Hybrid Precoding Design for Subarray-Structure-Enabled mmWave URLLC System

Abstract

Ultra-reliable low-latency communication (URLLC) is an important application scenario in fifth generation (5G) communication. With the increasing deployment of 5G, spectrum resources are becoming increasingly scarce, and millimeter wave (mmWave) operating in a high frequency has garnered significant attention. However, the short wavelength of the mmWave makes the signals susceptible to fading. Precoding has emerged as a promising solution for mitigating the severe path loss in mmWave, which can enhance the system capacity and communication performance. In our system, an achievable rate maximization problem is established by jointly optimizing the analog and digital precoders at the base station, subject to the constraint of the total power and the constant modulus constraint of the subarrays. The simulation results demonstrate that the algorithm employed in our system can successfully solve the optimization problem in this new scenario. Furthermore, the subarray-structure hybrid precoder has a higher energy efficiency than that of a fully connected hybrid precoder or all-digital precoder. The proposed algorithm also outperforms the MM (Majorization–Minimization) algorithm.

1. Introduction

Ultra-reliable low-latency communication (URLLC) is one of the most creative technologies in fifth generation (5G) communication, which is widely used in many fields such as industrial production, smart medical care, the internet of vehicles, electric power control, and so on [1,2]. According to the third generation project of the URLLC, the latency of the transmission should be less than 1 ms, and the block error rate should be less than ${10}^{-5}$ [3]. The rapid development of 5G has resulted in a significant increase in demand for spectrum resources, making them increasingly scarce and valuable.

This scarcity has led researchers to explore new techniques for enhancing the capacity of cellular networks without increasing the cell density [4]. One such technique is the use of millimeter-wave (mmWave) communication, which has garnered significant attention due to its ability to provide large bandwidth and a high transmission rate of up to gigabit-per-second [5]. To overcome the fading of the mmWave channel, precoding technology can be employed. Compared with traditional all-digital precoding, hybrid precoding can effectively reduce the system hardware complexity and improve the communication performance due to the lower number of radio frequency (RF) links [6]. The existing hybrid architecture schemes can be divided into two categories: fully connected architecture and subarray-connected architecture [7]. Compared with the fully connected architecture, the number of phase shifters (PSs) in the subarray-connected architecture is much lower.

In recent years, there has been a significant increase in research related to URLLC. Researchers have explored various approaches to optimize the design of the bandwidth, power allocation, beamforming, and other system parameters. One such approach was proposed in [8], which considered a multiuser URLLC scenario and aimed to optimize the maximum throughput of a downlink single-input single-output (SISO) and multiple-input multiple-output (MIMO) system. The proposed approach employed a path tracking algorithm to design the optimal digital precoding matrix, which could enhance the system capacity and improve the communication performance. A URLLC downlink MIMO communication model with the optimization objective of maximizing energy efficiency was adapted in [9], by constraining the block error rate and transmit power, and an iterative optimization algorithm was applied to design the all-digital precoding matrix. A hybrid precoding scheme for mmWave MIMO with a subarray-connected structure was designed in [10], which leveraged the Kronecker product to establish a coupled scheme of digital and analog precoders. The proposed scheme was shown to outperform traditional successive interference cancellation (SIC) algorithms in terms of performance, offering promising potential for enhancing the system capacity and communication performance in mmWave MIMO systems.

It can be observed that current research on URLLC typically focuses on the design of all-digital precoders, and the research on designing hybrid precoders in the mmWave frequency band is more oriented towards traditional communication scenarios [11,12]. As we can see, the research on the precoding design for a mmWave URLLC system is limited. Therefore, this paper investigates the precoding design for a subarray-structure-enabled mmWave URLLC system. The methods and results of this research can provide some reference for the practical application of technology in communication scenarios.

In order to facilitate the analysis of the nonconvex optimization problem, we utilize the rate optimization algorithm and the hybrid precoder alternating minimization algorithm. The simulation results show that our algorithms can solve the new scenario optimization problem, and the hybrid precoder with a subarray structure performs better than a fully connected hybrid precoder and all-digital precoder.

