Hybrid Precoding Based on a Switching Network in Millimeter Wave MIMO Systems

Abstract

Aiming at the algorithm difficult optimization of the fully-connected hybrid precoding structure and the complex hardware implementation in millimeter wave multiple-input multiple-output (MIMO) systems, this paper proposes a novel hybrid precoding structure based on a switching network (SNHBP). The structure enables the dynamic grou** of the sub-arrays in three ways by switching the network, which can greatly reduce the hardware complexity, simplify the optimization algorithm, and avoid the performance degradation defect caused by a partially-connected structure. Through simulation of different antenna sizes and different numbers of RF chains, experimental results show that SNHBP can approach the performance of the full digital precoder. Under the condition of more than 2 RF chains, the difference between the unit structure and the full digital precoding is less than 0.13 dB. The spectral efficiency of the low-precision phase optimization algorithm is better than that of the partially-connected structure when the quantization bit of the phase shifter is 3. The feasibility of the three dynamic grou** schemes is verified, which is more beneficial to engineering implementation than the fully-connected structure.

1. Introduction

Millimeter wave communication technology has become the most promising technology in the field of communication. For two main reasons: On one hand, the short wavelength of millimeter wave technology allows a large number of antennas to be installed in a small space. On the other hand, massive MIMO antenna arrays can provide a strong enough gain to compensate for the severe free-space path loss of the millimeter wave [1,2,3]. However, in the full digital precoding structure, each antenna needs to be connected to a separate and expensive RF chain, which mainly includes a digital-to-analog converter (DAC), an analog-to-digital converter (ADC), a power amplifier (PA), a mixer, a local oscillator, a data converter, etc. [4,5,6]. The high cost and high power consumption of the full digital precoding structure lead people to think about develo** different hardware structures. It has been proposed to use the hybrid precoding structure to reduce the number of RF chains, which can reduce the power consumption and hardware complexity of the system while ensuring the spectral efficiency of the system [7,8,9]. At present, there are two main structures of hybrid precoding; one is the fully-connected structure, as shown in Figure 1a. Each RF chain is connected to all antennas through phase shifters whose numbers are equal to the number of antennas. The other is the partially-connected structure, as shown in Figure 1b. Each RF chain is connected to a fixed antenna sub-array [10].

Related studies have proposed a zero forcing algorithm (ZF) [11], an orthogonal matching pursuit algorithm (OMP) [12], etc., for the fully-connected hybrid precoding structure. The fully-connected structure can make full use of the degrees of freedom (DOF) of precoding provided by the RF chains, and its spectral efficiency is similar to that of the full digital precoding structure under the condition of a certain number of RF chains. However, in the fully-connected structure, it is necessary to add $N_{RF}{N}_T$ analog phase shifters ($N_{RF}$ represents the number of RF chains, and $N_T$ represents the number of antennas), and the increase of analog devices and RF chains make it difficult to accurately control the amplitude and phase of the precoding vector in practical applications. In [13], a more mature MO-AltMin algorithm based on manifold optimization was proposed for the fully connected structure. Although MO-AltMin provides near-optimal spectral efficiency, it suffers from not only slowing down of the convergence speed of the procedure due to its nested-loop structure between $F_{BB}$ and $F_{RF}$, but also an extremely high complexity to handle big matrices due to the Kronecker product [14].

For the partially-connected structure, related optimization algorithms include the dynamic sub-array algorithm [15], the water-filling algorithm [16], the iterative precoding and combining algorithm [17], etc. In the partially-connected structure, although the hardware complexity of the system can be greatly reduced and the phase adjustment is performed by analog phase shifters, the amplitude of a sub-array is provided by an RF chain, and it is difficult to realize the precise control of the amplitude of the precoding vector by phase shifter change alone. Although the SDR-AltMin algorithm proposed in [13] can improve the spectral efficiency through optimization, due to the partially-connected structure itself, the spectral efficiency of the SDR-AltMin algorithm has a great performance loss compared with that of the full digital precoding structure and the fully-connected hybrid precoding structure.

