## 4 Proposed Methods
### Update of \(\mathbf{X}\)
\[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]4 Influence of different transmit antenna numbers on beamforming performance in symmetric scenarios with QPSK modulation. ### Assessing Radar and Communication Trade-Offs

For the communication and radar integrated system, balancing the importance of different functionalities is crucial. Figure 7 has demonstrated communication performance v.s. different weighting factors, where \(B=128\) and \(U=16\). In Figure 7, using the linear one-bit precoding MRT algorithm as a baseline, we observe significant BER improvements when \"\(\delta=0\)\", which allocates all weighting to the communication side. When \"\(\delta=0.5\)\", dividing weights equally between communication and radar sensing, our method matches the MRT performance at \(\text{SNR}=-2\,\text{dB}\) and outperforms it at higher SNRs. Figure 8 shows the radar performance, indicating that increasing the weight factor \(\delta\) (allocating more resources to radar) leads to better PSLR. This implies that the system can better suppress sidelobe interference as the weight factor increases, to further enhance radar's target detection performance. This strategy of weighting factor allocation and balancing different functional requirements allows us to adjust the system's performance according to actual needs for communication performance while ensuring radar performance. In Figure 9, we compare the average running time of algorithms with different numbers of antennas (in seconds), considering \"\(\delta=0.5\)\" as the performance criterion. For instance, BCDPG2 maintains the lowest average runtime among the nonlinear precoding algorithms, with values ranging from approximately \(2.5\times 10^{-3}\) s at 32 antennas to \(6\times 10^{-3}\) s at 256 antennas. The Proposed precoder, although slightly slower, achieves runtimes of approximately \(2.5\times 10^{-3}\) to \(1\times 10^{-2}\) s for the same range of antennas, indicating competitive efficiency. In contrast, the SQUID and ADMM algorithms exhibit significantly higher runtimes, surpassing \(10^{-2}\) s when the number of antennas reaches 256. The linear quantized precoders (MRT, ZF, ZFi, and WF) remain much faster due to their lower complexity, with runtimes consistently around \(10^{-4}\) s. However, this reduced complexity comes at the cost of poorer performance in high-precision scenarios. As shown in Figure 10, we present a comparison of the convergence metric versus the number of iterations, assuming the objective function defined in Equation (19) has a value at the \(n\)-th iteration denoted as \(Q^{n}\). The convergence metric is defined as \(\left|\,Q^{n+1}-Q^{n}\right|/Q^{n}\). It can be observed that increasing the quantization resolution results in a decrease in convergence

\[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]5 Influence of different transmit antenna numbers on beamforming performance in asymmetric scenarios with QPSK modulation. accuracy. However, the algorithm maintains robust performance, achieving convergence within approximately 230 iterations even when using a 3-bit DAC. \[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]6 Beamforming analysis with different weighting factors. \[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]7 BER analysis with different weighting factors. \[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]8 Impact of quantization resolution on algorithm convergence. \[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]9 Average runtime performance of precoding algorithms with varying numbers of transmit antennas.