2. System Model

Consider a point-to-point mmWave MISO URLLC communication system, as shown in Figure 1 [13]. The base station is equipped with $N_{\mathrm{t}}$ antennas and $N_{\mathrm{RF}}$ RF chains, while the user is equipped with one antenna. The data stream of a single user model is $N_{\mathrm{p}}=1$.

2.1. Millimeter-Wave Channel

Generally speaking, mmWave channels are typically sparse and dispersed. The paper [5] provided a widely used Saleh–Valenzuela (SV) channel model for mmWave communication [14]. In the actual communication system, considering the short wavelength of mmWave communication, when the signal is directly transmitted between the base station and the user, it degrades due to factors such as bad weather and building obstruction. The attenuation of the signal is not conducive to transmission, and reliable line-of-sight (LOS) communication cannot be achieved. Therefore, we consider that the direct path from the base station to the user is non-line-of-sight (NLOS). The NLOS channel is represented as

$$\mathbf{h}=\sqrt{\frac{N_{\mathrm{t}}}{L_{\mathrm{d}}}}\sum _{l=1}^{L_{\mathrm{d}}}{\alpha }_l{\mathbf{a}}_{\mathrm{r}}\left({\varphi }_{\mathrm{r}}\right){\mathbf{a}}_{\mathrm{t}}{\left({\varphi }_{\mathrm{t}}\right)}^H,$$

(1)

where $\mathbf{h}\in {\mathbb{C}}^{N_{\mathrm{t}}\times 1}$, $L_{\mathrm{d}}$ represents the total number of paths, ${\alpha }_l$ represents the complex gain of the lth channel, ${\varphi }_{\mathrm{t}}$ represents the angle of departure, ${\varphi }_{\mathrm{r}}$ represents the angle of arrival, while ${\mathbf{a}}_{\mathrm{t}}$ and ${\mathbf{a}}_{\mathrm{r}}$ represent the normalized transmission and achievable array response vectors, respectively. The complex gain following Gaussian distribution ${\alpha }_l$ can be expressed as ${\alpha }_l\sim \mathcal{N}(0,{10}^{-0.1\eta })$, where $\eta =\hat{a}+10\hat{b}\log \left(\hat{d}\right)=\xi$, $\hat{d}$ is the distance between the base station and the user, $\xi \sim \mathcal{N}(0,{\sigma }_{\xi }^2)$.

2.2. Problem Formulation

Let $\mathbf{f}\in {\mathbb{C}}^{N_{\mathrm{RF}}\times 1}$ represent the digital precoding matrix and $\mathbf{A}\in {\mathbb{C}}^{N_{\mathrm{t}}\times N_{\mathrm{RF}}}$ represent the analog precoding matrix; let $N_{\mathrm{x}}=N_{\mathrm{t}}/N_{\mathrm{RF}}$ denote the number of antennas of each subarray. Since phase shifters can only change the phases of the signal, the amplitude of the analog precoding matrix is constant. Specifically, the unit modulus constraint can be expressed by $|{\mathbf{a}}_b|=1$, where ${\mathbf{a}}_b\in {\mathbb{C}}^{N_{\mathrm{x}}\times 1}$ is the analog precoder for the b-th RF chain.

Assuming that each signal block has M symbols, the transmit signal is $x\left[m\right],$ $m=1,2,\dots ,M$, and the received signal of the m-th symbol can be denoted as [15]

$$y\left[m\right]={\mathbf{h}}^H\mathbf{A}\mathbf{f}x\left[m\right]+z\left[m\right],m=1,2,\dots ,M,$$

(2)

where $\mathbf{h}$ represents the BS-to-UE link channel matrix, $z\left[m\right]$ represents the additive white Gaussian noise (AWGN) with zero mean and a variance of ${\sigma }_k^2$, where $z\left[m\right]\sim (0,{\sigma }^2)$. For the received signal, the signal-to-noise ratio (SNR) is

$$\gamma =\frac{|{\mathbf{h}}^H{\mathbf{A}\mathbf{f}|}^{\mathbf{2}}}{{\sigma }^2}.$$

(3)

With the short-packet regime, the achievable rate of the signal can be written as [16]

$$R={log}_2(1+\gamma )-\sqrt{\frac{V\left(\gamma \right)}{M}}\frac{Q^{-1}\left(\epsilon \right)}{ln2},$$

(4)

where $V\left(\gamma \right)=1-{(1+\gamma )}^{-2}$ is the channel dispersion value, $Q^{-1}$ is the inverse of function Q, and $\epsilon$ indicates the packet error rate.

