# Architecture Exploration of a Backprojection Algorithm for Real-Time Video SAR
## 3 Architecture Exploration for Backprojection Based Video SAR  
### Backprojection Based Video SAR  
A video SAR system transmits pulses at regular intervals and samples the reflected pulses from the target area in a phase history format. Phase history represents streaming data that inherits a group of pulse signatures that correspond to the SAR frame at a specific moment. Because the cross-range resolution improves with the increasing length of the synthetic aperture, BP-VISAR sub-frames generated by a small group of pulses can be superposed to construct a full BP-VISAR frame having improved cross-range resolution. From the nature of the backprojection algorithm, the computation for BP-VISAR frame formation can be easily parallelized. The group of pulses for a full-synthetic-aperture is partitioned into smaller groups for sub-synthetic-aperture processing. Each smaller group can be used to construct a group of BP-VISAR frames with low cross-range resolutions. This sub-group processing is performed independently and can be parallelized. Unlike the non-overlapped mode in Figure 1(a), in the overlapped mode in Figure 1(b), multiple sub-aperture pulses in partial overlap intervals are reused to construct neighboring BP-VISAR frames. This enables the multiple BP-VISAR frames with a low cross-range resolution to be superposed into BP-VISAR frames with a higher resolution. This helps avoid redundant processing for the overlapped domain data, and the frame rate can be increased with reduced computational load in the overlapped BP-VISAR mode. The processing stages to implement BP-VISAR consist of I/O, range compression (RC), image formation (IF), and post-processing (PP), as illustrated in Figure 2(a). It is assumed that \(N_{p}\) pulses are employed to construct a single BP-VISAR frame. The \(n\)th pulse is transmitted at \(\tau_{n}\). The received signal is transformed into a frequency domain after A/D conversion, and the \(k\)th frequency sample for the \(n\)th reflected pulse is denoted by \(S(f_{k},\tau_{n})\). Given phase history data \(S(f_{k},\tau_{n})\), the range profile at range bin \(m\) for a received pulse at \(\tau_{n}\) is given as follows. \[s(m,\tau_{n})=\sum_{k=1}^{K}S(f_{k},\tau_{n})\exp\left(\frac{i4\pi f_{k}\Delta R (m,\tau_{n})}{c}\right), \tag{7}\]  
where \(\Delta R(m,\tau_{n})\) represents the range difference from the antenna phase center to the target and scene origin. \(K\) and \(c\) represent the number of frequency samples in the phase history data of each pulse and the speed of light, respectively. For the SAR frame generation with the range profile in Equation (7), interpolation is required to calculate the differential range accurately for every pixel from the discrete values  
Figure 2: Non-overlapped and overlapped apertures for video SAR frame construction. of the range profile. This process can be implemented by performing linear interpolation on the inverse fast Fourier transform (IFFT) output of the zero-padded phase history data. The final pixel response at location \(r\) is calculated by superposing all the interpolated values of the range profile, \(s_{int}(r,\tau_{n})\), with pulses within the synthetic aperture for the formation of the video frame. The pixel value at location \(r\) formed using \(N\) pulses starting from the \(i\)th pulse is given as follows. \[I(r,i,N)=\sum_{n=i}^{i+N-1}s_{int}(r,\tau_{n}). \tag{8}\]  
For BP-VISAR, we propose constructing a sub-aperture consisting of adjacent pulses with a size less than that of the full-aperture. If the number of pulses in a sub-aperture is \(N_{s}\) and \(M\) sub-apertures construct a full-aperture, the BP-VISAR frame for the sub-aperture starting from the \(i\)th pulse is defined as follows. \[I_{s}(r,i)=I(r,i,N_{s}). \tag{9}\]  
Then, in the post-processing stage, the BP-VISAR frame for the full-aperture frame constructed by \(M\) sub-apertures starting from the \(i\)th pulse is defined as follows.