{r}+k_{r}\left(\tau_{0}- \tau_{ref}\right)\right)\right)\cdot\exp\left(-j2\pi\tau_{0}f_{r}\right)\\ &\quad\cdot\exp\left[-j\frac{4\pi}{\lambda}r_{0}\right]\cdot\exp \left(j2\pi f_{c}\tau_{ref}\right)\cdot R_{az}\left(\theta\right)\cdot b_{n} \left(\theta\right)a_{m}\left(\theta\right)\end{split}  \tag{41}\]  
where \(\mathrm{F}_{\mathrm{r}}\left[\cdot\right]\) is the Fourier transform operator with respect to fast time. Using the fast time relationship, \(t_{r}=\frac{T_{d}}{B_{r}}f_{r}\), and the range relationship, \(r=\frac{cT_{d}}{2B_{r}}f_{r}\), we can define the point spread function in range as follows:  
\[\delta_{r}\left(r\right)\equiv\left.T_{d}\mathrm{sinc}\left(\pi T_{d}f_{r} \right)\right|_{f_{r}=\frac{2B_{r}}{cT_{d}}r} \tag{42}\]  
It is also assumed that all targets exist only within the range swath. The \(MN\times 1\) range compression vector, which is output from the virtual array, is given from Equation (40) as [3],  
\[ \begin{split}&\mathbf{w}\left(f_{r},\theta\right)\\ &\equiv\mathrm{Vec}\left\{\mathbf{R}\mathbf{R}_{RC}\left(f_{r}, \theta\right)\right\}\\ &=\left[\begin{array}{c}w_{0}\left(f_{r},\theta\right)\\ \vdots\\ w_{l}\left(f_{r},\theta\right)\\ \vdots\\ w_{MN-1}\left(f_{r},\theta\right)\end{array}\right]\end{split}  \tag{43}\]Using Equations (19), (41) and (42), this equation is represented in the range-azimuth frequency domain, \(\left(r,f_{a}\right)\), as:  
\[ \begin{array}{l}\mathbf{w}\left(r,f_{a}\right)\\ =\delta_{r}\left(r-\left(r_{0}-R_{a}\right)\right)\cdot\exp\left(-j2\pi\frac{4 B_{r}r_{0}}{c^{2}T_{d}}r\right)\\ \cdot\exp\left[-j\frac{4\pi}{\lambda}\left(r_{0}-R_{a}\right)\right]\cdot R_{ az}\left(f_{a}\right)\cdot g\left(f_{a}\right)\end{array}  \tag{44}\]  
This equation represents \(MN\) channel data of the virtual array formed by BFD FMCW demodulation described above, without taking into account the azimuth ambiguities. When the azimuth ambiguities are considered, the signal from the \(MN\) virtual elements can be expressed as an \(MN\times 1\) vector, using Equation (44), as follows [9]:  
\[ \begin{array}{c}\mathbf{c}\left(r,f_{a}\right)=\left\{\sum_{l=0}^{MN-1} \delta_{r}\left(r-\left(r_{0}-R_{a}\right)\right)\cdot\exp\left(-j2\pi\frac{4 B_{r}r_{0}}{c^{2}T_{d}}r\right)\cdot\exp\left[-j\frac{4\pi}{\lambda}\left(r_{0}-R_{a} \right)\right]\\ \cdot R_{az}\left(f_{a}+lf_{p}\right)\cdot\mathbf{\Phi}\left(f_{a}+lf_{p} \right)\right\}\cdot\mathbf{1}_{MN}\end{array}