Hybrid Precoding Based on Partial Connection for Millimeter-Wave Massive MIMO System

Abstract

Aiming at poor system performance in existing partially connected hybrid precoding schemes, in this paper, we propose a partially connected hybrid precoding scheme based on alternating optimization with the auxiliary precoder. In particular, the spectral efficiency optimization problem is transformed into a sub-optimization problem under certain constraints. For each sub-optimization problem, we use the singular value decomposition method to optimize the analog precoder and digital precoder alternately, and optimal solutions are provided for both digital and analog precoders in each alternate iteration. Moreover, we design a heuristic algorithm to calculate the initial analog precoder as the starting point for alternating optimization, which speeds up the convergence speed. The simulation results show that the proposed algorithm is at least 16.2% higher than the existing two traditional partially connected algorithms at 20 dB received SNR with almost no increase in complexity.

1. Introduction

Fifth-Generation Mobile Communication Technology (5G) and 6G mobile communication require the support of many technologies such as heterogeneous networks and mobile ad hoc networks [1,2,3,4]. They can greatly improve the efficiency of systems. Millimeter-wave (mm-Wave) multiple-input multiple-output (MIMO) technology is one of the foundations, so it has attracted extensive attention from scholars at home and abroad [5,6,7]. The high frequency of mm-Waves enables the size of large-scale antennas to be designed to be small, so a large-scale antenna array can be packaged into a small device. This feature can better realize the large-scale input and output in the mm-Wave system, to provide sufficient beam gain to counter the very serious path loss in the mm-Wave channel [8,9]. However, in the traditional MIMO system, the all-digital precoding architecture requires the same number of radio frequency (RF) chains (including analog-to-digital/analog converters (ADC/DAC), etc.) as the number of antennas. These large amounts of mm-Wave RF chains and other hardware components bring unbearable costs and power consumption to the system, so the full-digital precoding scheme cannot be practically applied to mm-Wave systems [10,11]. To reduce the number of RF chains and reduce hardware costs and power consumption, a hybrid precoder composed of an analog precoder and a digital precoder has been proposed [12,13,14].

The hybrid precoding is mainly divided into partially connected (PC) architecture, fully connected (FC) architecture, and hybrid-connected (HC) architecture according to the different mapping modes between RF chains and antennas. In [15], the authors proposed a fully connected scheme based on the orthogonal matching pursuit (OMP) algorithm, which selects the optimal column vectors with the greatest correlation with the residual matrix in the array response matrix column by column to update the analog precoder, and then calculates the digital precoder through the least-squares method each time. This scheme can achieve good performance, but the OMP algorithm involves a lot of matrix transposition, which makes it more complicated. A partially connected scheme based on a continuous interference cancellation (SIC) algorithm was proposed in [16]. The antenna is divided into multiple subarrays through each RF chain, and the total spectral efficiency is converted into the sum of the rates of each antenna subarray, which is calculated one by one to reduce the computational complexity. However, this method is limited to cases in which the number of RF chains is equal to the data flow, so its practical application is limited. In [17], a precoding algorithm based on geometric mean decomposition (GMD) was proposed, which avoids complex bit allocation. However, the GMD algorithm contains an inherent zero-forcing operation, which causes a large capacity loss. The authors of [18] proposed converting the design of the RF precoder and the combinator into a problem based on the maximization of channel gain. They completed the design of the analog precoder through multiple iterations by using the method of alternating minimization and then achieved the design of the baseband part by using the singular value decomposition (SVD) of the channel. However, due to the random generation of the initial analog combination precoder during the alternating optimization, it requires many iterations to achieve better performance, which leads to high computational complexity. Using the signal-to-leakage-and-noise ratio (SLNR) as the performance index, the authors of reference [19] jointly optimized the analog–digital hybrid precoding and power allocation to maximize the spectral efficiency of the system. The authors of [20] proposed effective alternating minimization algorithms based on the zero-gradient method to establish fully connected structures. Although the algorithms in references [19,20] have improved the system performance to some extent, the implementation complexity is high and the fully connected structure also results in higher hardware costs.

Based on the above research analysis and engineering application, we propose a partially connected hybrid precoding scheme based on an alternating optimization algorithm to improve system performance in this paper. We first propose that the digital precoder can be obtained by singular value decomposition of the auxiliary precoder, which is composed of the optimal full-digital precoder and the analog precoder. Furthermore, to approximate the optimal full-digital precoder, an alternating optimization algorithm to maximize the trace of the product of the optimal full-digital precoder and the hybrid precoder is designed. The optimization of the analog precoder and the update of the digital precoder are completed step by step. In addition, to achieve fast convergence, we designed a heuristic algorithm to calculate the initial value of the analog precoder as the starting point of alternating optimization. Compared with the simulation results in [18,21], the alternating optimization algorithm proposed in this paper can achieve performance comparable to existing full-digital precoding schemes. At the same time, the number of iterations is greatly reduced due to the design of the initial analog precoder, which not only ensures the system’s performance but also reduces the computational complexity.

