## 4 Proposed Methods
### Update of \(\mathbf{X}\)
### BER Performance in the Presence of CSI Imperfections

This section investigates the robustness of our proposed algorithm for channel estimation errors. According to [8], we assume that the BS has obtained CSI and the channel matrix is calculated as follows

\[\bar{\mathbf{x}}^{(k+1)} =\arg\min_{\bar{\mathbf{x}}}L(\bar{\mathbf{z}}^{(k+1)},\bar{\mathbf{x}},\mathbf{ \eta}^{(k)})\]0

where \(\varepsilon\in[0,1]\) indicates the channel estimation error level, \(\widetilde{\mathbf{H}}\) represents the estimated channel matrix, and \(\Delta\mathbf{H}\) denotes the error matrix. Figure 11 illustrates the proposed method's robustness against channel estimation errors based on simulation results. Since perfect CSI is typically unavailable, this validates the adaptiveness of the proposed method for practical DFRC systems. BER performance under imperfect CSI is evaluated. At an SNR of 5 dB, it is observed that nonlinear precoding exhibits significant advantages in performance when \(\varepsilon\leq 0.5\), while the BER performance of linear precoding approaches saturation. This indicates that when the channel estimation error is relatively small, nonlinear precoding can better handle the channel estimation errors, thereby improving system performance. For \(\varepsilon\leq 0.3\), nonlinear precoding methods demonstrate significant performance advantages, particularly within smaller ranges of channel estimation errors. This enables the system to better adapt to the complexities of practical communication environments, thereby enhancing system reliability and performance. ## 6 Conclusions

This paper proposes a joint optimization method for large-scale MU-MIMO DFRC downlink systems with low-resolution quantized DACs and frame-based precoding. The method balances radar sensing and communication by decomposing the optimization problem and using the DPR algorithm within the ADMM framework. Simulation results validate the effectiveness of the proposed method across diverse scenarios. Compared

\[\bar{\mathbf{x}}^{(k+1)} =\arg\min_{\bar{\mathbf{x}}}L(\bar{\mathbf{z}}^{(k+1)},\bar{\mathbf{x}},\mathbf{ \eta}^{(k)})\]1 Comparison of BER for different precoding schemes with channel estimation error at SNR = 5 dB. to traditional nonlinear precoding methods, the proposed algorithm delivers an average \(5\%\) improvement in BER performance for QPSK modulation in higher SNR scenarios. Moreover, practical systems with large antenna arrays inherently rely on parallel digital processing architectures and fast hardware accelerators (e.g., GPUs or FPGAs) to handle matrix operations efficiently. Thus, the proposed method can be implemented in real-time or near real-time radar-communication systems, making it applicable to next-generation wireless infrastructures where cost and energy constraints demand low-resolution hardware solutions. Conceptualization, X.F., Z.Z. ([PERSON]) and J.W. ([PERSON]); Data curation, Z.Z. ([PERSON]) and J.W. ([PERSON]); Formal analysis, X.F. and Z.Z. ([PERSON]); Funding acquisition, X.F., Z.Z. ([PERSON]) and J.W. ([PERSON]); Investigation, X.F., Z.Z. ([PERSON]), Z.Z. ([PERSON]) and J.W. ([PERSON]); Methodology, X.F., Z.Z. ([PERSON]) and Z.Z. ([PERSON]); Project administration, X.F., Z.Z. ([PERSON]) and Z.Z. ([PERSON]); Resources, Z.Z. ([PERSON]), Z.Z. ([PERSON]) and [PERSON] ([PERSON]); Supervision, X.F. and J.W. ([PERSON]); Validation, Z.Z. ([PERSON]), J.W. ([PERSON]) and [PERSON] ([PERSON]); Writing--original draft, X.F., Z.Z. ([PERSON]) and [PERSON] ([PERSON]); Writing--review and editing, X.F., [PERSON] ([PERSON]) and [PERSON] ([PERSON]). All authors have read and agreed to the published version of the manuscript. This research was supported in part by the major scientific and technological innovation projects of Shandong Province of China (Grant Nos. 2021 ZLGX05 and 2022 ZLGX04), and in part by the National Natural Science Foundation of China, Grant 42127804. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.