# Design of an Integrated System for Spaceborne SAR Imaging and Data Transmission
## 2 Construction of Mathematical Models
### Algorithms
#### 2.3.1 Alternating Direction Method of Multipliers
\begin{split}&(\mathcal{P}_{7})\underset{\mathbf{p}}{\max}\; \mathrm{SNR}_{81}(\mathbf{p}|\mathbf{p}^{t-1})\\ &\text{s.t.}\left\{\begin{array}{c}\sum\limits_{k=1}^{K}p_{k} \leq P\\ \xi\leq p_{k}\leq\frac{\eta P}{K}\\ \sum\limits_{k=1}^{K}\kappa u_{k}-\log_{2}\biggl{(}1+\frac{|\alpha_{k}|_{2}(1-u _{k})p_{k}}{\sigma_{\mathrm{c}}^{2}}\biggr{)}\geq\sum\limits_{k=1}^{K}\kappa u _{k}\\ -\log_{2}\Biggl{(}1+\frac{\left|\alpha_{k}\right|_{2}(1-u_{k})p_{k}^{t -1}}{\sigma_{\mathrm{c}}^{2}}\Biggr{)}+\frac{(1-u_{k})(p_{k}-p_{k}^{t-1})}{\ln 2 \biggl{(}(1-u_{k})p_{k}^{t-1}+\frac{\sigma_{\mathrm{c}}^{2}}{\left|\alpha_{k} \right|^{2}}\biggr{)}}\end{array}\right.\end{split}  \tag{25}\]

As \(\mathcal{P}_{2}\) is a mixed-integer nonlinear optimization problem, a genetic algorithm is employed to alternately solve for the approximate solution of \(\mathcal{P}_{7}\) by fixing \(u\) and \(p\). The alternating direction genetic algorithm is as follows (Algorithm 2):

By solving \(\mathcal{P}_{5}\), \(\mathcal{P}_{6}\), and \(\mathcal{P}_{7}\), the integrated waveform allocation scheme \(\mathbf{u}\) and the transmission power \(\mathbf{p}\) for each subcarrier are obtained. In the next pulse, the optimized integrated waveform is transmitted. Using the most recent echo data, we gather information about the communication channel and recalibrate the adaptive integrated waveform. By adjusting the parameter \(\kappa\), we balance the weights assigned to the radar and transmission systems to meet mission requirements. ```
Input:\(\xi\), \(\eta\), \(P\) x, Crossover probability \(P_{cross}\), Mutation probability \(P_{mutate}\) Output: Channel allocation \(\boldsymbol{u}\); Power allocation \(\boldsymbol{p}\) Initialization: Initialize population size: 50, max generations: 100, Tolerance: \(1\times 10^{-9}\)\(\boldsymbol{p}^{(0)}\), \(\boldsymbol{u}^{(0)};\alpha\) and \(\beta\) randomly, let \(t=0\); repeat \(t=t+1\);  According to fitness function (24) with \(\boldsymbol{p}^{(t)}\), update \(\boldsymbol{u}^{(t)}\), through crossover and mutation obtain \(\boldsymbol{u}^{(t+1)}\);  According to fitness function (25) with \(\boldsymbol{u}^{(t)}\), update \(\boldsymbol{p}^{(t)}\), through crossover and muttion obtain \(\boldsymbol{p}^{(t+1)}\);  Update \(\boldsymbol{p}\), \(\boldsymbol{u}\); until\(t\geq\text{max}\) generations
```

**Algorithm 2** Alternating Direction Genetic Algorithm

## 3 Simulation Experiment and Result Analysis

We used the MATLAB simulation package in this experiment, with computational hardware consisting of an Intel Core i7-7700 HQ CPU and a GTX 1060 ti GPU. The effectiveness of the proposed method was validated using the simulation data package, with specific simulation parameters listed in Table 1.

\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Parameters** & **Values** \\ \hline \multicolumn{3}{c}{Bandwidth \(B_{w}\)} & 180 MHz \\ \cline{2-3}  & Carrier frequency \(f_{\text{c}}\) & 5 GHz \\ \cline{2-3}  & Sampling frequency \(f_{\text{s}}\) & 220 MHz \\ \cline{2-3}  & Number of subcarriers \(K\) & 16 \\ \cline{2-3} Satellite & Orbital altitude \(R\) & 500 km \\ \cline{2-3}  & Azimuth resolution of a \(\,\) sing\(\,\)gle\(\,\)range\(\,\)image\(



The prototype filter used in the FBMC-CQAM system for the experiment is as follows:

\[q(t)=1+2\sum_{n=1}^{N-1}{(-1)}^{n}H_{n}\cos\left(\frac{2\pi nt}{T}\right) \tag{26}\]

where \(N\) represents the overlapping factor in FBMC and \(H_{n}\) denotes an FBMC parameter, as detailed in [PERSON] et al. [39]. Due to the high transmission power of spaceborne radar and communication systems, ground-based user receivers do not need high antenna gain. As a result, mobile terminals use omnidirectional antennas that are a quarter-wavelength of the signal carrier. The SAR satellite utilizes a planar phased array antenna, providing benefits such as convenient beamforming, flexible beam scanning, and reliable electronic control. These characteristics make it ideal for the wide scanning range necessary in SAR imaging [40].