In this paper, we develop a scheme for the mm-Wave system based on the hybrid precoder. We use alternate optimization as the main design principle, which helps to decouple the design problem of the precoder into two sub-problems. The novel contributions of this paper are summarized as follows:

• We propose a new alternate optimization framework. The hybrid precoder is effectively designed by SVD variation of the auxiliary matrix, and optimal solutions are provided for both digital and analog precoders in each alternate iteration. Compared with the traditional non-alternate optimization precoding scheme, the system performance is greatly improved
• Aiming at the disadvantage of slow convergence in alternate optimization algorithms, we propose a heuristic algorithm to calculate the initial values of analog precoders. Compared with the traditional alternative optimization method, which randomly sets the initial value, the proposed algorithm converges faster and is less likely to fall into local suboptimal solutions.
• Simulation is performed in a more realistic scenario, which is controlled by the accuracy factor of the channel. In addition, the proposed algorithm does not require an equal relationship between the number of RF chains and the transmitted data streams, which increases the flexibility of its application scenarios.

Notations: ${\mathit{I}}_n$ represents the identity matrix of $N\times N$, {${\mathit{X}}^{\mathit{H}}$,${\mathit{X}}^T$,${\mathit{X}}^{-1}$} represent the conjugate transpose matrix, transpose matrix, and inverse matrix of $\mathit{X}$ respectively. $tr\left(\mathit{X}\right)$ represents the trace of $\mathit{X}$, $E\left(\mathit{X}\right)$ represents the expectation of $\mathit{X}$, $\left|\mathit{X}\right|$ represents the determinant of $\mathit{X}$, ${\Vert \mathit{X}\Vert }_2$ represents the 2 norms of $\mathit{X}$.

2. System and Channel Model

2.1. System Model

Although the performance of the partially-connected hybrid beamforming architecture is inferior to that of the fully connected hybrid beamforming architecture, its energy consumption is low and the implementation is simple. We consider a point-point mm-Wave MIMO system with antenna sub-array hybrid beamforming architecture at the transmitting end, where a transmitter with $N_t$ antennas sends $N_s$ data streams to a receiver with $N_r$ antennas. The number of RF chains at the transmitter is $N_{RF}$, which satisfies $N_s\le N_{RF}\le N_t$. Each RF chain has only one independent antenna sub-array connected to it, and each independent sub-array has $M$ antenna, where $M=N_t/N_{RF}$. Generally, the transmitted signal is given by

$$x={\mathit{F}}_{RF}{\mathit{F}}_{BB}\mathit{s}$$

(1)

where $\mathit{s}\in {ℂ}^{N_s\times 1}$ denotes the transmitted signal with $E\left(\mathit{s}{\mathit{s}}^{\mathit{H}}\right)=\frac{1}{N_s}{\mathit{I}}_{N_{\mathit{S}}}$, ${\mathit{F}}_{RF}\in {ℂ}^{N_t\times N_{RF}}$ denotes the RF precoder and ${\mathit{F}}_{BB}\in {ℂ}^{N_{RF}\times N_s}$ denotes the digital precoder.

In addition, we mainly focus on two constraints based on the partially connected architecture. One is total power constraints at the base station (BS), which can be presented as

$$\Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}{\Vert }^2=N_s$$

(2)

The other is that according to the system characteristics of partial connection architecture, the RF precoder has hardware constraints, which can be presented as

$${\mathit{F}}_{RF}=\left(\begin{array}{ccc}{\mathit{f}}_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {\mathit{f}}_N\end{array}\right)$$

(3)

where $N=N_{RF}$, each subarray ${\mathit{f}}_n\in {ℂ}^{M\times 1}$, $n=1,2,\cdots , \mathit{a}nd N$ in the ${\mathit{F}}_{RF}$ satisfies $\left|{\mathit{f}}_n\right|=\frac{1}{\sqrt{M}}$. The received baseband signal $\mathit{y}\in {ℂ}^{{\mathit{N}}_r\times \mathit{1}}$ can be written as

$$\mathit{y}=\sqrt{\rho }\mathit{H}{\mathit{F}}_{RF}{\mathit{F}}_{BB}\mathit{s}+\mathit{n}$$

(4)

where $\rho$ denotes the average received power, $\mathit{H}\in {ℂ}^{N_r\times N_t}$ denotes the channel matrix, and $\mathit{n}\in {ℂ}^{{\mathit{N}}_{\mathit{r}}\times \mathit{1}}$ is the vector of the independent and identically distributed (i.i.d) $\mathcal{C}\mathcal{N}\left(0,{\sigma }^2\right)$ noise.