In this paper, the maximum achievable rate of the signal is taken as the optimization objective, and the optimization problem is established as

$$\begin{array}{cc}\hfill \underset{\mathbf{A},\mathbf{f}}{max}& R\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \mathrm{such}\mathrm{that}& {\parallel \mathbf{A}\mathbf{f}\parallel }^{\mathbf{2}}\le {\mathbf{P}}_{\mathrm{tot}}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & |{\mathbf{a}}_b|=1,\hfill \end{array}$$

(7)

where Equation (6) represents the limitation of the transmit power, and $P_{\mathrm{tot}}$ represents the total transmit power of the system. As we can see, it is intractable to obtain the global optimal solution of problem (5) because of the nonconvexity of Equation (7); here, we convert the form of Equation (7) in the following part.

3. Problem Solution

In Section 3.1, we calculate the optimal all-digital precoding matrix, and the optimal solution will be substituted into the next subsection as the reference value. In Section 3.2, the Euclidean distance between the hybrid and all-digital precoding matrix is expected to be minimal to obtain a hybrid precoding matrix.

3.1. Optimal Digital Pre-Encoder Design

From Equation (4), we find that the achievable rate R contains two parts; the first part is the spectral efficiency of the Shannon theorem, and the second part is the channel dispersion function. Both of them are related to $\gamma$, which cannot be solved directly [17]. For ease of solving, we split R into two parts and analyze them separately, Let $\mathbf{w}=\mathbf{A}\mathbf{f}$, and we have [18,19]

$$\begin{array}{ccc}\hfill & & \underset{\mathbf{w}}{\max }R\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & {\mathrm{such}\mathrm{that}\parallel \mathbf{w}\parallel }^2\le P_{\mathrm{tot}}.\hfill \end{array}$$

(9)

Here, Equation (4) can be rewritten as

$$R=f\left(\mathbf{w}\right)-a^{\prime }g\left(\mathbf{w}\right),$$

(10)

where $f\left(\mathbf{w}\right)={log}_2(1+\gamma )$, $a^{\prime }=Q^{-1}\left(\epsilon \right)/ln2\sqrt{M}$, and $g\left(\mathbf{w}\right)=\sqrt{1-{(1+\gamma )}^{-2}}$. To ensure that R is a convex function for easy solving, we analyze the convexity of the two expressions on the right side of (8) separately.

(1)

Convex analysis for $f\left(\mathbf{w}\right)$

Expanding expression $f\left(\mathbf{w}\right)$, we have $f\left(\mathbf{w}\right)={log}_2(1+|{\mathbf{h}}^H{\mathbf{w}|}^2/{\sigma }^2)$. According to the principle of convex optimization, $|{\mathbf{h}}^H{\mathbf{w}|}^2$ can be judged to be a convex expression; namely, $(1+|{\mathbf{h}}^H{\mathbf{w}|}^2/{\sigma }^2)$ is convex. As a composition function of logarithms and powers, $f\left(\mathbf{w}\right)$ meets the characteristics of a logarithmic concave function.

(2)

Convex approximation for $g\left(\mathbf{w}\right)$

Expanding expression $g\left(\mathbf{w}\right)$, we have $g\left(\mathbf{w}\right)=\sqrt{(1-(1+|{\mathbf{h}}^H\mathbf{w}{{|}^2/{\sigma }^2)}^{-2})}$. It can be found that $(1-(1+|{\mathbf{h}}^H\mathbf{w}{{|}^2/{\sigma }^2)}^{-2})$ is a concave function, whose value is within interval $(0,1)$, which means $g\left(\mathbf{w}\right)$ is supposed to be concave. However, there exists a minus sign before $g\left(\mathbf{w}\right)$, which means Equation (8) can not be solved by the optimization algorithm.

