## 4 Experiments
Their constellation diagrams are shown in Figure 8. Among them, the amplitude of 16 APSK satisfies the constant envelope characteristic, enabling it to combat nonlinear channels and making it suitable for power-constrained scenarios. The constellation points of 16 QAM are evenly distributed, offering high-frequency utilization, but it cannot effectively resist nonlinear distortion. The 16 spiral constellation exhibits a spiral structure, possessing excellent PN resistance properties. These constellation di  
Figure 6: BER of CO-IM-OFDM under different weighting factors. Figure 7: CRB of CO-IM-OFDM under different weighting factors. agrams cannot simultaneously satisfy the requirements of resisting PN, PA nonlinearity, and ensuring communication and sensing performance. Figure 9 illustrates BER performance curves of the proposed constellation and classical constellations when applied to IM-OFDM systems. When sensing performance is prioritized (\(\alpha=1\)), the proposed scheme exhibits degraded BER performance, even underperforming 16 QAM and 16 APSK. However, for \(0\leq\alpha\leq 0.8\), the proposed constellation achieves remarkable BER improvements. For instance, at \(\alpha=0.8\), it attains a 5 dB gain over 16 QAM under the same BER conditions, demonstrating its adaptability to channel constraints and system requirements. Furthermore, smaller \(\alpha\) values correlate with better BER performance, with optimization effects becoming more pronounced in high-SNR regions (\(E_{b}/N_{0}>18\,\mathrm{dB}\)). These results confirm that the proposed algorithm dynamically balances signal randomness and deterministic structures to enhance demodulation robustness. Figure 10 illustrates throughput performance for different training results. The simulation results indicate that, in terms of throughput performance, all the values of \(\alpha\), except for 1, significantly outperformed the traditional 16 QAM and 16 APSK. It is noteworthy that the result for \(\alpha=1\) showed a decrease in throughput compared to the baseline scheme. Under full sensing conditions, the training process solely maximizes the sensing performance without considering the communication performance. Consequently, the network parameters under this mode cannot effectively support efficient data transmission, leading to lower communication throughput than the baseline scheme that balances both communication and sensing. The significant improvement in throughput performance observed at other values of \(\alpha\) fully demonstrates that the proposed joint optimization architecture can effectively achieve performance gains in terms of throughput. Figure 8: The constellation: (**a**) 16 APSK; (**b**) 16 QAM; (**c**) 16 spiral. Figure 9: The comparison results of BER under different weighting factors. Figure 11 depicts CRB performance curves of the proposed constellation and benchmark constellations in IM-OFDM systems. When communication performance is prioritized (\(\alpha=0\)), the CRB degrades significantly due to the signal's high randomness, which introduces amplitude fluctuations detrimental to parameter estimation accuracy. However, as \(\alpha\) increases within \(0.2\sim 1\), the CRB decreases, reaching its minimum at \(\alpha=1\). This trend aligns with the theoretical trade-off between deterministic signal structures and stochastic modulation diversity in ISAC systems. The joint analysis of BER and CRB performance reveals that, except for the communication-optimal (\(\alpha=0\)) and sensing-optimal (\(\alpha=1\)) configurations, all trained constellations achieve significant performance gains compared to conventional 16 QAM, 16 APSK, and 16 spiral schemes.