## 3 Proposed Method
### Loss Function  
To evaluate the communication performance while considering the impact of symbol confidence on system robustness, cross-entropy is selected as a refined communication performance metric [36], given by  
\[\mathcal{I}_{\text{ce}}=-\sum_{i=1}^{M}P_{\text{true}}(i)\log P_{\text{pred}}(i) \tag{10}\]  
where \(P_{\text{true}}\) is the true probability distribution of transmitted symbols and \(P_{\text{pred}}\) is the predicted probability distribution from the receiver. The cross-entropy loss penalizes low-confidence correct classifications by assigning higher loss values and encourages output distributions closer to ground truth through gradient updates. Under dynamic channel conditions, this metric effectively quantifies receiver sensitivity to channel impairments. The core objective of sensing systems lies in high-precision parameter estimation, such as range, velocity, and angle. The CRB, serving as the theoretical performance lower bound, characterizes the minimum mean square error (MSE) of unbiased estimators and evaluates the optimal achievable estimation accuracy. When the estimation parameter is \(\mathbf{\theta}=[\tau,f_{\text{D}}]\), we obtain \(\text{CRB}(\mathbf{\theta})=\mathbf{J}^{-1}(\mathbf{\theta})\), where \(\mathbf{J}(\mathbf{\theta})\) denotes the Fisher information matrix (FIM), yielding  
\[\mathbf{J}(\mathbf{\theta})=\frac{2}{\sigma^{2}}\sum_{j=0}^{N_{\text{s}}-1}\sum_{ m=0}^{M-1}\left(|\mathbf{X}_{M\times N_{\text{s}}}|^{2}\frac{\partial\mathbf{F}_{ \text{Tx}}^{*}}{\partial\theta_{i}}\frac{\partial\mathbf{F}_{\text{Tx}}}{ \partial\theta_{j}}\right), \tag{11}\]  
where \(F_{\text{Tx}}(j,m)=H_{p}\text{e}^{\text{j}2\pi(f_{\text{D}})T_{\text{s}}-m \Delta f\tau)}\) is the element of matrix \(\mathbf{F}_{\text{Tx}}\). \(\{\theta_{i},\theta_{j}\}\in\mathbf{\theta}\) corresponds to \(\theta_{1}=\tau\) and \(\theta_{2}=f_{\text{D}}\). Then, \(\mathbf{J}(\mathbf{\theta})\) is a matrix whose elements are as follows:  
\[\mathbf{J}(\mathbf{\theta})=\Bigg{[} \begin{array}{cc}\mathbf{J}(\theta_{1,1})& \mathbf{J}(\theta_{1,2})\\ \mathbf{J}(\theta_{2,1})&\mathbf{J}(\theta_{2,2})\end{array} \Bigg{]}, \tag{12}\]  
where  
\[ \begin{array}{c}\mathbf{J}(\theta_{1,1})=\frac{2}{\sigma^{2}}\sum_{j=0}^{N_{ \text{s}}-1}\sum_{m=0}^{M-1}|x_{m}(j)|^{2}\big{(}2\pi m\Delta f\big{)}^{2},\\ \mathbf{J}(\theta_{1,2})=J(\theta_{2,1})=\frac{2}{\sigma^{2}}\sum_{j=0}^{N_{ \text{s}}-1}\sum_{m=0}^{M-1}-|x_{m}(j)|^{2}\big{(}4\pi jmT_{\text{s}}\Delta f \big{)},\\ \mathbf{J}(\theta_{2,2})=\frac{2}{\sigma^{2}}\sum_{j=0}^{N_{\text{s}}-1}\sum _{m=0}^{M-1}|x_{m}(j)|^{2}\big{(}2\pi jT_{\text{s}}\big{)}^{2}.\end{array}  \tag{13}\]  
Then, the CRB function of parameters \(\tau\) and \(f_{\text{D}}\) is given by  
\[ \begin{array}{c}\text{CRB}(\tau)=\frac{\mathbf{J}(\theta_{2,2})}{\mathbf{J} (\theta_{1,1})\big{(}\theta_{2,2})-J(\theta_{1,2})\big{(}\theta_{2,1})},\\ \text{CRB}(f_{D})=\frac{\mathbf{J}(\theta_{1,1})}{\mathbf{J}(\theta_{1,1}) \big{(}\theta_{2,2})-J(\theta_{1,2})J(\theta_{2,1})}.\end{array}  \tag{14}\]  
To achieve a dynamic balance between communication and sensing performance, a weighted joint loss function \(Loss\) is constructed. For the joint performance of the communication-sensing system, the weighting factor \(\alpha\) is employed to combine communication performance and sensing performance, yielding  
\[Loss=(1-\alpha)\underbrace{\mathcal{L}_{\text{CS}}(\mathcal{I}_{\text{ce}}; \mathbf{c})}_{\text{Communication}}+\alpha\underbrace{\mathcal{L}_{\text{SC}}(\text{CRB}( \mathbf{\theta});\mathbf{c})}_{\text{Sensing}}, \tag{15}\]where the weighting factor \(\alpha\in[0,1]\) controls the priority between communication and sensing objectives, enabling adaptive constellation design for different scenarios. In communication-priority mode (\(\alpha\to 0\)), high-density irregular constellation mappings are adopted to maximize BER. In sensing-priority mode (\(\alpha\to 1\)), regular sparse constellation mappings are designed to minimize CRB.