2.2. Channel Model

Assuming that the channel state information is known, here we adopt a simplified cluster channel model based on the extended Saleh–Valenzuela model [22]. The discrete-time narrowband channel $\mathit{H}$ can be expressed as:

$$\mathit{H}=\sqrt{\frac{M_r{M}_t}{L}}{\displaystyle \sum }_{l=1}^L{\alpha }_l{\mathit{a}}_r\left({\theta }_l\right){\mathit{a}}_t^{\mathit{H}}\left({\phi }_l\right)$$

(5)

where $L$ is the number of propagation paths between the BS and the mobile station (MS). ${\alpha }_l$ represents the complex gain of the $l$-th path and ${\theta }_l\in \left[0,2\pi \right]$ and ${\phi }_l\in \left[0,2\pi \right]$ denote the azimuth (elevation) angles of arrival and departure, respectively. The vectors ${\mathit{a}}_t\left({\phi }_l\right)$ and ${\mathit{a}}_r\left({\theta }_l\right)$ are the received and transmitted array response vectors, respectively. Considering the uniform linear array (ULA) with N antennas, the forms of ${\mathit{a}}_t\left(\phi \right)$ and ${\mathit{a}}_r\left(\theta \right)$ are

$${\mathit{a}}_t\left(\phi \right)=\frac{1}{\sqrt{M_t}}{\left[1,e^{jkd\sin \phi },\cdots ,e^{j\left(M_t-1\right)kd\sin \phi }\right]}^T$$

(6)

$${\mathit{a}}_r\left(\theta \right)=\frac{1}{\sqrt{M_r}}{\left[1,e^{jkd\sin \theta },\cdots ,e^{j\left(M_r-1\right)kd\sin \theta }\right]}^T$$

(7)

where $k=\frac{2\pi }{\lambda }$ and $\lambda$ is the signal wavelength and $d$ is the spacing between adjacent antennas.

3. Hybrid Precoding Design Based on Alternating Optimization

In this section, based on the system model in the previous section, we will propose an alternating optimization-based scheme to implement partially connected hybrid precoding.

3.1. Problem Description

In this section, the goal of designing the partially connected hybrid precoding is to improve the system’s spectral efficiency and reduce the complexity of the algorithm. The system model is shown in Figure 1.

We only seek to design the hybrid precoders at the BS, that is, to solve the hybrid precoder $\left({\mathit{F}}_{RF},{\mathit{F}}_{BB}\right)$. Therefore, the spectral efficiency of the mm-Wave massive MIMO system can be expressed as

$$R={\log }_2\left(\left|{\mathit{I}}_{N_r}+\frac{\rho }{N_s{\sigma }_n^2}\mathit{H}{\mathit{F}}_{RF}{\mathit{F}}_{BB}\times {\mathit{F}}_{BB}^{\mathit{H}}{\mathit{F}}_{RF}^{\mathit{H}}{\mathit{H}}^{\mathit{H}}\right|\right)$$

(8)

The digital precoder ${\mathit{F}}_{BB}$ and the analog precoder ${\mathit{F}}_{RF}$ are designed to maximize the spectral efficiency in Equation (8). The optimization objective is transformed into:

$$\left({\mathit{F}}_{RF},{\mathit{F}}_{BB}\right)=\arg max{\log }_2\left(\left|{\mathit{I}}_{N_r}+\frac{\rho }{N_s{\sigma }_n^2}\mathit{H}{\mathit{F}}_{RF}{\mathit{F}}_{BB}\times {\mathit{F}}_{BB}^{\mathit{H}}{\mathit{F}}_{RF}^{\mathit{H}}{\mathit{H}}^{\mathit{H}}\right|\right)$$

(9)

$$s.t.\Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}\Vert {}_F^2=N_s$$

$$\left|{\mathit{f}}_n\right|=\frac{1}{\sqrt{M}};n=1,2,\cdots N$$

We can transform Equation (8) into finding the minimum Euclidean distance between the optimal full-digital precoder and the product of the hybrid precoders ${\mathit{F}}_{RF}{\mathit{F}}_{BB}$ [15]:

$$\left({\mathit{F}}_{RF}^{opt},{\mathit{F}}_{BB}^{opt}\right)=\arg \min \Vert {\mathit{F}}_{opt}-{\mathit{F}}_{RF}{\mathit{F}}_{BB}\Vert {}_F$$

(10)

$$s.t.\Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}\Vert {}_F^2=N_s$$

$$\left|{\mathit{f}}_n\right|=\frac{1}{\sqrt{M}};n=1,2,\cdots N$$

We define the singular value decomposition (SVD) of $\mathit{H}$ as $\mathit{H}=\mathit{U}\mathit{\Sigma }{\mathit{V}}^{\mathit{H}}$, where $\mathit{U}$ and $\mathit{V}$ correspond to the left and right singular matrices of $\mathit{H}$, respectively $\mathit{\Sigma }$ is the characteristic matrix of $\mathit{H}$ . The optimal full-digital precoder is ${\mathit{F}}_{opt}=\mathit{V}\left(:,1:N_s\right)$  [15].

