## 4 Azimuth-Variant Doppler Characteristics Analysis
### Two-Dimensional Echo Spectrum Characteristics  
[PERSON] _et al._[38] divided the bistatic phase history into two quasi-monostatic phase histories and expanded them into two Taylor series around the individual points of the stationary phase. In this way, an approximation of the stationary-phase azimuth time is constructed, and a two-dimensional BiSAR spectrum is deduced. Several algorithms have been developed based on this model [39]. Another BiSAR spectrum model using series expansion to express azimuth time as a function of azimuth frequency during azimuth Fourier transform was proposed in [40], and a chirp scaling algorithm was developed based on this model [41]. However, the accuracy is controlled by keeping enough terms in the power series, but only the first several series can be used. Using the Fresnel approximation, a two-dimensional BiSAR spectrum model was derived in [42]. This model can be used only in small a squint angle. Moreover, it has no advantages in approximate error because the Fresnel approximation is used in the algorithm. In summary, most BiSAR spectrum models cannot be used in the BiSAR with a large difference between the transmitter-to-target distance and target-to-receiver distance or a large squint angle. To evaluate the applicability of the [PERSON]'s model for azimuth-variant BiSAR, we suppose the transmitted signal is  
\[s_{t}(t)=p(t)\exp(j2\pi f_{c}t) \tag{7}\]  
where \(p(t)\) is an encoded wide bandwidth signal, for example chirp signal, and up-converted by the transmitter to the carrier frequency \(f_{c}\). The radar echo reflected from a point target experiences a time-delay that is proportional to the range history \(R(\tau)\)  
\[s_{r}(t,\tau;R_{r0},\tau_{r0})=s_{t}\left(t-\frac{R(\tau)}{c_{0}}\right) \tag{8}\]  
Figure 3: Targets with the same range delay at the zero-Doppler but will have different range histories. where \(\tau_{r0}\) is the azimuth time when the point target is seen perpendicularly to the receiver track. Note that the antenna pattern and constant amplitude terms in the signal have been omitted. After being down-conversed, the demodulated signal is  
\[s_{r}(t,\tau;R_{r0},\tau_{r0})=p\left(t-\frac{R(\tau)}{c_{0}}\right)\exp\left(j2 \pi\frac{R(\tau)}{\lambda}\right) \tag{9}\]  
After Fourier transforming from range-time domain to range-frequency domain, we then have  
\[s_{r}\left(f_{r},\tau;R_{r0},\tau_{r0}\right)=P\left(f_{r}\right)\exp\left(-j2 \pi\frac{f_{r}+f_{c}}{c_{0}}R(\tau)\right) \tag{10}\]  
where \(P(f_{r})\) is the Fourier transform of \(p(t)\). Transforming Equation (10) from azimuth-time to azimuth-frequency domain yields  
\[s_{r}\left(f_{r},f_{a};R_{r0},\tau_{r0}\right)=P\left(f_{r}\right)\int_{- \infty}^{\infty}\exp\left(-j2\pi\frac{f_{r}+f_{c}}{c_{0}}R(\tau)-j2\pi f_{a} \tau\right)\mathrm{d}\tau \tag{11}\]  
where \(f_{a}\) is the azimuth Doppler frequency. [PERSON] _et al._[38] expanded the phase histories into two Taylor series around the individual stationary phase points \(\tau_{T}^{*}\) and \(\tau_{R}^{*}\)  
\[ \begin{split}\Phi_{T}(\tau)&=-2\pi\left[\left(f_{r }+f_{c}\right)\frac{R_{t}(\tau)}{c_{0}}-f_{a}\tau\right]\\ &\approx\Phi_{T}(\tau_{T}^{*})+\Phi_{T}^{\prime}(\tau_{T}^{*}) \left(\tau-\tau_{T}^{*}\right)+\frac{1}{2}\Phi_{T}^{\prime\prime}(\tau_{T}^{* })\left(\tau-\tau_{T}^{*}\right)^{2}+\ldots\\ \Phi_{R}(\tau)&=-2\pi\left[\left(f_{r}+f_{c}\right) \frac{R_{r}(\tau)}{c_{0}}-f_{a}\tau\right]\\ &\approx\[PERSON]_{R}(\tau_{R}^{*})+\[PERSON]_{R}^{\prime}(\tau_{R}^{*}) \left(\tau-\tau_{R}^{*}\right)+\frac{1}{2}\[PERSON]_{R}^{\prime\prime}(\tau_{R}^{* })\left(\tau-\tau_{R}^{*}\right)^{2}+\ldots\end{split}