## 4 Azimuth-Variant Doppler Characteristics Analysis
### Doppler Characteristics  
At an azimuth time \(\tau\), the range history sum to an arbitrary point target can be expressed as  
\[R(\tau)=\sqrt{R_{t0}^{2}+(v_{t}\tau)^{2}}+\sqrt{R_{r0}^{2}+(v_{r}\tau)^{2}} \tag{5}\]  
where \(R_{t0}\) and \(R_{r0}\) denote, respectively, the closest ranges from the transmitter and the receiver to a given point target when they move along their trajectories. We can notice that the range history of a point target does not depend any more on the zero-Doppler distance and the relative distance from the target to the transmitter, but also on the absolute distance to the receiver. The Doppler chirp rate can be easily derived as  
\[k_{d}(\tau)=-\frac{1}{\lambda}\cdot\frac{\partial^{2}R(\tau)}{\partial^{2} \tau}\approx-\frac{1}{\lambda}\cdot\left[\frac{v_{t}^{2}}{R_{t0}}\cos\left( \frac{v_{t}\tau}{R_{t0}}\right)+\frac{v_{r}^{2}}{R_{r0}}\cos\left(\frac{v_{r} \tau}{R_{r0}}\right)\right] \tag{6}\]  
Using the parameters listed in Table 1, Figure 2 shows the calculated Doppler chirp rate of the three example BiSAR configurations. Note that in Cases A and B the synthetic aperture times are determined by the transmitter, but in Case C it is determined by the receiver. We can notice that the transmitting signal can be available for a short time span due to the large speed difference between the transmitter and the receiver [33, 34]. This disadvantage can be overcome by using a dedicated illuminating system such as a modified version of GNSS and a constellation of low Earth orbit (LEO) satellites. It can also be noticed that it has azimuth-variant Doppler chirp rate. This azimuth-variant phenomenon brings a great challenge towards developing an efficient image formation algorithm. Consequently, the locus of the slant ranges at the beam center crossing times of all targets that are parallel to the azimuth axis follow an approximate hyperbola. As an example, we consider a typical geometry shown in Figure 3(left), where the \((x,y)\) plane is locally tangent to the Earth surface. The targets are assumed to lie on this plane, and the transmitter velocity vector is parallel to the \(y\)-axis. Figure 3(right) shows the trajectories of three targets \(A\), \(B\), and \(C\) follow a hyperbolic locus. As a consequence, the imaged targets \(A\), \(B\) and \(C\) will have an equal slant range, _i.e._, the target \(B\) will be shifted to the position \(D\). This is different from general monostatic case, which has a linear locus instead of a hyperbolic one. General range Doppler image formation algorithms cannot handle this problem in a high-precision manner. Consequently, there will be a range ambiguity that does not exist in monostatic SARs. It can be observed from Figure 3 that two and more targets (e.g., \(A\), \(C\) and \(D\)) locating at different positions with the same range delay at the zero-Doppler but will have different range histories (curvature). Similar phenomena was investigated in [35], where a ground-based stationary receiver is assumed. However, for nonstationary receiver this problem will be much more complex, because the transmitter follows a rectilinear trajectory, while the receiver follows also a rectilinear trajectory and has a different velocity. The azimuth-variant problem will become more complicated for unflat digital Earth model (DEM) topography. In monostatic or azimuth-invariant BiSAR systems, scene topography can be ignored in developing image formation algorithms because the measured range delay is related with the double target distance and the observed range curvature; hence, topography is only used to project the compressed image from the slant range to the ground range. However, for azimuth-variant BiSAR, it is mandatory to know both the transmitter-to-target distance and the target-to-receiver distance to properly focus the raw data, but they clearly depend also on the target height. In this case, conventional imaging