According to reference [23], the Euclidean distance between ${\mathit{F}}_{RF}{\mathit{F}}_{BB}$ and ${\mathit{F}}_{opt}$ is the closest when the trace of the product of them is the largest. And Equation (10) is equivalent to

$$\left({\mathit{F}}_{RF}^{opt},{\mathit{F}}_{BB}^{opt}\right)=\arg \max tr\left({\mathit{F}}_{opt}^{\mathit{H}}{\mathit{F}}_{RF}{\mathit{F}}_{BB}\right)$$

(11)

$$s.t.\Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}\Vert {}_F^2=N_s$$

$$\left|{\mathit{f}}_n\right|=\frac{1}{\sqrt{M}};n=1,2,\cdots N$$

3.2. Alternating Iterative Design

In this section, an iterative optimization method is proposed to design the analog precoder ${\mathit{F}}_{RF}$ and the digital precoder ${\mathit{F}}_{BB}$ to maximize the system spectral efficiency. The analog precoder ${\mathit{F}}_{RF}$ satisfying the hardware constraint is randomly generated as the initial precoder. Defining the auxiliary precoder $\mathit{G}={\mathit{F}}_{opt}^{\mathit{H}}{\mathit{F}}_{RF}$, through the product property of matrixes $tr\left(AB\right)=tr\left(BA\right)$, Equation (11) is transformed into

$${\mathit{F}}_{BB}^{opt}=\arg \max tr\left({\mathit{F}}_{BB}\mathit{G}\right)$$

(12)

$$s.t.\Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}{\Vert }_F^2=N_s$$

$$\left|{\mathit{f}}_n\right|=\frac{1}{\sqrt{M}};n=1,2,\cdots N$$

Let $\mathit{G}$ be decomposed by SVD as $\mathit{G}={\mathit{U}}_{\mathit{G}}{\mathit{\Sigma }}_{\mathit{G}}{\mathit{V}}_{\mathit{G}}^{\mathit{H}}$ and let $\mathit{S}=\sqrt{{\mathit{\Sigma }}_{\mathit{G}}}$. Deduce Equation (12) as follows:

$$\begin{array}{ll}tr\left({\mathit{F}}_{BB}\mathit{G}\right)& =tr\left({\mathit{F}}_{BB}{\mathit{U}}_{\mathit{G}}{\mathit{\Sigma }}_{\mathit{G}}{\mathit{V}}_{\mathit{G}}^{\mathit{H}}\right)\\ & =tr\left({\mathit{F}}_{BB}{\mathit{U}}_{\mathit{G}}{\mathit{S}}^2{\mathit{V}}_{\mathit{G}}^{\mathit{H}}\right)\\ & =tr\left(\left({\mathit{F}}_{BB}{\mathit{U}}_{\mathit{G}}\mathit{S}\right){\left({\mathit{V}}_{\mathit{G}}\mathit{S}\right)}^{\mathit{H}}\right)\\ & =\langle {\mathit{F}}_{BB}{\mathit{U}}_{\mathit{G}}\mathit{S},{\mathit{V}}_{\mathit{G}}\mathit{S}\rangle \end{array}$$

(13)

Through Cauchy-Schwarz inequality $|\langle x,y\rangle |\le \Vert x\Vert ·\Vert y\Vert$ , then Equation (13) can be written as:

$$\begin{array}{ll}tr\left({\mathit{F}}_{BB}\mathit{G}\right)& \le \Vert {\mathit{F}}_{BB}{\mathit{U}}_{\mathit{G}}\mathit{S}{\Vert }_2·\Vert {\mathit{V}}_{\mathit{G}}\mathit{S}{\Vert }_2\\ & =\Vert \mathit{S}{\Vert }_2·\Vert \mathit{S}{\Vert }_2\\ & =tr\left(\mathit{S}{\mathit{S}}^{\mathit{H}}\right)\\ & =tr\left({\mathit{\Sigma }}_{\mathit{G}}\right)\end{array}$$

(14)