The function g at ${\gamma }^k$ can be expanded with the first-order Taylor expansion as

$$\begin{array}{cc}\hfill g\left(\gamma \right)=& \sqrt{1-\frac{1}{{(1+\gamma )}^2}}\hfill \\ \hfill \approx & \sqrt{1-\frac{1}{{(1+{\gamma }^k)}^2}}+\frac{\gamma -{\gamma }^k}{{(1+{\gamma }^k)}^3\sqrt{1-\frac{1}{{(1+{\gamma }^k)}^2}}}\hfill \\ \hfill \triangleq & \hat{g}\left(\gamma \right).\hfill \end{array}$$

(11)

The achievable rate maximization optimization problem can be written as

$$R=\psi \left(\gamma \right)-a^{\prime }\hat{g}\left(\gamma \right).$$

(12)

In traditional communication scenarios, the optimal value of the digital precoding vector can be obtained by using the MRT solution when the transmit power is maximized. However, in the context of the URLLC, the achievable rate function R of the system is not a jointly concave function, and the maximum spectral efficiency may not be achieved based on the MRT solution. Therefore, we use an MRT solution that contains an unknown variable $P_x$, and substitute it into the original optimization problem. Before further analysis, we first present the following theorem [20].

Theorem 1. 

The digital precoding vector of problem (8) is in the form of a Maximum-Ratio Transmission (MRT)

$${\mathbf{w}}^{*}=\sqrt{P_0}\frac{{\mathbf{h}}^H}{\parallel \mathbf{h}\parallel },$$

(13)

where $P_0$ represents the actual transmit power at the base station.

Proof. 

We assume there exists an arbitrary solution ${\mathbf{w}}_1=P_1\overline{\mathbf{a}}$ that satisfies $P_1\le P_{\mathrm{tot}}$, where $|\overline{\mathbf{a}}|=1$, and $P_1$ represents the power of the digital precoding vector ${\mathbf{w}}_1$. In this case, we can always find another feasible solution ${\mathbf{w}}_2=\sqrt{P_2}{\overline{\mathbf{h}}}^H/\left|\overline{\mathbf{h}}\right|$ with power $P_2=P_1\overline{\mathbf{h}}\overline{\mathbf{a}}/\left|\overline{\mathbf{h}}\right|$, where $P_2$ represents the power of the beamforming vector ${\mathbf{w}}_2$. Since $|\overline{\mathbf{h}}\overline{\mathbf{a}}{|}^2\le {\left|\overline{\mathbf{h}}\right|}^2$, we have $P_1\le P_2$. Since $P_1\le P_{\mathrm{tot}}$, there exists $P_2\le P_{\mathrm{tot}}$, which means that we can always obtain an MRT solution for the precoding vector. Therefore, for any feasible solution to the above optimization problem, we can obtain an MRT-form solution with respect to it.    □

In this case, the digital precoding vector can be written as ${\mathbf{w}}^{*}=\sqrt{P_x}\frac{{\mathbf{h}}^H}{\parallel \mathbf{h}\parallel }$, where $P_x$ is an unknown variable representing the actual transmission power at the base station. By substituting ${\mathbf{w}}^{*}$ into the signal-to-noise ratio expression, we can express the signal-to-noise ratio as

$$\begin{array}{cc}\hfill \gamma =& \frac{|{\mathbf{h}}^H{\mathbf{w}|}^2}{{\sigma }^2}\hfill \\ \hfill =& \frac{{\mathbf{h}}^H{\mathbf{w}}^{*}{{\mathbf{w}}^{*}}^H\mathbf{h}}{{\sigma }^2}\hfill \\ \hfill =& \frac{{\mathbf{h}}^H\sqrt{P_x}\frac{\mathbf{h}}{\parallel \mathbf{h}\parallel }\dotplus \sqrt{P_x}\frac{{\mathbf{h}}^{\mathbf{H}}}{\parallel \mathbf{h}\parallel }\mathbf{h}}{{\sigma }^2}\hfill \\ \hfill =& \frac{P_x{\parallel \mathbf{h}\parallel }_2^2}{{\sigma }^2}.\hfill \end{array}$$

