# 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 SNR for the downlink power of each subcarrier can be expressed as:

\[\text{SNR}_{k}=\frac{p_{k}h_{k}G_{\text{sat}}G_{\text{ut}}c^{2}}{(4\pi f_{ \text{c}}R)^{2}N_{\text{c}}}={\alpha_{k}}^{2}p_{k} \tag{27}\]

where \(\text{SNR}_{k}\) represents the SNR for the downlink of the \(k\)-th subcarrier; \(G_{\text{sat}}\) and \(G_{\text{ut}}\) denote the satellite antenna gain and the receiving antenna gain, respectively; \(N_{\text{c}}\) indicates the system noise power; and \(N_{\text{c}}=k_{\text{B}}\Delta BT_{u}\). \(h_{k}\) represents the stochastic channel parameter. Based on Equations (5) and (7), the image data volume for SAR with 1 m azimuth resolution, 1 m range resolution, and a coverage area of 5 km by 15 km, using 14 and 15 subcarrier channels, ranged between 84 and 90 Mb. The synthetic aperture time was 1.97 s, requiring the data transmission system to achieve a transfer rate of 42.6 to 45.6 Mbps. ### Simulation Result

In 100 Monte Carlo experiments, the BER of Rayleigh fading and additive white Gaussian noise (AWGN) channels at various SNR levels for CP-OFDM and FBMC modulation schemes are shown in Figure 4. The experiments were conducted under ideal conditions where Doppler effect and multipath effect were not considered. The convergency performance of the objective functions of Algorithms 1 and 2 under different \(\eta\) at \(\kappa=0.5\) is illustrated in Figure 5.

Figure 4: BER of Rayleigh fading and AWGN channels across various signal-to-noise ratio levels. At \(\eta=6\) dB, the convergency performance of the objective functions of Algorithms 1 and 2 under different \(\kappa\) is illustrated in Figure 6.

In the simulation, Algorithm 1 had an iteration time of 1.22 s, while Algorithm 2 took 6.52 s. Algorithm 1 achieved convergence in 2.44 s, whereas Algorithm 2 required between 39.12 and 52.16 s. For high-velocity low Earth orbit (LEO) SAR satellites traveling at approximately 7 km/s, Algorithm 1 is more suitable due to rapid changes in satellite-to-ground relationships and channel states. The short synthetic aperture time necessitates a fast algorithm speed to handle these dynamics, leading to the exclusive use of Algorithm 1 in subsequent experiments. The equal power allocation scheme prioritizes communication by allocating superior channels to the communication system and assigning the remaining channels to the SAR system. Figure 7 illustrates the spectral efficiency versus total power for Algorithm 1's allocation scheme and the equal power allocation scheme with a communication priority. Figure 5: Convergence performance of the objective function under different values of \(\eta\). (**a**) Algorithm 1. (**b**) Algorithm 2. Figure 6: Convergence performance of the objective function under different values of \(\kappa\). (**a**) Algorithm 1. (**b**) Algorithm 2. The two proposed improved algorithms can be compared with the classic power allocation algorithm. The water-filling algorithm [41] allocates power across different channels based on their specific states, following [PERSON]'s capacity formula. This algorithm prioritizes channels with lower noise power to maximize overall system performance. The greedy algorithm, as described in the literature [42], initially assigns data transmission power to meet required data rates. It then looks for the best power allocation for radar, considering the objective function parameters. Figure 8 illustrates the noise equivalent power of echo channels and the subcarrier power allocation for an integrated spaceborne SAR and data transmission system, both under a PAPR of \(\eta=2\) dB. Figure 9 displays the performance of algorithms for the integrated spaceborne SAR and data transmission system. It shows the comparisons of echo SNR and frequency efficiency across various algorithms. Figure 8: Subcarrier power allocation for integrated spaceborne SAR and data transmission system. \[q(t)=1+2\sum_{n=1}^{N-1}{(-1)}^{n}H_{n}\cos\left(\frac{2\pi nt}{T}\right) \tag{26}\]0 Spectral efficiency versus total power. (**a**) \(\eta=6\) dB scheme. (**b**) \(\eta=3\) dB scheme.