Through the above calculation, the best digital precoder is obtained ${\mathit{F}}_{BB}^{opt}={\mathit{V}}_{\mathit{G}}^{{}^{\prime }}{\mathit{U}}_{\mathit{G}}^{\mathit{H}}$ where ${\mathit{V}}_{\mathit{G}}^{{}^{\prime }}$ is the first $N_s$ columns of ${\mathit{V}}_{\mathit{G}}$ that is, ${\mathit{V}}_{\mathit{G}}^{{}^{\prime }}={\mathit{V}}_{\mathit{G}}\left(:,1:N_s\right)$ .

$${\mathit{F}}_{BB}^{opt}={\mathit{V}}_{\mathit{G}}^{{}^{\prime }}{\mathit{U}}_{\mathit{G}}^{\mathit{H}}$$

(15)

Then, fix the digital precoder ${\mathit{F}}_{BB}$, and the optimal design of the analog precoder ${\mathit{F}}_{RF}$ is transformed into:

$${\mathit{F}}_{RF}^{opt}=\arg \max tr\left({\mathit{F}}_{opt}^{\mathit{H}}{\mathit{F}}_{RF}{\mathit{F}}_{BB}\right)$$

(16)

$$s.t.\Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}{\Vert }_F^2=N_s$$

$$\left|{\mathit{f}}_n\right|=\frac{1}{\sqrt{M}};n=1,2,\cdots N$$

While the analog precoder has block diagonal characteristics under partial connection architecture, ${\mathit{F}}_{RF}$ can be obtained by block diagonalization of ${\mathit{F}}_{opt}{\mathit{F}}_{BB}^{\mathit{H}}$, i.e., ${\mathit{F}}_{RF}=blkdi\mathit{a}g\left({\mathit{F}}_{opt}{\mathit{F}}_{BB}^{\mathit{H}}\right)$. Moreover, the analog precoder needs to satisfy the constant modulus limit, so the ${\mathit{F}}_{RF}$ phase information is extracted and updated to obtain the ${\mathit{F}}_{RF}^{opt}$, so the analog precoder ${\mathit{F}}_{RF}^{opt}$ can be obtained:

$${\mathit{F}}_{RF}^{opt}=\frac{1}{\sqrt{N_t}}{e}^{j\times angle\left({\mathit{F}}_{RF}\right)}$$

(17)

where $angle\left({\mathit{F}}_{RF}\right)$ represents the phase value of each element selected for ${\mathit{F}}_{RF}$.

3.3. Design of Initial Analog Precoder

To make the hybrid precoder converge to the optimal full-digital precoder quickly in the alternating optimization algorithm, we propose to design the initial analog precoder based on the maximum equivalent channel gain to speed up the convergence speed and reduce the number of iterations, so as to reduce the complexity of the algorithm.

In the mm-Wave massive MIMO system, the capacity of the system is positively correlated with the channel gain. When the channel gain is larger, the capacity of the system is larger. Take $\mathit{H}{\mathit{F}}_{RF}$ as a whole; then, for ${\mathit{F}}_{BB}$, $\mathit{H}{\mathit{F}}_{RF}$ is equivalent to an equivalent channel, so let ${\mathit{H}}_e=\mathit{H}{\mathit{F}}_{RF}$. Therefore, maximizing the channel gain can be transformed into

$${\mathit{F}}_{RF}=\arg \max tr\left({\mathit{H}}_e^{\mathit{H}}{\mathit{H}}_e\right)$$

(18)

$$s.t.\left|{\mathit{f}}_n\right|=\frac{1}{\sqrt{M}};n=1,2,\cdots N$$

Channel matrix $\mathit{H}=\left[{\mathit{H}}_1,{\mathit{H}}_2,\cdots ,{\mathit{H}}_N\right]$, where ${\mathit{H}}_n\in {ℂ}^{N_t\times M}$. The equivalent matrix is

$$\begin{gathered} {\mathit{H}}_e=\mathit{H}{\mathit{F}}_{RF}=\left[{\mathit{H}}_1,{\mathit{H}}_2,\cdots ,{\mathit{H}}_N\right]\left(\begin{array}{ccc}{f}_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & f_N\end{array}\right) \\ =\left[{\mathit{H}}_1{\mathit{f}}_1,{\mathit{H}}_2{\mathit{f}}_2,\cdots ,{\mathit{H}}_N{\mathit{f}}_N\right] \end{gathered}$$

(19)