(14)

The original optimization problem (8) can be written as an optimization problem related to the unknown variable $P_x$ as follows

$$\begin{array}{cc}\hfill \underset{P_x}{\max }& R\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \mathrm{such}\mathrm{that}& P_x\le P_{\mathrm{tot}}.\hfill \end{array}$$

(16)

Problem (15) can be solved by an iterative optimization algorithm, whose main idea is to find a solution through continuous iteration that satisfies the constraint and maximizes the objective function. If the convergence condition of optimization is satisfied, the obtained solution can be regarded as the result of the original problem. The iteration process of our algorithm is shown in Algorithm 1.

Algorithm 1 Digital Precoder Design (DPD) Algorithm

Input: Initialize the transmit power that satisfies $P_x^0\le P_{\mathrm{tot}}$, convergence threshold $ϵ$, and maximum iteration number $L_k$. Step1: Set the iteration coefficient $k=0$. Step2: Calculate $P_x^{k+1}$ according to (15). Step3: $P_x^k=P_x^{k+1}$. Step4: $k\leftarrow k+1$. Step5: The result converges or the maximum iteration number $L_k$ is reached. Step6: Substitute the optimal value $P_x^k$ into Equation (13) to obtain ${\mathbf{w}}^k$. Step7: The optimal all-digital precoding vector $\mathbf{w}={\mathbf{w}}^k$.

3.2. Digital and Analog Precoder Design [21]

Based on the optimal all-digital precoding matrix ${\mathbf{w}}^{\mathrm{opt}}$ obtained in Section 3.1, we let the Euclidean distance between the hybrid precoding matrix and ${\mathbf{w}}^{\mathrm{opt}}$ be the shortest and analyze the constant-mode constraint on the subarray structure. Since the joint optimization of two unknown variables is complicated, the alternating minimization (Alt-min) algorithm is used to solve $\mathbf{A}$ or $\mathbf{f}$ alternately, while fixing the other. Based on problem (6) and (7), the hybrid precoding matrix computational problem can be written as follows

$$\begin{array}{cc}\hfill \underset{\mathbf{A},\mathbf{f}}{\min }& \parallel {\mathbf{w}}^{\mathrm{opt}}{-\mathbf{A}\mathbf{f}\parallel }^2\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \mathrm{such}\mathrm{that}& |{\mathbf{a}}_b|=1.\hfill \end{array}$$

(18)

Since the subarray structure deploys fewer phase shifters, and each RF chain is connected to $N_{\mathrm{t}}/N_{\mathrm{RF}}$ antennas, the analog precoder $\mathbf{A}$ belongs to a set of block matrices ${\mathrm{\Phi }}_s$, where each matrix block is a $N_{\mathrm{t}}/N_{\mathrm{RF}}$ constant module dimension matrix, and the structure of $\mathbf{A}$ can be expressed as

$$\begin{array}{c}\hfill \mathbf{A}=\left[\begin{array}{cccc}{\mathbf{s}}_1& 0& \dots & 0\\ 0& {\mathbf{s}}_2& & 0\\ ⋮& & \ddots & ⋮\\ 0& 0& \dots & {\mathbf{s}}_{N_{\mathrm{RF}}}\end{array}\right],\end{array}$$

where ${\mathbf{s}}_i=[exp\left(j{\theta }_{(i-1)\frac{N_{\mathrm{t}}}{N_{\mathrm{RF}}}+1}\right),\dots ,exp\left(j{\theta }_{i\frac{N_{\mathrm{t}}}{N_{\mathrm{RF}}}}\right)]$, and ${\theta }_i$ represents the phase of the i-th phase shifter. The optimization objective can be rewritten as

$$\begin{array}{cc}\hfill \underset{\mathbf{A},\mathbf{f}}{\min }& \parallel {\mathbf{w}}^{\mathrm{opt}}{-\mathbf{A}\mathbf{f}\parallel }^{\mathbf{2}}\end{array}$$

