## 4 Azimuth-Variant Doppler Characteristics Analysis
### Two-Dimensional Echo Spectrum Characteristics
 \tag{12}\]  
Supposing that the second-order range rate contributions are negligible against the linear terms in the common stationary phase point and that the fourth-order Lagrangian error terms (summing up all the higher order terms) of the Taylor series expansion are negligible against the third-order term, the common stationary phase point can be derived as  
\[\tau^{*}=\frac{\Phi_{T}^{\prime\prime}(\tau_{T}^{*})\tau_{T}^{*}+\Phi_{R}^{ \prime\prime}(\tau_{R}^{*})\tau_{R}^{*}}{\Phi_{T}^{\prime\prime}(\tau_{T}^{* })+\Phi_{R}^{\prime\prime}(\tau_{R}^{*})} \tag{14}\]  
However, as noted in [38], the validity of Equation (14) is constrained by  
\[\left|\tau_{T}^{*}-\tau_{0T}\right|^{2} =\frac{2R_{0t}^{2}}{7v_{t}^{2}} \tag{15a}\] \[\left|\tau_{R}^{*}-\tau_{0R}\right|^{2} =\frac{2R_{0r}^{2}}{7v_{r}^{2}} \tag{15b}\]  
where \(\tau_{0T}\) is the azimuth time when the point target is seen perpendicularly to the transmitter track. Figure 4 shows the constraints expressed in Equation (15) for several typical BiSAR velocity configurations. Note that this figure means that, to satisfy the requirements expressed in Equation (15), the instantaneous transmitter and receiver squint angles must be limited. We can see that the Loffeldmodel can only be used in a small squint angle or a small difference between the transmitter velocity \(v_{t}\) and the receiver velocity \(v_{r}\). Hence, the Loffeld model is not valid for general azimuth-variant BiSAR imaging, particularly for the HAP-borne passive BiSAR system. **Figure 13.** The constraints of the [PERSON]'s BiSAR spectrum model. (**a**) In small difference between \(v_{t}\) and \(v_{t}\). **Figure 14.** The constraints of the [PERSON]'s BiSAR spectrum model. (**a**) In small squint-angle cases; (**b**) In small difference between \(v_{t}\) and \(v_{t}\). Considering the phase history in Equation (11) and supposing its stationary phase point is \(\tau^{*}\), from the principle of stationary phase, we can get  
\[S(f_{r},f_{a})=R(f_{r})\exp\left[-j2\pi\frac{f_{0}+f_{r}}{c_{0}}R(\tau^{*})-j2 \pi f_{a}\tau^{*}\right] \tag{16}\]  
Using the instantaneous squint angles \(\xi_{t}\) (for the transmitter) and \(\xi_{r}\) (for the receiver), \(f_{a}\) can be represented by  
\[f_{a}=-\frac{f_{r}+f_{a}}{c_{0}}\left.\frac{\partial R(\tau)}{\partial\tau} \right|_{\tau=\tau^{*}}=-\frac{f_{r}+f_{a}}{c_{0}}\left[v_{t}\sin(\xi_{t})+v_{r }\sin(\xi_{r})\right] \tag{17}\]  
As the instantaneous bistatic range history at the azimuth time, \(\tau\) can be presented by  
\[ \begin{split} R(\tau)=& R_{t0}\cos(\xi_{t})+[R_{t0} \tan(\xi_{t0})-v_{t}\tau]\sin(\xi_{t})+\\ & R_{r0}\cos(\xi_{r})+[R_{r0}\tan(\xi_{r0})-v_{r}\tau]\sin(\xi_{r })\end{split}  \tag{18}\]  
where \(\xi_{t0}\) and \(\xi_{r0}\) denote, respectively, the instantaneous squint angles of the transmitter and receiver at their zero-Doppler frequencies. We then have  
\[ \begin{split} S\left(f_{r},f_{a}\right)=& P(f_{r})\exp\left\{-j2\pi\frac{f_{0}+f_{r}}{c_{0}}\left[R_{t0} \cos(\xi_{t})+R_{t0}\tan(\xi_{t0})\sin(\xi_{t})\right.\right.+\\ &\left.R_{r0}\cos(\xi_{r})+R_{r0}\tan(\xi_{r0})\sin(\xi_{r}) \right]\}\end{split}  \tag{19}\]  
This is just for a general BiSAR. For a general monostatic SAR system, a focused SAR image can be obtained from a two-dimensional inverse Fourier transform of Equation (19). However, this method is not effective for the azimuth-variant BiSAR configurations. One reason is that azimuth Doppler frequency is non-uniformly sampled for the azimuth-variant BiSAR configurations, and the other reason is that its precision two-dimensional spectrum is difficult to be obtained in a high-precision manner. This is the reason why azimuth-variant BiSAR imaging algorithm has not been thoroughly researched [39, 43, 44], while many azimuth-invariant BiSAR imaging algorithms have been proposed [45, 46, 47, 48].