# Low-Resolution Precoding for Multi-Antenna Downlink Channels and OFDM +
## 5 Numerical Results  
We evaluate the GMIs of four systems. The main parameters are listed in Table 2 and we provide a few more details here. * System A: the DFT has length \(T_{F}=256\) and the channel has either \(L=1\) or \(L=15\) taps of Rayleigh fading with a uniform PDP. The minimum cyclic prefix length for the latter case is \(T_{c}=14\) so the minimum OFDM blocklength is \(T=270\);
* System B: MSM is applied to PSK. However, the MSM complexity limited the simulations to smaller parameters than for System A. The channel now has \(L=4\) taps of Rayleigh fading with a uniform PDP. The \(T=35\) OFDM symbols include a DFT of length \(T_{F}=32\) and a minimum cyclic prefix length of \(T_{c}=3\);
* System C: System C is actually two systems because we compare the performance under Rayleigh fading to the performance with the Winner2 model [51] whose number \(L\) of channel taps varies randomly. For the Winner2 channel, the choice \(T_{c}=30\) suffices to ensure that \(T_{c}\geq L-1\). The Rayleigh fading model has \(L=22\) taps with a uniform PDP, where \(L\) was chosen as the maximum Winner2 channel length that has almost all the channel energy;
* System D: similar to System A but for a 5G NR LDPC code with code rate \(8/9\) and 64-QAM for an overall rate of 5.33 bpcu. The LDPC code uses the BG1 base graph of the 3 GPP Specification 38.212 Release 15, including puncturing and shortening as specified in the standard. The code length is 9504 bits or 1584 symbols of 64-QAM; this corresponds to 4 frames of \(T_{F}=396\) symbols.The codewords were transmitted using at least \(T=410\) symbols that include a DFT of length \(T_{F}=396\) and a minimum cyclic prefix length of \(T_{c}=14\). The average GMIs for Systems A-C were computed using \(S=256\), \(B=200\), and a data-aided detector. The coded results of System D instead have \(S=1584\) symbols to fit the block structure determined by the LDPC encoder. For System D we considered both PAT and a data-aided detector. For all cases, the GMI was computed by averaging over the sub-carriers, i.e., channel coding is assumed to be applied over multiple sub-carriers and OFDMsymbols. The MAGIQ and QCM algorithms were both initialized with a time-domain quantized solution of the transmit matched filter (MF). Figures 3 and 4 show the average GMIs for System A with \(b=2\) and \(b=3\), respectively. In Figure 3, MAGIQ performs four iterations for each OFDM symbol while QCM performs six iterations. Observe that MAGIQ and QCM are best at all SNRs and they are especially good in the interesting regime of high SNR and rates. The gap to the rates over flat fading channels (\(L=1\)) is small. SQUID with 64-QAM requires 100-300 iterations for SNR \(>15\) dB and a modified algorithm with damped updates, otherwise SQUID diverges. In addition, we show the broadcast channel capacity with uniform power allocation and Gaussian signaling as an upper bound for the considered scenario [52;  
Figure 5 compares achievable rates of QCM, SQUID, and MSM for a smaller system studied in [42]. We use PSK because the MSM algorithm was designed for PSK. The figure shows that MSM outperforms SQUID and QCM at low to intermediate SNR and rates, but QCM is best at high SNR and rates. This suggests that modifying the cost function (8) to include a safety margin will increase the QCM rate at low to intermediate SNR, and similarly modifying the MSM optimization to more closely resemble QCM will increase the MSM rate at high SNR. We tried to simulate MSM for System A but the algorithm ran into memory limitations (we used 2 AMD EPYC 7282 16-Core processors, 125 GB of system memory, and Matlab with both dual-simplex and interior-point solvers). Consider next the Winner2 non-line-of-sight (NLOS) C2 urban model [51], which is more realistic than Rayleigh fading. The model parameters are as follows.