## 3 Signal Processing
### Signal Model
\begin{array}{l}ss_{ref}\left(t,t_{a}\right)\\ =s\left.s_{m}\left(t-\tau_{ref},t_{a}\right)\right|_{m=0}\\ =\text{rect}\left(\frac{t-t_{a}-\tau_{ref}}{T_{d,ref}}\right)\exp\left[j2\pi f _{c}\left(t-t_{a}-\tau_{ref}\right)+j\pi k_{r}\left(t-t_{a}-\tau_{ref}\right) ^{2}\right]\end{array}  \tag{13}\]  
where, \(\tau_{ref}\) is the reference delay time to the center of the swath, and \(T_{d,ref}\) is the sweep duration of the reference signal. The dechirp-on-receive technique is widely used with FMCW systems. The dechirped signal that results from mixing the received signal in Equation (12) with the reference signal in Equation (13) is:  
\[ \begin{split}& rr_{n,dc}\left(t,t_{a}\right)=rr_{n}\left(t,t_{a} \right)ss_{ref}^{*}\left(t,t_{a}\right)\\ &=\sum_{m=0}^{M-1}a_{nm}\text{rect}\left(\frac{t-t_{a}-\tau_{nm} }{t_{d}}\right)\cdot\text{rect}\left(\frac{t-t_{a}-\tau_{ref}}{T_{d,ref}} \right)\\ &\quad\cdot\exp\left[j2\pi\left(m\Delta f_{b}-k_{r}\left(\tau_{ nm}-\tau_{ref}\right)\right)\left(t-t_{a}-\tau_{nm}\right)\right]\cdot\exp\left[-j\pi k _{r}\left(\tau_{nm}-\tau_{ref}\right)^{2}\right]\\ &\quad\cdot\exp\left[-j2\pi f_{c}\left(\tau_{nm}-\tau_{ref} \right)\right]\end{split}  \tag{14}\]  
The round-trip range from the \(m\)-th Tx antenna to a point scatterer and to the \(n\)-th Rx antenna, can be expanded and approximated using Taylor series expansion as,  
\[ \begin{split}& r_{n,m}\left(t;r_{0}\right)=c\frac{\tau_{nm}}{2}=r_{0} \sqrt{1+\left(\frac{vt-\Delta x_{tx,m}}{r_{0}}\right)^{2}}+r_{0}\sqrt{1+\left( \frac{vt-\Delta x_{rx,n}}{r_{0}}\right)^{2}}\\ &\approx r_{0}\sqrt{1+\left(\frac{vt_{a}-\Delta x_{tx,m}}{r_{0}} \right)^{2}}+r_{0}\sqrt{1+\left(\frac{vt_{a}-\Delta x_{rx,n}}{r_{0}}\right)^{ 2}}\\ &\approx r_{0}\left(1+\frac{1}{2}\left(\frac{vt_{a}-\Delta x_{ tx,m}}{r_{0}}\right)^{2}\right)+r_{0}\left(1+\frac{1}{2}\left(\frac{vt_{a}-\Delta x_{rx,n}}{r_{0}} \right)^{2}\right)\\ &=2r_{0}+\frac{v^{2}\left(t_{a}-\frac{\Delta x_{tx,m}+\Delta x_ {rx,n}}{2v}\right)^{2}}{r_{0}}+\frac{\left(\Delta x_{tx,m}-\Delta x_{rx,n} \right)^{2}}{4r_{0}}\end{split}  \tag{15}\]  
where, \(r_{0}\) is the closest distance between the antenna and the scatterer, \(\Delta x_{tx,m}\) is the distance between the reference Tx antenna and the \(m\)-th Tx antenna, and \(\Delta x_{rx,n}\) is the distance between the reference Rx antenna and the \(n\)-th Rx antenna. The range can be interpreted as a phase given by:  
\[ \begin{split}&\frac{2\pi}{\lambda}r_{n,m}\left(t;r_{0}\right)\\ &=\frac{2\pi}{\lambda}\left[2r_{0}+\frac{v^{2}\left(t_{a}-\frac{ \Delta x_{tx,m}+\Delta x_{rx,n}}{2v}\right)^{2}}{r_{0}}+\frac{\left(\Delta x_{ tx,m}-\Delta x_{rx,n}\right)^{2}}{4r_{0}}\right]\\ &=\frac{4\pi}{\lambda}r_{0}+\frac{2\pi}{\lambda}\frac{v^{2}\left( t_{a}-\frac{\Delta x_{tx,m}+\Delta x_{rx,n}}{2v}\right)^{2}}{r_{0}}+\frac{\pi \left(\Delta x_{tx,m}-\Delta x_{rx,n}\right)^{2}}{2\lambda r_{0}}\end{split}  \tag{16}\]  
where \(\lambda\) is the wave length. After substituting Equation (15) or (16) into Equation (14), the dechirped signal at the \(n\)-th Rx channel can be rewritten as,  
\[ \begin{split}& rr_{n,dc}\left(t_{r},t_{a}\right)\\ &\approx\sum_{m=0}^{M-1}a_{nm}\text{rect}\left(\frac{t_{r}-\tau_{ nm}}{T_{d}}\right)\cdot\text{rect}\left(\frac{t_{r}-\tau_{ref}}{T_{d,ref}} \right)\\ &\quad\cdot\exp\left[j2\pi m\Delta f_{b}\left(t_{r}-\tau_{0} \right)\right]\\ &\quad\cdot\exp\left[-j2\pi k_{r}\left(t_{r}-\tau_{0}\right) \left(\tau_{0}-\tau_{ref}\right)-j\pi k_{r}\left(\tau_{0}-\tau_{ref}\right)^{ 2}\right]\\ &\quad\cdot\exp\left[-j\frac{4\pi}{\lambda}r_{0}-j\frac{2\pi}{ \lambda}\frac{v^{2}\left(t_{a}-\frac{\Delta x_{tx,m}+\Delta x_{rx,n}}{2v} \right)^{2}}{r_{0}}-j\frac{\pi\left(\Delta x_{tx,m}-\Delta x_{rx,n}\right)^{ 2}}{2\lambda r_{0}}\right]\\ &\quad\cdot\exp\left(j2\pi f_{d}t_{r}\right)\cdot\exp\left(j2\pi f _{c}\tau_{ref}\right)\end{split}  \tag{17}\]where, \(t_{r}=t-t_{a}\) is fast time, \(\tau_{0}\) is the time delay corresponding to \(r_{0}\), and \(f_{d}\) is the Doppler frequency shift within the transmission of one sweep.