## 5 Azimuth-Variant Bistatic SAR Imaging Algorithm  
For the HAP-borne BiSAR with fast-moving transmitter and slow-moving receiver,  
\[v_{t}\gg v_{r} \tag{20a}\] \[R_{t0}\gg v_{r}\tau,\hskip 14.226378 pt\tau\in[-T_{s}/2,T_{s}/2] \tag{20b}\]  
where \(T_{s}\) is the synthetic aperture time, we can make the following approximation  
\[ \begin{split} R_{b}(\tau)&=\sqrt{R_{t0}^{2}+(v_{t }\tau)^{2}}+\sqrt{R_{r0}^{2}+(v_{r}\tau)^{2}}\\ &\approx\sqrt{R_{t0}^{2}+(v_{eq}\tau)^{2}}+R_{r0}\end{split}  \tag{21}\]  
where \(\vec{v}_{eq}=\vec{v}_{t}+\vec{v}_{r}\) is the equivalent velocity between the transmitter and the receiver, as shown in Figure 5. Note that the transmitter and receiver may have non-parallel flying tracks. Suppose the transmitter slant range and receiver slant range are fixed to be \(800\) km and \(30\) km, Figure 6 gives the range errors caused by the equivalent BiSAR system model. As the relevant phase errors that are smaller than \(\pi/4\) can be ignored for subsequent imaging algorithms, we can conclude that the equivalent model for small \(v_{r}\) is satisfactory for subsequent image formation processing. The equivalent instantaneous azimuth Doppler frequency \(f_{a}\) can then be expressed as  
\[f_{a}=\frac{v_{eq}}{\lambda}\sin(\phi_{t}) \tag{22}\]  
where \(\phi_{t}\) is the instantaneous angle between the transmitter and the point target. Since \(\lambda f_{a}/v_{eq}\ll 1\), for small \(\phi_{t}\), we have  
\[R_{b}(f_{a};R_{t0})=\frac{R_{t0}}{\cos(\phi_{t})}+R_{r0}\approx R_{t0}\left[1+ \frac{\lambda^{2}f_{a}^{2}}{2v_{eq}^{2}}\right]+R_{r0} \tag{23}\]  
For simplicity, we denote \(R_{b}(f_{a};R_{t0})\) by  
\[R_{b}(f_{a};R_{t0})=R_{t0}\left[1+C_{s}\right]+R_{r0} \tag{24}\]  
where  
\[C_{s}=\frac{1}{\sqrt{1-(\lambda f_{a}/v_{eq})^{2}}}-1 \tag{25}\]  
is the scaling factor of the chirp scaling (CS) algorithm. Next, a NCS algorithm, which is similar to the algorithms developed in [35, 41], can then be applied. Our algorithm has the processing steps similar to [35, 41], but has different compensation phase terms, \(\Phi_{i}\) (\(i=[1,2,3,4,5]\)), because the azimuth-variant Doppler characteristics are considered in deriving the algorithm. Suppose the transmitted signal is  
\[s_{t}(t)=\exp\left[j\pi(2f_{c}t+k_{r}t^{2})\right] \tag{26}\]  
where \(k_{r}\) is the chirp rate. Note that the amplitude terms are ignored. The demodulated baseband signal in the receiver is  
\[s_{r}(t)=\exp\left\{-j\pi\left[k_{r}\left(t-\frac{R_{b}(\tau)}{c_{0}}\right)^ {2}+2\frac{R_{b}(\tau)}{c_{0}}\right]\right\} \tag{27}\]  
Figure 5: Azimuth-variant BiSAR geometry and its equivalent model with stationary receiver. Applying the Fourier transform to the azimuth time \(\tau\) yields  
\[ \begin{split} S_{r}\left(t,f_{a}\right)=&\exp\left\{-j \pi k_{eq}\left[t-\frac{R_{b}(f_{a};R_{t0})}{c_{0}}\right]^{2}\right\}\times\\ &\exp\left(-j\pi\frac{\lambda R_{b0}f_{a}^{2}}{v_{eq}}\right) \cdot\exp\left(-j2\pi f_{a}\frac{y_{p}}{v_{eq}}\right)\end{split}  \tag{28}\]  
Figure 6: Range errors caused by the equivalent BiSAR geometry model. (**a**) Spaceborne transmitter (\(R_{t0}=800\) km, \(v_{t}=7,600\) m/s); (**b**) Airborne transmitter (\(R_{t0}=15\) km, \(v_{t}=100\) m/s). with  
\[k_{eq}=\frac{1}{\frac{1}{k_{r}}-\frac{\lambda R_{tot}}{c_{0}^{2}}\cdot\frac{( \lambda f_{a}/v_{eq})^{2}}{[\sqrt{1-(\lambda f_{a}/v_{eq})^{2}}]^{3}}} \tag{29}\]  
where \((x_{p},y_{p},0)\) is the point target's coordinate and \(R_{b0}\) is the smallest bistatic range. We perform the Chirp scaling processing with the phase term  
\[\Phi_{1}=\exp\left\{-j\pi k_{eq}\left[\frac{1}{\sqrt{1-(\lambda f_{a}/v_{eq})^ {2}}}-1\right]\left[t-\frac{R_{b}(f_{a};R_{\text{bref}})}{c_{0}}\right]\right\} \tag{30}\]  
where \(R_{\text{bref}}\) is the reference range. Equation (28) is transformed into two-dimensional frequency-domain though range FFT.