(19)

$$\begin{array}{cc}\hfill \mathrm{such}\mathrm{that}& \mathbf{A}\in {\mathrm{\Phi }}_s.\end{array}$$

(20)

3.2.1. Analog Precoder Design

Assuming that $P_t={\parallel {\mathbf{w}}^{\mathrm{opt}}\parallel }^2$, $\mathbf{A}\mathbf{f}$ can be treated as every nonzero element in $\mathbf{A}$ multiplied by the corresponding row in $\mathbf{f}$, and there exists

$${\parallel \mathbf{A}\mathbf{f}\parallel }^2=\frac{N_{\mathrm{t}}}{N_{\mathrm{RF}}}{\parallel \mathbf{f}\parallel }^2=P_t.$$

(21)

Moreover, the optimization goal of problem (19) can be rewritten as

$$\underset{{\left({\theta }_i\right)}_{i=1}^{N_{\mathrm{t}}}}{\min }{\parallel {\left({\mathbf{w}}^{\mathrm{opt}}\right)}_{i,:}-e^{j{\theta }_i}{\left(\mathbf{f}\right)}_{l,:}^H\parallel }^2,$$

(22)

where $l=⌈i\frac{N_{\mathrm{RF}}}{N_{\mathrm{t}}}⌉$, which means the minimum integer that is no less than $\left(i\frac{N_{\mathrm{RF}}}{N_{\mathrm{t}}}\right)$ is assigned to l. $\mathbf{A}$ is set up as a matrix approximation problem for the phase rotation as

$$arg{\left(\mathbf{A}\right)}_{i,l}=arg{\left({\mathbf{w}}^{\mathrm{opt}}\right)}_{i,:}{\left(\mathbf{f}\right)}_{l,:}^H,i\le N_{\mathrm{t}},l=⌈i\frac{N_{\mathrm{RF}}}{N_{\mathrm{t}}}⌉.$$

(23)

From Equation (23), we note that the special characteristic of $\mathbf{A}$ simplifies the analog precoder design; thus, the constant mode constraint can be easily solved according to the above transformation.

3.2.2. Digital Precoder Design

Substituting Equation (21) into problem (19), the optimization problem can be expressed as

$$\begin{array}{ccc}\hfill \underset{\mathbf{f}}{\min }& & \parallel {\mathbf{w}}^{\mathbf{k}}{-\mathbf{A}\mathbf{f}\parallel }^2\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill \mathrm{such}\mathrm{that}& & {\parallel \mathbf{f}\parallel }^2=\frac{N_{\mathrm{RF}}{P}_t}{N_{\mathrm{t}}}.\hfill \end{array}$$

(25)

Problem (24) is a nonconvex quadratically constrained quadratic programming (QCQP) problem, which can be reformulated as a homogeneous QCQP problem

$$\begin{array}{cc}\hfill \underset{\mathbf{Y}\in {\mathbb{H}}^n}{\min }& \mathrm{Tr}\left(\mathbf{G}\mathbf{Y}\right)\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \mathrm{such}\mathrm{that}& \mathrm{Tr}\left({\mathbf{A}}_1\mathbf{Y}\right)=\frac{N_{\mathrm{RF}}{P}_t}{N_{\mathrm{t}}}\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & \mathrm{Tr}\left({\mathbf{A}}_2\mathbf{Y}\right)=1\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & \mathbf{Y}\succcurlyeq 0,\mathrm{rank}\left(\mathbf{Y}\right)=1,\hfill \end{array}$$