The above hybrid precoding algorithm is assumed to be implemented under the condition of infinite or high-resolution phase shifters, but high-resolution phase shifters will significantly increase power consumption and hardware complexity in the application of the millimeter wave frequency spectrum. Therefore, it is more realistic to use low-resolution phase shifters in practical applications [18]. In [19], the use of phase shifters and switching networks in a hybrid precoding structure was compared, but the two cannot be combined in the hybrid precoding system. In [20], the effectiveness of introducing a switching network into a hybrid precoding system is demonstrated. In [21], the sub-array is dynamically optimized through the switching network, but the fixed sub-array is taken as the optimization unit, and the system performance still suffers a great loss. In [22], the switching network is applied to a large-scale antenna array, and the application position is at the RF side to solve the antenna selection problem under the condition of low-resolution phase shifters. However, the application of the switching network in hybrid precoding cannot effectively solve the problem of amplitude precision of the sub-array, so the performance loss of spectral efficiency is large.

The main contribution of this paper: A novel hybrid precoding structure is proposed to effectively balance the relationship between hardware complexity (the number of RF chains and the use of low-resolution phase shifters) and spectral efficiency. The hybrid precoding structure based on a switching network which dynamically groups the sub-arrays through the switching network, reduces the hardware complexity while ensuring the spectral efficiency of the system, effectively approaches the spectral efficiency of full digital precoding under the condition of high-resolution phase shifters, and develops an optimization algorithm for low-resolution phase shifters. When the quantization accuracy of the phase shifter is 3-bit, the spectral efficiency can still be better than that of the partially-connected structure.

Other structures of this paper are as follows. The second part introduces the structure and problem formation of hybrid precoding based on a switching network. The third part introduces the design of a hybrid precoding algorithm based on a switching network. The fourth part introduces the simulation experiment and experimental results.

Table 1. Table of Acronym.

Acronym	Definition
MIMO	Multiple-Input Multiple-Output
SNHBP	Hybrid Precoding Structure Based on a Switching Network
DAC	Digital-to-Analog Converter
ADC	Analog-to-Digital Converter
PA	Power Amplifier
ZF	Zero Forcing Algorithm
OMP	Orthogonal Matching Pursuit Algorithm
DOF	Degrees of Freedom
MO-AltMin	Manifold Optimization Based Hybrid Precoding for the fully-connected [13]
SDR-AltMin	Semidefinite Relaxation Based Hybrid Precoding for the partially-connected [13]
SVD	Singular Value Decomposition
USPAs	Uniform Square Arrays
AOD	Angles of Departure
AOA	Angles of Arrival

lists the acronyms used in this paper.

Notation: Lower-case and upper-case boldface letters denote vectors and matrices, respectively; ${(\cdot )}^H$ and $\det (•)$ denote the conjugate and determinant of a matrix, respectively; ${\Vert ●\Vert }_F$ denotes the Frobenius norm of a vector; $|•|$ denote the absolute operator; $\mathrm{{\rm E}}(•)$ denotes the expectation; finally, $I_N$ is the $N\times N$ identity matrix.

2. System Model and Problem Formation

2.1. System Model

We consider a large point-to-point MIMO system, as shown in Figure 2. When there are $N_s$ data streams at the transceiver end, the unit structure in Figure 3 is multiplexed $N_s$ times so as to realize the communication transmission of multiple data streams.

The system model is analyzed by taking the hybrid precoding unit structure based on a switching network as an example. In the unit structure, the transmitting end has $N_T$ transmitting antennas and $N_{RF}^t$ RF chains, and the corresponding receiving end has $N_R$ receiving antennas and $N_{RF}^t$ RF chains. The number of data streams in the unit structure satisfies $N_S=1$, $N_S\le N_{RF}^t\ll N_T$, and $N_S\le N_{RF}^r\ll N_R$. In the structure of Figure 3, the input signals are the precoding by $N_{RF}^t\times 1$ digital baseband precoder $F_{BB}\in {ℂ}^{N_{RF}^t\times 1}$, and then the dynamic grou** of RF chains and antennas is realized through the transmitting end $N_T\times N_{RF}^t$ switching network matrix $S_{WT}\in {ℝ}^{N_T\times N_{RF}^t}$. Finally, phase modulation of signals is achieved through the $N_T\times N_T$ analog RF precoder $F_{RF}\in {ℂ}^{N_T\times N_T}$, where $F_{RF}$ is the diagonal matrix of which the number of non-zero elements is $N_T$. The number of non-zero elements in $S_{WT}$ is also $N_T$, and they are all 1. The transmitted signal $t$ can be written as

