## 4 Proposed Methods
### Update of \(\mathbf{X}\)
Consequently, for \(n_{\max}\) outer iterations and \(k_{\max}\) inner DPR updates, the total complexity of the proposed algorithm is \(\mathcal{O}\big{(}n_{\max}(BUL+BL^{2}+B^{2}L+B^{3})+n_{\max}k_{\max}\big{(}L^ {3}(U+B)B^{2}\big{)}\big{)}\). Assuming that all algorithms execute the same number of iterations \(T\), for our proposed algorithm, the computational complexity per iteration is dominated by updating \(\rho\), \(\mathbf{X}_{0}\), \(c\), \(\bar{\mathbf{z}}\), and \(\bar{\mathbf{x}}\), leading to a total complexity of \(\mathcal{O}(T(BUL+BL^{2}+B^{2}L+B^{3}+k_{\max}(L^{3}(U+B)B^{2})))\). For the ADMM algorithm, the per-iteration complexity includes a costly matrix inversion operation with complexity \(\mathcal{O}\big{(}B^{3}L^{3}\big{)}\), resulting in a total complexity of \(\mathcal{O}(T(BUL+L^{3}(U+B)B^{3}))\). The BCDPG2 algorithm involves additional operations such as Cholesky decomposition and SVD, leading to a per-iteration complexity of \(\mathcal{O}(T(BUL+B^{2}L^{2}))\), and a total complexity of \(\mathcal{O}(T(BUL+B^{2}L^{2}+L^{3}(U+B)B^{2}))\). ## 5 Simulation Results and Discussion

This section assesses the radar and communication efficiency of our proposed method in a downlink setting. By default, the SNR is defined as \(P_{T}/N_{0}\) with \(P_{T}=1\). The communication frame or radar pulse length is fixed at \(L=20\), and we employ QPSK modulation with up to \(n_{\max}=100\) iterations. Each transmitter antenna uses a pair of low-resolution quantized DACs, with an extreme configuration of a single one-bit DAC per antenna. The user's reception employs QPSK symbol modulation. We compare the DPR algorithm (referred to as \"Proposed\") presented in this paper with various precoding algorithms, based on 1000 Monte Carlo trials. ### Communication Performance

We analyze the communication efficiency of large-scale MIMO DFRC downlink systems utilizing one-bit quantization. For Figure 2, the number of transmit antennas is set to \(B=64\) with \(U=8\) users. We set \"\(\delta=0\)\" to focus solely on communication performance. For Figure 3, we increase the setup to \(B=128\) transmit antennas and \(U=16\) users, using infinite-resolution zero-forcing (ZFi) precoding as a baseline for comparison. By comparing various precoding schemes, quantized linear precoding methods (namely, WF and MRT) exhibit lower complexity than the proposed nonlinear precoding but with a higher BER. For nonlinear precoding methods (namely, SQUID, ADMM, and projected gradient-based block coordinate descent (BCDPG2) [40]) (MRT and ZF are classified as linear precoding methods because they directly map transmitted symbols to antenna signals via a fixed linear filter, typically derived from the channel matrix; in contrast, techniques such as SQUID, ADMM-based algorithms, BCDPG2, and our proposed method incorporate iterative optimization and discrete quantization steps, resulting in a nonlinear input-output relationship), we observe that nonlinear precoding becomes significantly more advantageous as the SNR increases, for both the \(64\times 4\) and \(128\times 16\) systems. Specifically, simulation results demonstrate that, compared to ZFi with infinite precision quantization, the proposed method in the system with \(64\) transmit antennas incurs only a \(3\) dB loss at a BER of \(10^{-4}\). Furthermore, in a large-scale MIMO system with \(128\) transmit antennas, the SNR loss is under \(2\) dB. Employing a large antenna array effectively reduces the challenges associated with low-resolution quantization and RF hardware imperfections. As depicted in Figures 2 and 3, our proposed approach enhances BER performance in large-scale MIMO systems. Notably, when \(\delta=0\), the method achieves the highest BER efficiency. \[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]1 Performance comparison of BER for various precoding algorithms with \(B=64\), \(U=8\), QPSK. In previous trials, the system performance was evaluated using one-bit DACs.