# Hybrid Precoding Design for Subarray-Structure-Enabled mmWave URLLC System
## 3 Problem Solution
### Digital and Analog Precoder Design [21]
#### 3.2.2 Digital Precoder Design  
Substituting Equation (21) into problem (19), the optimization problem can be expressed as  
\[\min_{\mathbf{f}} \|\mathbf{w}^{\mathbf{k}}-\mathbf{Af}\|^{2}\] (24) such that \[\|\mathbf{f}\|^{2}=\frac{N_{\mathrm{RF}}P_{t}}{N_{\mathrm{t}}}. \tag{25}\]  
The parameters of the mmWave channel model were set as \(a=72\), \(b=2.92\), \(\zeta=8.7\)[5]. The noise power \(\sigma^{2}=-104\) dBm, and the maximum decoding error probability was \(10^{-7}\). Figure 2 shows the convergence diagram of the DPD algorithm. By transforming the upper-bound transmit power or the number of antennas, we detected a tendency towards convergence. In total, 10 iterations were conducted in the algorithm, and it can be seen that after 3-4 iterations, the achievable rate began to converge and remained stable in the subsequent iterations. Through observation, it can be seen that the convergence was fast, indicating that the algorithm had good performance, which means in practical communication, the system performance can be improved by changing the number of transmit antennas. Figure 3 shows the relationship between the transmit power and the maximum achievable rate under the different numbers of transmitting antennas; the number of antennas was \(N_{\text{t}}=32\), 64. In this figure, by transforming the number of transmitting antennas at the base station, the achievable rate of the signal increased as well. If the transmit power remained unchanged, the more antennas the base station had, the higher the signal achievable rate was, which illustrates the scalability of the system. The achievable rate between the proposed algorithm and the MM (Majorization-Minimization) algorithm [24] of the subarray-connected structure was compared. It was found that the HPD algorithm in this system had better performance than the MM algorithm. As we can see, the achievable rate between the finite blocklength (FBL) and the infinite blocklength (IFBL) was compared. It was found that the achievable rate of the system under a finite packet length was slightly lower than that under an infinite packet length. A finite packet length means URLLC, and an infinite packet length is similar to the Shannon upper limit. To achieve higher reliability and a lower delay, the URLLC sacrifices some achievable rate of the signal. Figure 4 compares the impact of using a fully digital precoder and hybrid precoder. As the number of RF links increased, the signal' achievable rate in the subarray-connected structure also improved, but the rate of the all-digital precoder did not change. This is because the number of RF chains in the all-digital precoder is equal to the number of transmit antennas, which is not affected in this scenario. Fixing the number of RF chains, we found that the signal's achievable rate obtained under the all-digital precoder was the highest, because the all-digital precoder has the highest connection level and the best performance. The subarray-connected structure optimization algorithm we used had a better performance compared to the MM algorithm in [24], indicating that the algorithm we used was more suitable for the current scenario. Figure 2: Convergence diagram of the DPD algorithm. For fully connected structures and subarray-connected structures, their difference is the number of phase shifters; so, we calculated the energy efficiency of the different kinds of precoders. The energy efficiency ratio is  
\[EE_{h}=\frac{R}{P_{c}+N_{\text{RF}}P_{\text{RFC}}+N_{\text{PS}}P_{\text{PS}}+P_{0 }}, \tag{34}\]  
where \(P_{c}\) represents the transmit power of the base station, \(P_{\text{RFC}}\) represents the power of the RF chains and amplifier, \(P_{\text{PS}}\) represents the power of the phase shifters, and \(P_{0}\) represents the power consumption for other hardware. For a subarray-connected structure, the number of phase shifters is equal to the number of transmitting antennas, which is \(N_{t}\).