## 4 Proposed Methods
### Update of \(\mathbf{X}\)
However, multi-bit precoding techniques can provide higher signal precision quantization than one-bit DACs. By employing multi-bit precoding at the output of DACs, the input signal can be decomposed into more levels, thus achieving finer quantization. This can significantly reduce quantization errors and improve the signal quality of the system. We thus shifted our focus to multi-bit precoding with low-resolution quantization DACs. Compared to traditional nonlinear precoding methods, simulation comparisons show that our algorithm achieves a lower BER at the same SNR. Moreover, as shown in Figure 4, it is observed that in large-scale MIMO systems, our algorithm improves BER performance by an average of 5% compared with traditional nonlinear precoding methods. \[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]2 Performance comparison of BER for various precoding algorithms in low-resolution quantization precoding with \(B=128\), \(U=16\), QPSK. \[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]3 Performance comparison of BER for various precoding algorithms with \(B=128\), \(U=16\), QPSK. ### Radar Performance

Further investigation was conducted on the impact of different transmit antennas on the beampatterns in various radar scenarios. In the symmetric scenario illustrated in Figure 5, we considered three targets located at angles \([-30^{\circ}\), \(0^{\circ}\), \(30^{\circ}]\). For beampattern design, \"Ideal\" refers to the optimal radar precoding, while \"Radar-Only\" denotes the beampattern achieved using our proposed method with \"\(\delta=1\)\". Here, the \"Ideal\" curve is obtained by first solving the radar-only problem described in Equation (10). We then decompose the resulting covariance matrix \(\mathbf{R}\) to obtain the radar precoding matrix \(\mathbf{X}_{0}\), which serves as the reference \"Ideal\" performance. When \"\(\delta=1\)\", the algorithm exclusively focuses on approximating the radar precoder \(\mathbf{X}_{0}\), thereby reducing its flexibility in adapting to communication symbols. This constraint results in a slight degradation in communication performance due to the limited degrees of freedom after quantization. The results demonstrate that, even under a one-bit constraint, the designed beampatterns generate primary lobes towards the specified target directions, ensuring effective radar detection and perception. In the asymmetric scenario depicted in Figure 6, where targets are positioned at \([-30^{\circ}\), \(0^{\circ}]\), our proposed approach significantly improves the PSLR for radar precoding towards these target angles. Notably, with \(B=128\), the performance exhibits only a minor degradation compared to the ideal radar precoding. This indicates that our method consistently enhances radar precoding performance in target directions within asymmetric settings as the number of antennas increases. To further quantify radar performance, we measure the PSLR for each number of antennas in both the symmetric and asymmetric scenarios. In the symmetric scenario (Figure 5), the PSLR values for \(B=8\), \(64\), \(128\) are approximately \(11.8\,\mathrm{dB}\), \(12.1\,\mathrm{dB}\), and \(13.2\,\mathrm{dB}\), respectively. Similarly, in the asymmetric scenario (Figure 6), the PSLR increases from about \(11.7\,\mathrm{dB}\) (\(B=8\)) to \(12.4\,\mathrm{dB}\) (\(B=64\)) and reaches \(12.8\,\mathrm{dB}\) (\(B=128\)). These results confirm that as the number of transmit antennas increases, our proposed method more effectively suppresses sidelobes, thus enhancing radar performance (Note that the 'Ideal' curve is omitted in Figure 6 to highlight the performance of the practical precoding strategies under the asymmetric scenario. The radar-only 'Ideal' reference, although it achieves slightly lower sidelobes, does not significantly change the comparative insights for these asymmetrically spaced targets).