$$t=F_{RF}{S}_{WT}{F}_{BB}s$$

(1)

where the input signal is $s\in {ℂ}^{1\times 1}$, and satisfies $E\left[s{s}^H\right]=1$. The normalized power limit for the entire system is ${\Vert F_{RF}{S}_{WT}{F}_{BB}\Vert }_F^2=1$. The structure of the receiving end is similar to that of the transmitting end. $W_{BB}\in {ℝ}^{N_{RF}^r\times 1}$, ${\mathrm{S}}_{\mathrm{WR}}\in {ℂ}^{N_R\times N_{RF}^r}$, and $W_{RF}\in {ℂ}^{N_R\times N_R}$ are the digital baseband decoder, the switching network matrix, and the analog RF decoder of the receiving end, respectively. Analog RF precoder and decoder control the phase of the signal by phase shifters, so all non-zero elements in $F_{RF}$ and $W_{RF}$ should satisfy the unit modulus constraints, i.e., $\left|{\left(F_{RF}\right)}_{i,j}\right|=1$, $\left|{\left(W_{RF}\right)}_{i,j}\right|=1$ for nonzero elements. $H\in {ℂ}^{N_R\times N_T}$ is the channel matrix and $n$ is the additive Gaussian white noise (AWGN), which is independently and identically distributed in $\mathcal{C}\mathcal{N}\left(0,{\sigma }_n^2\right)$. Therefore, the received signal after decoding is given as:

$$y=\sqrt{P_T}{W}_{BB}^H{S}_{WR}^H{W}_{RF}^{H}H{F}_{RF}{S}_{WT}{F}_{BB}s+W_{BB}^H{S}_{WR}^H{W}_{RF}^{H}n$$

(2)

The spectral efficiency in the unit structure can be expressed as the following expression:

$$R={\log }_2\det \left(\begin{array}{l}{I}_{N_T}+\frac{P_T}{{\sigma }_n^2{N}_S}{\left(W_{RF}{S}_{WR}{W}_{BB}\right)}^{H}H{F}_{RF}{S}_{WT}{F}_{BB}\\ \times F_{BB}^H{S}_{WT}^H{F}_{RF}^H{H}^H\left(W_{RF}{S}_{WR}{W}_{BB}\right)\end{array}\right)$$

(3)

The objective function is

$$\begin{array}{l}\underset{\left(F_{RF}{S}_{WT}{F}_{BB}\right)\left(W_{RF}{S}_{WR}{W}_{BB}\right)}{\arg \max }{\log }_2\det \left(\begin{array}{l}{I}_{N_T}+\frac{P_T}{{\sigma }_n^2{N}_S}{\left(W_{RF}{S}_{WR}{W}_{BB}\right)}^{H}H{F}_{RF}{S}_{WT}{F}_{BB}\\ \times F_{BB}^H{S}_{WT}^H{F}_{RF}^H{H}^H\left(W_{RF}{S}_{WR}{W}_{BB}\right)\end{array}\right)\\ subject to \left|{\left(F_{RF}\right)}_{i,j}\right|=1,\left|{\left(W_{RF}\right)}_{i,j}\right|=1,{\Vert F_{RF}{S}_{WT}{F}_{BB}\Vert }_F^2=1\end{array}$$

(4)

2.2. Channel Model

The mmWave propagation environment is characterized by a clustered channel mode [23]. In this paper, we chose the Saleh–Valenzuela mode. In this mode the mmWave channel matrix $H$ is expressed as

$$H=\sqrt{\frac{N_T{N}_R}{N_{cl}{N}_{ray}}}{\displaystyle \sum _{i=1}^{N_{cl}}{\displaystyle \sum _{l=1}^{N_{ray}}{\alpha }_{il}{\mathrm{a}}_r({\varphi }_{il}^r,{\theta }_{il}^r)}}{\mathrm{a}}_t{({\varphi }_{il}^t,{\theta }_{il}^t)}^H$$