$tr\left({\mathit{H}}_e^{\mathit{H}}{\mathit{H}}_e\right)$ can be simplified as follows:

$$\begin{gathered} tr\left({\mathit{H}}_e^{\mathit{H}}{\mathit{H}}_e\right)=tr\left(\begin{array}{c}{\left[{\left({\mathit{H}}_1{\mathit{f}}_1\right)}^{\mathit{H}},{\left({\mathit{H}}_2{\mathit{f}}_2\right)}^{\mathit{H}},\cdots {\left({\mathit{H}}_N{\mathit{f}}_N\right)}^{\mathit{H}}\right]}^{\mathit{H}}\\ \left[\left({\mathit{H}}_1{\mathit{f}}_1\right),\left({\mathit{H}}_2{\mathit{f}}_2\right),\cdots \left({\mathit{H}}_N{\mathit{f}}_N\right)\right]\end{array}\right) \\ ={\mathit{f}}_1^{\mathit{H}}{\mathit{H}}_1^{\mathit{H}}{\mathit{H}}_1{\mathit{f}}_1+{\mathit{f}}_2^{\mathit{H}}{\mathit{H}}_2^{\mathit{H}}{\mathit{H}}_2{\mathit{f}}_2+\cdots {\mathit{f}}_N^{\mathit{H}}{\mathit{H}}_N^{\mathit{H}}{\mathit{H}}_N{\mathit{f}}_N \\ ={\displaystyle \sum }_{m=1}^N{\mathit{f}}_m^{\mathit{H}}{\mathit{H}}_m^{\mathit{H}}{\mathit{H}}_m{\mathit{f}}_m \end{gathered}$$

(20)

Therefore, Equation (18) can be transformed into:

$${\mathit{F}}_{RF}=\arg \max {\displaystyle \sum }_{m=1}^N{\mathit{f}}_m^{\mathit{H}}{\mathit{H}}_m^{\mathit{H}}{\mathit{H}}_m{\mathit{f}}_m$$

(21)

$$s.t.\left|{\mathit{f}}_n\right|=\frac{1}{\sqrt{M}};n=1,2,\cdots N$$

Node ${\mathit{W}}_m={\mathit{H}}_m^{\mathit{H}}{\mathit{H}}_m$, and let ${\mathit{W}}_m$ be decomposed by SVD as ${\mathit{W}}_m={\mathit{U}}_m{\mathit{\Sigma }}_m{\mathit{V}}_m^{\mathit{H}}$. ${\mathit{\Sigma }}_m$ is a diagonal matrix with decreasing values of diagonal elements. The solution of the$m$-th optimal unconstrained vector is

$${\mathit{v}}_m={\mathit{V}}_m\left[:,1\right]$$

(22)

Due to the special structure of the constraint on ${\mathit{F}}_{RF}$, it is necessary to find a vector conforming to the constraint condition and make it as close as possible to the optimal unconstrained vector

$${\overline{\mathit{v}}}_m^{opt}=\arg \min \Vert {\mathit{v}}_m-{\overline{\mathit{v}}}_m\Vert {}_2^2$$

(23)

The objective function of Problem (23) can be rewritten as

$$\begin{gathered} \Vert {\mathit{v}}_m-{\overline{\mathit{v}}}_m\Vert {}_2^2={\left({\mathit{v}}_m-{\overline{\mathit{v}}}_m\right)}^{\mathit{H}}\left({\mathit{v}}_m-{\overline{\mathit{v}}}_m\right) \\ ={\mathit{v}}_m{}^{\mathit{H}}{\mathit{v}}_m+{\overline{\mathit{v}}}_m{}^{\mathit{H}}{\overline{\mathit{v}}}_m-2Re\left({\mathit{v}}_m{\overline{\mathit{v}}}_m\right) \\ =2-2Re\left({\mathit{v}}_m{\overline{\mathit{v}}}_m\right) \end{gathered}$$

(24)

According to the simplification results, the optimal solution of Equation (23) to maximize $Re\left({\mathit{v}}_m{\overline{\mathit{v}}}_m\right)$ is

$${\overline{\mathit{v}}}_m^{opt}=e^{j\times angle\left({\mathit{v}}_m\right)}$$

(25)

The $m$-th optimal analog precoding vector is ${\mathit{f}}_m^{opt}=\frac{1}{\sqrt{M}}{\overline{\mathit{v}}}_m^{opt}$. After the analog precoding vectors are calculated one by one, the initial analog precoder is finally obtained by substituting it into Equation (2)

$${\mathit{F}}_{RF}^0=\left(\begin{array}{ccc}{\mathit{f}}_1^{opt}& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {\mathit{f}}_N^{opt}\end{array}\right)$$

(26)

In this section, a hybrid precoding scheme based on alternate optimization is designed. The specific steps are shown in Algorithm 1:

Algorithm 1 Alternately optimized hybrid precoding scheme.