(29)

where ${\mathbb{H}}^n$ is a set of $n=N_{\mathrm{RF}}+1$ dimensional complex Hermitian matrices. Let $\hat{\mathbf{y}}={\left[\mathrm{vec}\left(\mathbf{f}\right)\mathbf{t}\right]}^{\mathrm{T}}$, where $\mathbf{t}$ is an auxiliary variable, $\mathbf{Y}=\hat{\mathbf{y}}{\hat{\mathbf{y}}}^{\mathrm{H}}$, ${\mathbf{w}}_b=\mathrm{vec}\left({\mathbf{w}}^{\mathrm{opt}}\right)$, and

$$\begin{array}{c}\hfill {\mathbf{A}}_1=\left[\begin{array}{cc}{\mathbf{I}}_{n-1}& 0\\ 0& 0\end{array}\right],{\mathbf{A}}_2=\left[\begin{array}{cc}{\mathbf{0}}_{n-1}& 0\\ 0& 1\end{array}\right],\\ \hfill G=\left[\begin{array}{cc}{({\mathbf{I}}_{P_{\mathrm{t}}}⨂\mathbf{A})}^H({\mathbf{I}}_{P_{\mathrm{t}}}⨂\mathbf{A})& -{({\mathbf{I}}_{P_{\mathrm{t}}}⨂\mathbf{A})}^H{\mathbf{w}}_b\\ -{\mathbf{w}}_b({\mathbf{I}}_{P_{\mathrm{t}}}⨂\mathbf{A})& {\mathbf{w}}_b^H{\mathbf{w}}_b\end{array}\right].\end{array}$$

The tensor product of ${\mathbf{I}}_{P_{\mathrm{t}}}$ and $\mathbf{A}$ is denoted as $({\mathbf{I}}_{P_{\mathrm{t}}}⨂\mathbf{A})$. For the matrix $\mathbf{Y}$, the rank constraint is partially nonconvex, so problem (26) can be omitted by semidefinite relaxation, by

$$\begin{array}{cc}\hfill \underset{\mathbf{Y}\in {\mathbb{H}}^n}{\min }& \mathrm{Tr}\left(\mathbf{G}\mathbf{Y}\right)\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \mathrm{such}\mathrm{that}& \mathrm{Tr}\left({\mathbf{A}}_1\mathbf{Y}\right)=\frac{N_{\mathrm{RF}}{P}_{\mathrm{t}}}{N_{\mathrm{t}}}\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \mathrm{Tr}\left({\mathbf{A}}_2\mathbf{Y}\right)=1\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & \mathbf{Y}\succcurlyeq 0.\hfill \end{array}$$

(33)

For complex flush QCQP problems, the SDR can be assumed to be tight if the number of constraints is less than 3. Thus, the above problem can be reduced to a semidefinite programming (SDP) problem to obtain a globally optimal solution for the digital precoder. The implementation of the digital and analog precoder design algorithm is summarized in Algorithm 2.

For Algorithm 1, the algorithm reduces the dimension of the complex digital precoding optimization problem into a low-complexity transmit power optimization problem. Assuming that the algorithm converges after $l_1$ iterations, the complexity of the optimization problem is expressed as $\mathcal{O}\left(l_1\right)$. Assuming that the number of iterations of Algorithm 2 is $l_2$, its complexity is $\mathcal{O}\left(l_2(N_{\mathrm{RF}}^3+N_{\mathrm{t}})\right)$. Considering that the complexity of Algorithm 1 is much lower than that of Algorithm 2, the whole complexity of the algorithm is $\mathcal{O}\left(l_2(N_{\mathrm{RF}}^3+N_{\mathrm{t}})\right)$ [22,23].

Algorithm 2 Hybrid Precoding Design (HPD) Algorithm

Input: Optimal all-digital pre-encoder ${\mathbf{w}}^{\mathrm{opt}}$. Step1: Initialization: Initialize a feasible ${\mathbf{A}}_0$ with random phrase, set k = 0. Step2: Fix ${\mathbf{A}}_k$, updating ${\mathbf{f}}_{\mathbf{k}+\mathbf{1}}$ according to problem (30). Step3: Fix ${\mathbf{f}}_{\mathbf{k}+\mathbf{1}}$, updating ${\mathbf{A}}_{k+1}$ according to Equation (23). Step4: $k\leftarrow k+1$. Step5: Repeat $Steps$ 2–4 until the result convergences. Step6: Output digital and analog precoder $\mathbf{A}$ and $\mathbf{f}$.

