# 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
Table 2 displays the performance of quantitative analysis of different algorithms. We have defined as follows: Weighted score = Echo SNR + \(\rho\) * Spectral Efficiency. ### Results Analysis

Figure 4 indicates that the BER of FBMC was slightly lower than that of OFDM at low SNR levels. When the power threshold \(\xi\) was set to 30 W, 40 W, and 50 W, the BER excluding channel encoding and decoding were \(7.4\times 10^{-3}\), \(2.4\times 10^{-3}\), and \(7.4\times 10^{-4}\), respectively. Efficient channel encoding and decoding techniques, such as low-density parity-check codes or turbo codes, can be employed to significantly reduce the BER, often achieving a BER of \(1\times 10^{-6}\) or lower. This ensures the reliability and integrity of data transmission, thereby meeting the BER requirements of satellite communication systems. Figure 5 shows that Algorithm 1 reached its convergence value after the second iteration under varying maximum PAPR \(\eta\). In Algorithm 2, a higher value of \(\eta\) leads to faster convergence. The larger the value of \(\eta\) = 6 dB, the faster the convergence. This suggests that allocating more power to subcarriers with better channel conditions expands the feasible solution space, making convergence easier. In Algorithm 1, the objective functions at 6 dB and 9 dB were similar, showing that the system's objective function approached its upper limit at 6 dB under the constraint of \(\xi\). As a result, changes in \(\eta\) had limited impact on objective functions. Algorithm 2 converged in eight iterations and may face scenarios where internal iterations exceed the maximum population iteration count, halting the process prematurely. Theoretically, higher values of \(\eta\) lead to faster convergence. However, at 6 dB, convergence may reach local optima, yielding slightly worse performance than Algorithm 2. Increasing population iterations can enhance convergence outcomes at the expense of longer computational time. In both algorithms, when \(\eta\ =\ 0\) dB, power \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Echo SNR** & **Spectral Efficiency** & **Weighted Score** \\ \hline Genetic Algorithms & \(-12.832\) dB & 4.262 & 0.0564 \\ \hline ADMM Algorithms & \(-12.037\) dB & 3.752 & 0.0663 \\ \hline greedy algorithm & \(-12.218\) dB & 4.085 & 0.0641 \\ \hline water-filling algorithm & \(-12.626\) dB & 4.085 & 0.0587 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results for Algorithm performance. Figure 9: Algorithm performance for the integrated spaceborne SAR and data transmission system. is distributed evenly among subcarriers, with the optimal power being \(P/K\), resulting in suboptimal power allocation to better channels and lower objective function values compared to other \(\eta\) schemes. Normalized waveforms require increased PAPR in power allocation to boost the SNR of echo signals and algorithm convergence speed. The objective function results of Algorithm 1 outperform those of Algorithm 2 but require more iterations. Figure 6 illustrates the convergence performance of the algorithm under varying transmission weights \(\kappa\) and its impact on SNR. The variable \(\kappa\) represents the weight of the task data volume for each transmission, as defined by Equation (7). When \(\kappa=0.5\), it indicates the complete transmission of the entire data volume in each instance. When \(\kappa=0.25\), it signifies the transmission of half the image data in each instance, while \(\kappa=0.05\) corresponds to the transmission of one-tenth of the image data per instance. Algorithms 1 and 2 achieve convergence after two and six iterations, respectively. As the value of \(\kappa\) increases, the number of subcarriers required for communication also increases, which in turn leads to a degradation in radar performance. Consequently, the system needs to allocate more power and subcarrier channels to communication.