Input: channel matrix $\mathit{H}$, optimal precoder ${\mathit{F}}_{opt},$ preset number of iterations n, send data stream NS. 1. The initial analog precoder ${\mathit{F}}_{RF}^0$ is obtained from Equations (18)–(26) 2. Let k = 0, and k is the current number of iterations 3. When $k\le n$, proceed to steps 3–6 4. Fix the analog precoder ${\mathit{F}}_{RF}^k$ and obtain ${\mathit{F}}_{BB}^k={\mathit{V}}_{\mathit{G}}^{{}^{\prime }}{\mathit{U}}_{\mathit{G}}^{\mathit{H}}$ from Equation (13) 5. Fix the digital precoder ${\mathit{F}}_{BB}^k$, make block diagonal output for ${\mathit{F}}_{opt}{\mathit{F}}_{BB}^k{}^{\mathit{H}}$, extract its phase and update ${\mathit{F}}_{RF}^{k+1}$ 6. Let k = k + 1 and return to step 3 7. Obtain the digital precoder ${\mathit{F}}_{BB}$ and analog precoder ${\mathit{F}}_{RF}$ 8. Normalize the digital precoder,${\mathit{F}}_{BB}=\frac{\sqrt{N_s}}{\Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}{\Vert }_F}{\mathit{F}}_{BB}$ Output: $\mathit{F}={\mathit{F}}_{RF}{\mathit{F}}_{BB}$

4. Complexity Analysis

This section mainly analyzes the algorithm complexity of the proposed partially connected hybrid precoding scheme in detail and compares it with Algorithm 1 in reference [18].

The precoding scheme proposed in this paper is mainly divided into two parts: the design of the initialization analog precoder and alternating optimization. For the design of the initial analog precoder, its complexity mainly focuses on the design of the equivalent matrix, and the computational complexity is $O\left(N_t{N}_{RF}{N}_r\right)$. After obtaining the initial analog precoder, the digital precoder is updated through SVD, so the computational complexity of this part is $O\left(N_t{N}_{RF}{N}_s+N_s^2\left(N_s+N_{RF}\right)\right)$. For the analog precoder, the computational complexity of each iterative update is $O\left(N_t{N}_{RF}{N}_s\right)$. Therefore, the total computational complexity is $O\left(N_t{N}_{RF}\left(N_s+N_r\right)+N_s^2\left(N_s+N_{RF}\right)\right)$. In Algorithm 1 proposed in reference [18], the complexity of SVD for each block matrix in the first stage is $O\left(N_r{M}^2\right)$. The complexity of designing the analog precoder is $O\left(N_{RF}{N}_r{M}^2\right)$. In the second stage, the complexity of calculating the equivalent channel is $O\left(N_t{N}_{RF}{N}_r\right)$. In the third stage, the complexity of designing digital precoding is $O\left(N_{RF}^2{N}_r\right)$. The total computational complexity is $O\left(N_{RF}{N}_r{M}^2\right)$. Since the partial connection system satisfies $N_s\le N_{RF}\ll M\le N_r<N_t$, the complexity of the algorithm proposed in this paper is less than that of Algorithm 1 in reference [18].

5. Simulation and Analysis

To verify the feasibility of the proposed algorithm, we used an mm-Wave massive MIMO environment with a single-cell partial connection architecture to simulate the algorithm. The simulation curves of the spectral efficiency of the proposed algorithm under a low signal-to-noise ratio (SNR) and high SNR are given, respectively, and compared with the optimal full-digital precoding scheme, pure analog precoding scheme, reference [18]’s partial connection precoding scheme based on equivalent channel and reference [21]’s partial connection precoding scheme based on SIC under different RF link numbers and receiving antennas. Additionally, we used MATLAB to conduct a detailed simulation evaluation of the performance of the proposed algorithm. The processor used is Intel(R) Core(TM) I5-7300HQ. Considering the single-user scenario, the simulation parameters are shown in

Table 1. Simulation parameters.

Parameter Name	Parameter Value
Number of transmitting antennas $N_t$	128
Antenna spacing d	0.5$\lambda$ mm
Number of channel scattering clusters $N_{cl}$	5
Number of paths in each scattering cluster $N_{r\mathit{a}y}$	25
Channel model	Saleh-Valenzuela
Antenna array	ULA

:

Figure 2 shows the comparison of the spectral efficiency with the increasing SNR for various precoding schemes. The system simulation parameters are set as the number of RF links $N_{RF}=8$ and the number of transmitted data streams $N_s=8$, and the antenna of the transceiver is configured as $N_t\times N_r=128\times 16$. It can be seen from the simulation diagram that the pure analog precoding scheme with hardware constraints and changing phase information only through a low-cost phase shifter has the worst performance, while the optimal full-digital precoding scheme has the best performance because the transmitter adopts the same number of antennas and RF links. To reduce the hardware complexity, the proposed algorithm uses fewer phase shifters, which brings performance loss, so the spectral efficiency is lower than that of full-digital precoding schemes. In the area with a low SNR (SNR is about −16 dB), the performance of this algorithm is close to but slightly better than that of reference [18] and reference [21]. The proposed algorithm provides a 0.645 bits/s/Hz gain over the corresponding algorithm in reference [18]. Compared with reference [18], we considered the design of the initial analog precoding matrix, so its performance is further improved. With the increase in SNR, the performance advantage of the algorithm is greater.

