# 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
In practical tasks, selecting \(\kappa=0.25\) enhances the objective function and improves the overall radar and communication performance of the system compared to \(\kappa=0.5\). In Figure 7, at \(\kappa=0.5\), \(\kappa|u|_{1}\) represents the ratio of the data transmission volume required to the subcarrier bandwidth. The communication capacities of the two schemes of Algorithm 1 met the channel threshold and showed an improvement compared to the equal power allocation scheme. Compared to \(\eta~{}=~{}3\) dB, the communication capacity at \(\eta~{}=~{}6\) dB was closer to the channel threshold, resulting in greater power savings. Based on Figures 8 and 9 and Table 2, the genetic algorithm showed similar performance to the ADMM-based subcarrier allocation algorithm. The power allocation of the ADMM algorithm was the same as that of the greedy algorithm. The ADMM algorithm achieved the optimal SNR of echoes, while the genetic algorithm excelled in spectral efficiency but reduced the SNR of echoes. Although the ADMM algorithm did not greatly affect frequency efficiency, it improved the SNR of echoes. In general, the ADMM algorithm achieved the highest weighted score and displayed superior performance. ## 4 Conclusions

We proposed a unified design for power allocation methods in an integrated space-borne multicarrier radar and communication system. This method enabled independent subcarriers for communication and radar while sharing the same frequency band, RF transmission system, and transmitting antenna via FBMC-OQAM modulation. Algorithms 1 and 2, under certain PAPR constraints, successfully maximized the SNR of echoes while meeting data throughput requirements for transmission. This study explored the relationship between tolerable BER and required power in OFDM and FBMC communications. By utilizing two algorithms, we established a mathematical model that clarified the connections between peak power, image transmission weight, the SNR of echoes, and communication capacity. We then developed a strategy to enhance the SNR of echoes within specified limitations. Simulation results demonstrated that for images with 1 m azimuth and range resolutions, covering a 5 km by 15 km area, the required transmission rate for a single echo image ranged from 42.6 Mbps to 45.6 Mbps. In comparison to equal power allocation, Algorithm 1 enhanced the SNR of echo images by 2.79 dB and increased the data transmission rate by 24.075 Mbps under identical transmission. Conceptualization, Q.W. and J.Y. ; methodology, J.Y. and P.G. ; software, Q.W. ; validation, Q.W. and J.Y. ; formal analysis, Q.W. ; investigation, Q.W. and Z.X. ; resources, Q.W. ; data curation, Q.W. ; writing--original draft preparation, Q.W. ; writing--review and editing, Q.W. and J.Y. ; visualization, Q.W. ; supervision, J.Y. ; project administration, Q.W. and J.Y. ; funding acquisition, Z.X. All authors have read and agreed to the published version of the manuscript. The authors declare that this study received funding from National Key Research and Development Program of China under Grant 2024 YFE0200200. The funder had the following in volvement with the study: the integrated communication-sensing-computing framework oriented to satellite-terrestrial integration. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Data are contained within the article. **Conflicts of Interest:** Author [PERSON] was employed by the company NariARI Group Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. ## References

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