# High-Resolution and Wide-Swath 3D Imaging for Urban Areas Based on Distributed Spaceborne SAR
## Subsatellite
#### 2.3.3 Obtaining Range Ambiguity Resolution Based on Multi-Beam Forming  
The investigation of range ambiguity resolution can be conducted after the coarse image registration. A multi-channel range ambiguity resolution technology based on multi-beam forming is proposed in this paper, as shown in Figure 7. First, the prior knowledge of the target location is obtained by introducing an external DEM. The accuracy of external DEM can reach 3-10 m, which can provide prior knowledge such as accurate target location for DBF.  
The target echo signal received from the \(n\)th SAR in pitch can be expressed as:  
\[\begin{array}{l}S_{n}(t)=A_{0}rect(\frac{t-t_{0}}{T_{p}})\exp(-j2\pi f_{c}t_{0} )\exp(j\pi k_{r}(t-t_{0})^{2})\\ \exp(\frac{(2\pi f_{c}d_{n}\sin(\theta_{r}-\theta_{c})}{c})\end{array} \tag{11}\]  
where \(A_{0}\) represents the constant related to the gain of the antenna pattern and the backscattering coefficient of the target; \(t_{0}\) represents the round trip delay of the target; \(T_{p}\) represents the pulse width of the transmitting signals; \(f_{c}\) represents the carrier frequency of the radar signal; \(k_{r}\) represents the frequency modulation; \(\theta_{c}\) represents the bottom view of the antenna normal; \(rect(\cdot)\) denotes the normalized rectangular window with a range of [\(-0.5\), \(0.5\)]; and \(d_{n}=(n-\frac{N_{e}-1}{2})d\) represents the distance between the nth SAR and the middle SAR.  
To reduce the pulse-broadening loss of the target echo, each SAR echo signal is matched and filtered.  
\[\begin{array}{l}\overline{S}_{n}(t)=\sqrt{|k_{r}|}T_{p}A_{0}\exp(-j2\pi f_{c }t_{0})\\ \sin c(krTp(t-t0))\exp(\frac{(2\pi f_{c}d_{n}\sin(\theta_{r}-\theta_{c})}{c}) )\end{array} \tag{12}\]  
The matched filtering results of all SAR received echoes are:  
\[S(t)=C_{0}(t)a(\theta_{r})+e(t) \tag{13}\]  
\[C_{0}(t)=\sqrt{|k_{r}|}T_{p}A_{0}\exp(-j2\pi f_{c}t_{0})\sin c(k_{r}T_{p}(t-t_ {0})) \tag{14}\]  
\[a(\theta_{r})=\left[\exp(\frac{j2\pi f_{c}d_{1}\sin(\theta_{r}-\theta_{c})}{c })\cdot\cdot\cdot\exp(\frac{j2\pi f_{c}d_{Ne}\sin(\theta_{r}-\theta_{c})}{c}) \right]^{T} \tag{15}\]  
where \(a(\theta_{r})\) represents the guidance vector of the target echo and \(e(t)\) denotes the noise vector. The matched filtering results of these echoes received by the SAR are weighted and  
Figure 7: The algorithm for obtaining range ambiguity resolution based on multi-beam forming.  
accumulated in the digital beamforming processor to generate an equivalent high-gain DBF beam pointing at the target, thereby achieving a high-gain received target echo signal. To ensure that the DBF receiving beam center points to the target echo direction, the pitch DBF weighting vector of each range gate unit should be calculated according to the target angle of arrival \(\theta_{r}\) as [34]:  
\[\omega(t)=a^{*}(\theta_{r}(t)) \tag{16}\]  
The elevation value of the target is needed to calculate the angle of view under the target corresponding to the distance gate unit. Therefore, it is necessary to calculate the target elevation corresponding to the range gate unit according to the digital elevation map of the real terrain based on the spaceborne SAR imaging geometry model.  
Note that there will be overlapping targets in the 3D imaging of urban areas studied in this paper, and their DOA is not unique. Therefore, the method proposed in this paper first performs multi-direction arrival filtering and multi-beam synthesis.