In Figure 3, the system simulation parameters are set to the number of RF links $N_{RF}=8$ and the number of transmitted data streams $N_s=8$, and the antenna of the transceiver is configured as $N_t\times N_r=128\times 16$. The simulation diagram of the relationship between spectral efficiency and SNR of each precoding scheme is under the condition of a high SNR (about more than 10 dB). It can be seen from Figure 3 that the performance of the algorithm proposed in this paper is significantly better than that proposed in reference [18] and reference [21]. When the SNR gradually increases to the state of a high SNR, the difference between them is more obvious. It can be seen that the proposed algorithm produces at least 16.2% higher performance than the algorithm in reference [18] at 20 dB received SNR. In the case of a high SNR, the error caused by noise decreases with each iteration and the system performance improvement brought by the proposed algorithm is more significant.

Then, we considered a more realistic scenario in which the BS has imperfect channel state information (CSI) and evaluated the impact of the proposed algorithm. Let $\widehat{\mathbf{H}}$ be the estimated channel, and it can be written as [21]

$$\widehat{\mathbf{H}}=\mathsf{\delta }\mathbf{H}+\sqrt{1-{\mathsf{\delta }}^2}\Xi$$

(27)

where $\delta \in \left[0,1\right]$ denotes the accuracy factor of $\widehat{\mathbf{H}}$ and $\Xi$ denotes the matrix of i.i.d $\mathcal{C}\mathcal{N}\left(0,1\right)$ noise. Figure 4 shows the comparison of spectral efficiency with different CSI conditions for mm-Wave massive MIMO system, where $N_t\times N_r=128\times 16$, N_RF=8, N_s=8. It can be seen from Figure 4 that the proposed algorithm is insensitive to CSI accuracy at a low SNR. When $\delta =0.9$, the spectral efficiency of the proposed algorithm is only about 1–2 bits/s/Hz lower than that of the perfect channel. Moreover, when $\delta =0.7$, the proposed algorithm still achieves about 88% of the rate compared with the perfect channel. This shows that the proposed algorithm has strong robustness.

Figure 5 shows the simulation diagram between the spectral efficiency of each precoding scheme and the number of receiving antennas. The system simulation parameters are set to SNR = 1 dB, the number of RF chains $N_{RF}=8$ and the number of transmitted data streams $N_s=8$. As can be seen from Figure 5, with the increase in the number of antennas at the receiving end, the spectral efficiency of each precoding scheme also increases. This is the mm-Wave massive MIMO system that has a strong beamforming gain, which can transmit a large amount of data in parallel. The spectral efficiency of the proposed algorithm increases significantly with the increase in the number of antennas, and the proposed algorithm produces about 13.1% higher performance than the corresponding algorithm in reference [18] and reference [21] at $N_r=30$.

In order to verify the feasibility of the algorithm proposed in this paper, its convergence was simulated. We compared the scheme without designing the initial analog precoder in the same alternating optimization framework. The simulation results are shown in Figure 6. In the simulation, the SNR was set as SNR = 5 dB, the number of RF links $N_{RF}=8$, the number of transmitted data streams $N_s=8$, and the antenna of the transceiver was configured as $N_t\times N_r=128\times 16$. It can be seen from Figure 6 that the algorithm proposed in this paper converges gradually, and it can be clearly found that rapid convergence can be achieved only after three iterations. In contrast, the scheme without an initial analog precoder has poor coding performance and the spectral efficiency is about 7 bits/s/Hz lower than that of the proposed algorithm after three iterations.

6. Conclusions and Future Work

6.1. Conclusions

In order to improve the performance of the mm-Wave massive MIMO system and reduce the complexity of the algorithm, we proposed a partially connected hybrid precoding scheme based on alternating optimization, in this paper. Furthermore, we designed a heuristic algorithm to calculate the initial value of the analog precoder as the starting point of alternating optimization to prevent the algorithm from converging too slowly. Simulation results show that the proposed scheme is superior to the other two comparison schemes. Meanwhile, the performance of the proposed algorithm will not suffer a great loss due to the error of channel estimation, and it has strong robustness.

6.2. Future Work

The proposed scheme is based on a fixed connection structure. The allocation of the antenna array is fixed. Therefore, the solution we obtained is only a static local optimal solution. For future work, it will be interesting to study the design of precoding algorithms under a dynamic subarray structure.