4. Simulation and Numerical Results

The number of antennas at the base station was $N_{\mathrm{t}}=32$, and the upper bound of the transmit power was $P_{\mathrm{tot}}$ = 0.1 W. The NLOS distance between the base station and the user equipment was 30 m, and each data block emitted by the base station had 500 symbols. The parameters of the mmWave channel model were set as $a=72$, $b=2.92$, $\xi$ = 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 $N_{\mathrm{t}}$ = 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.

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

$$E{E}_h=\frac{R}{P_c+N_{\mathrm{RF}}{P}_{\mathrm{RFC}}+N_{\mathrm{PS}}{P}_{\mathrm{PS}}+P_0},$$

(34)

where $P_c$ represents the transmit power of the base station, $P_{\mathrm{RFC}}$ represents the power of the RF chains and amplifier, $P_{\mathrm{PS}}$ 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 $N_t$. For a fully connected structure, the number of phase shifters is equal to the product of the number of transmitting antennas and the number of RF chains, which is $N_{\mathrm{t}}{N}_{\mathrm{RF}}$.

For all-digital precoders, the energy efficiency can be written as

$$E{E}_d=\frac{R}{P_c+N_{\mathrm{t}}{P}_{\mathrm{RFC}}+P_0},$$

(35)

where $P_c$ represents the transmit power of the base station, $P_{\mathrm{RFC}}$ represents the power consumption for the RF chains and amplifier, and $P_0$ represents the power consumption for other hardware.

Figure 5 compares the energy efficiency of the all-digital precoder with the fully connected hybrid precoder and the subarray-connected hybrid precoder, where $P_{\mathrm{RFC}}=100$ mW, $P_{\mathrm{PS}}=10$ mW, and $P_0=200$ mW [24,25].

As the number of RF chains increased, the energy efficiency of the hybrid precoder decreased, while the energy efficiency of the all digital precoder remained unchanged. This is because the number of RF links in the all-digital precoder is constant, and the average rate tended to stabilize even with other parameters unchanged, ensuring that the energy efficiency of the all-digital precoder remained unchanged. For the subarray-connected structure, due to the single user, the increase in the RF chains had little significant impact on the achievable rate, while the total power consumption of the system gradually increased, resulting in a downward trend in energy consumption. For fully connected structures, the increase in the achievable rate was far less than the increase in the energy consumption, resulting in a decreasing energy efficiency. Comparing the fully connected structure with the subarray-connected structure, the energy efficiency of the subarray-connected structures was significantly higher than that of the fully digital structures, indicating that using subarray-connected structures is a relatively better choice in terms of energy efficiency.

5. Conclusions

This paper investigated the design of hybrid precoders for an mmWave-channel-based URLLC communication system and explored a low-complexity, high-performance, high-reliability, and low-latency communication architecture. An achievable rate maximization problem was proposed by considering the total power, the precoder design, and the constant mode constraint of the subarray. To solve the nonconvex optimization problem, we employed two efficient iterative algorithms, DPD and HPD. This model can be applied to small-scale point-to-point communication scenarios and can effectively guarantee the communication in some delay-sensitive scenarios. The simulation results of our proposed algorithm demonstrated its ability to successfully solve the optimization problem in the new scenario, and our algorithm had better convergence and calculation results. Compared with another algorithm MM, it was found that the proposed algorithm had a better performance in the achievable rate and energy efficiency. Meanwhile, in the URLLC scenario with a single user and single antenna, the approximate solution of the MRT can also be used to solve the digital precoding. However, the research in this paper also had certain limitations. The algorithm used in this paper is only applicable to a URLLC with a single user and a single antenna. In multiuser scenarios, due to the interference between users, it is necessary to deal with interference items. In view of the characteristics of millimeter-wave fading, other technologies such as reconfigurable intelligent surface technology [26] can be used in addition to precoding technology to improve the quality of communication.