# Design of an Integrated System for Spaceborne SAR Imaging and Data Transmission
## 2 Construction of Mathematical Models
### Modeling Based on Joint Allocation System  
To optimize performance, each subcarrier is independently assigned to either the SAR system or the data transmission system. \[ \begin{split}\text{C}_{\text{s}}(\mathbf{p},\mathbf{u})&=\Delta B \sum\limits_{k=1}^{K}\text{log}_{2}\Bigg{(}1+\frac{\left|d_{k}\Big{|}2\Big{|} \mathbf{a}_{k}\Big{|}2\|\Phi_{\text{c},k}\right|^{2}(1-u_{k})p_{k}}{\sigma_{\text{ r}}^{2}}\Bigg{)}\\ &=\Delta B\sum\limits_{k=1}^{K}\text{log}_{2}\Bigg{(}1+\frac{| \alpha_{k}|_{2}(1-u_{k})p_{k}}{\sigma_{\text{c}}^{2}}\Bigg{)}\end{split}  \tag{8}\]  
where \(\text{C}_{\text{s}}(\mathbf{p},\mathbf{u})\) represents the data transmission capacity and \(\mathbf{u}\) is a binary selection vector [30], \(\mathbf{u}=\left[u_{1},\dots,u_{K}\right]^{\text{T}}\), where \(u_{k}\in\{0,1\}\). \(u_{k}=0\) denotes that the data transmission signal uses the \(k\)-th subcarrier and \(u_{k}=1\) indicates that the radar signal uses the \(k\)-th subcarrier. The power vector \(\mathbf{p}\) represents the power of each subcarrier, given by \(\mathbf{p}=\left[p_{1},p_{2},\dots,p_{K}\right]^{\text{T}}\). The radar power is given by \(\mathbf{p}_{\text{r}}=\mathbf{u}\odot\mathbf{p}\), while the data transmission power is \(\mathbf{p}_{\text{c}}=(\mathbf{1}_{K\times 1}-\mathbf{u})\odot\mathbf{p}\). \(\sigma_{\text{c}}^{2}\) denotes the power of noise in the data transmission channel. Considering the normalization of the modulation waveform, we assume \(\left\|\Phi_{k}\right\|^{2}=1\). SNR is a basic metric to measure the performance of radar echo, and it is also a common design quality in power allocation algorithms. This study does not involve the design of beamform or a precoding matrix, which correspond to different channel noise environments and reflect the direct relationship between subcarrier power and channels. On the other hand, PSLR and ISLR are related to the main lobe and sidelobe waveforms of the SAR signal itself, so they cannot directly reflect different channel noise environments. When the normalized backscattered signal intensity is set to 1, the SNR of SAR echo will be inversely proportional to the noise-equivalent sigma zero, which reflects the ability to detect the minimum backscattered signal [31]. The larger the SNR, the smaller the detectable backscatter signal becomes without being overwhelmed by noise, thereby enhancing recognition capability. Given that this study is focused mainly on the transmission quality of the overall SAR signals in channels, SNR is therefore used as the indicator for radar. \[\text{SNR}_{\text{s}}(\mathbf{p},\mathbf{u})=\sum\limits_{k=1}^{K}\frac{\left|\beta_{ i}\big{|}2\|\Phi_{k}\|^{2}u_{k}p_{k}\right.}{\sigma_{\text{r}}^{2}}=\sum \limits_{k=1}^{K}\frac{\left|\beta_{i}\big{|}2u_{k}p_{k}\right.}{\sigma_{ \text{r}}^{2}} \tag{9}\]  
where \(\text{SNR}_{\text{s}}(\mathbf{p},\mathbf{u})\) represents the SNR of SAR echoes and \(\sigma_{\text{r}}^{2}\) denotes the power of noise in the imaging channel. \[\text{Par}_{\text{s}}(\mathbf{p},\mathbf{u})=\frac{\text{max}(\left\|\Phi_{k}\right\|^ {2}p_{k})}{\frac{1}{K}\sum\limits_{k=1}^{K}\left(\left\|\Phi_{k}\right\|^{2}p_ {k}\right)}=\frac{\text{max}(p_{k})}{\frac{1}{K}\sum\limits_{k=1}^{K}\left(p_ {k}\right)}\leq\eta \tag{10}\]  
where \(\text{Par}_{\text{s}}(\mathbf{p},\mathbf{u})\) represents the PAPR of a channel and \(\eta\) indicates the constraint on the maximum PAPR. Given the fixed total power condition, the PAPR constraint can be reformulated as a peak constraint [32], and Equation (10) can be rewritten as follows:  
\[\left\{ \begin{array}{l}p_{k}\leq\frac{\eta P}{K}\\ \sum\limits_{k=1}^{K}p_{k}\leq P\end{array} \right. \tag{11}\]  
where \(P\) represents the total power of the transmitted signal. The optimization problem \(\mathcal{P}_{1}\) can be formulated as:  
\[