# 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
The problem \(\mathcal{P}_{4}\) is transformed into:  
\[(\mathcal{P}_{5})\underset{\mathbf{u},\mathbf{w},\mathbf{p},\mathbf{v}}{\text{max }}\sum_{k=1}^{K}\frac{|\beta_{k}|_{2}u_{k}p_{k}}{\sigma_{\text{r}}^{2}}+G(\bm {u},\mathbf{w},\mathbf{p},\mathbf{v})\] (21 a) \[\text{s.t.} \begin{cases}\begin{array}{l}\sum_{k=1}^{K}p_{k}\leq P \\ \xi\leq p_{k}\leq\frac{\eta P}{K}\\ w_{k}=min(max(\frac{1}{\
u}-1,\frac{|\alpha_{k}|_{2}\xi}{\sigma_{\text{c}}^{2} }),\frac{\eta P|\alpha_{k}|_{2}(1-u_{k})p_{k}}{K\sigma_{\text{c}}^{2}})\\ v_{k}=v_{k}+\mu(\frac{|\alpha_{k}|_{2}(1-u_{k})p_{k}}{\sigma_{\text{c}}^{2}}- w_{k})\end{array}&(\ref{eq:1}.\text{d})\end{cases}  \tag{21}\]  
where  
\[G(\mathbf{u},\mathbf{w},\mathbf{p},\mathbf{v})=\rho(\ln(1+w_{k}))-v_{k}w_{k}+v_{k}\frac{| \alpha_{k}|_{2}(1-u_{k})p_{k}}{\sigma_{\text{c}}^{2}}-\rho\kappa u_{k}) \tag{22}\]  
The steps for implementing ADMM with a penalty function are as follows (Algorithm 1):  
```
Input:\(\xi\), \(\eta\), \(P\)x Output: Channel allocation \(\mathbf{u}\): Power allocation \(\mathbf{p}\)  Initialization: Initialize \(\mathbf{p}^{(0)},\mathbf{u}^{(0)}\); \(\alpha\) and \(\beta\) randomly, let \(t=0\); repeat \(t=t+1\);  With \(\mathbf{p}^{(t)}\), \(\mathbf{u}^{(t)}\) and \(\mathbf{w}^{(t)}\), update \(\mathbf{u}^{(t+1)}\) by solving (21);  With \(\mathbf{p}^{



#### 2.3.2 Alternating Direction Genetic Algorithms

Genetic algorithms (GAs) simulate natural selection and genetics by iteratively searching a population of candidate solutions using crossover, mutation, and selection [36]. GAs are well-suited for solving complex nonlinear integer programming problems, especially in cases where the parameter space is vast and intricate [37; 38]. This conversion also involves transforming non-convex constraints into convex ones. The iterative residual of the objective function determines when the iterations should terminate. The non-convex constraint (17) can also be relaxed using the convex relaxation method. \(\mathbf{p}^{t-1}\) represents the subcarrier allocation power in the previous iteration. Therefore, the Taylor series expansion of constraint (12.3) around \(\mathbf{p}^{t-1}\) is given by:

\[ \begin{split}\sum\limits_{k=1}^{K}\kappa u_{k}-\log_{2}\biggl{(} 1+\frac{|\alpha_{k}|_{2}(1-u_{k})p_{k}}{\sigma_{\mathrm{c}}^{2}}\biggr{)}\geq \sum\limits_{k=1}^{K}\kappa u_{k}-\log_{2}\left(1+\frac{\left|\alpha_{k}\right| _{2}(1-u_{k})p_{k}^{t-1}}{\sigma_{\mathrm{c}}^{2}}\right)\\ +\frac{(1-u_{k})(p_{k}-p_{k}^{t-1})}{\ln 2\biggl{(}(1-u_{k})p_{k}^{t -1}+\frac{\sigma_{\mathrm{c}}^{2}}{\left|\alpha_{k}\right|^{2}}\biggr{)}} \end{split}  \tag{23}\]

Given a fixed power \(\mathbf{p}\), \(\mathcal{P}_{2}\) is transformed into the following form:

\[ \begin{split}&(\mathcal{P}_{6})\underset{\mathbf{u}}{\max}\; \mathrm{SNR}_{81}(\mathbf{u})\\ &\text{s.t.}\left\{\begin{array}{c}\sum\limits_{k=1}^{K}\kappa u _{k}-\log_{2}\biggl{(}1+\frac{|\alpha_{k}|_{2}(1-u_{k})p_{k}}{\sigma_{\mathrm{ c}}^{2}}\biggr{)}\geq\sum\limits_{k=1}^{K}\kappa u_{k}\\ -\log_{2}\Biggl{(}1+\frac{\left|\alpha_{k}\right|_{2}(1-u_{k})p_{k}^{t -1}}{\sigma_{\mathrm{c}}^{2}}\Biggr{)}+\frac{(1-u_{k})(p_{k}-p_{k}^{t-1})}{\ln 2 \biggl{(}(1-u_{k})p_{k}^{t-1}+\frac{\sigma_{\mathrm{c}}^{2}}{\left|\alpha_{k} \right|^{2}}\biggr{)}}\\ & u_{k}(1-u_{k})\leq 0\end{array}\right.\end{split}  \tag{24}\]

Given the fixed values of \(\mathbf{u}\) and \(\mathbf{p}^{t-1}\), \(\mathcal{P}_{2}\) can be expressed as:

\[