(5)

where $N_{cl}$ and $N_{ray}$ represent the number of clusters and the number of rays in each cluster, respectively, and ${\alpha }_{il}$ denotes the gain of the $l\mathrm{th}$ ray in the $i\mathrm{th}$ propagation cluster. ${\alpha }_{il}$ are i.i.d. are random variables following the complex Gaussian distribution $\mathcal{C}\mathcal{N}\left(0,{\sigma }_{\alpha ,i}^2\right)$, and ${\displaystyle {\sum }_{i=1}^{N_{cl}}{\sigma }_{\alpha ,i}^2}=\stackrel{\wedge }{\gamma }$ is the normalization factor to satisfy $E\left[{\Vert H\Vert }_F^2\right]=N_T{N}_R$. In addition, ${\mathrm{a}}_r({\varphi }_{il}^r,{\theta }_{il}^r)$ and ${\mathrm{a}}_t({\varphi }_{il}^t,{\theta }_{il}^t)$ represent the receive and transmit array response vectors, respectively, where ${\varphi }_{il}^r({\varphi }_{il}^r)$ and ${\theta }_{il}^r({\theta }_{il}^t)$ stand for azimuth and elevation angles of arrival and departure (AOAs and AODs), respectively. In this paper, we consider the uniform square planar array (USPA) with $\sqrt{N}\times \sqrt{N}$ antenna elements. Therefore, the array response vector corresponding to the $l\mathrm{th}$ ray in the $i\mathrm{th}$ cluster can be written as

$$\mathrm{a}({\varphi }_{il},{\theta }_{il})=\frac{1}{\sqrt{N}}{\left(1,\dots ,e^{j\frac{2\pi }{\lambda }d(p\sin {\varphi }_{il}\sin {\theta }_{il}+q\cos {\theta }_{il})},\dots ,e^{j\frac{2\pi }{\lambda }d((\sqrt{N}-1)\sin {\varphi }_{il}\sin {\theta }_{il}+(\sqrt{N}-1)\cos {\theta }_{il})}\right)}^T$$

(6)

where $d$ and $\lambda$ are the antenna spacing and the signal wavelength, respectively, and $0\le p\le \sqrt{N}$ and $0\le q\le \sqrt{N}$ are the antenna indices in the 2D plane, respectively. Our structures and precoder algorithm can be used for more general models. We assume that perfect channel information can be obtained.

2.3. Problem Formation

In [8], it was proved that the maximization objective function will approximate the following expression,

$$\begin{array}{l}\underset{F_{RF},S_{WT},F_{BB}}{\min }{\Vert F_{opt}-F_{RF}{S}_{WT}{F}_{BB}\Vert }_F\\ subject to \left|{\left(F_{RF}\right)}_{i,j}\right|=1,{\Vert F_{RF}{S}_{WT}{F}_{BB}\Vert }_F^2=1\end{array}$$

(7)

where $F_{opt}$ and $W_{opt}$ are $V$ and $U$ in the $N_S\mathrm{th}$ column, respectively, where both $V$ and $U$ are the singular value decomposition (SVD) of the unitary matrix from the channel matrix, i.e., $H=U\sum V^H$. To ensure the unit modulus constraints of analog RF precoding and the limit of transmitting power at the transmitting end, only the unit modulus constraints of the analog RF decoder need to be considered at the receiving end.

In the second part, the hybrid precoding based on the switching network structure and its unit structure are established, and the signal mathematical model and objective function of the unit structure are derived. The Saleh–Valenzuela model of mmWave channel is also introduced.

3. Hybrid Precoding Based on a Switching Network

3.1. High Resolution Phase Shifter Precoding Algorithm Based on a Switching Network

In the hybrid precoding structure, whether it is a fully-connected structure or a partially-connected structure, each antenna is connected to at least one analog phase shifter, so the phase accuracy in the precoding vector can be guaranteed, and the main goal is to improve the amplitude accuracy of the system as much as possible. The main method is to divide the antennas into $N_{RF}$ groups according to the number of RF chains according to the maximum and minimum amplitude of the optimal digital precoding vector. Each group of antennas is dynamically connected to an RF chain through the switching network, that is, the RF chain provides the amplitude information of the connected sub-array precoding vector. When grou** the amplitude of the optimal digital precoding vector, three dynamically grou** sub-array methods are selected:

• Grou** according to the amplitude of the optimal digital precoding vector $F_{opt}$, which is called optimal dynamic grou**;
• Grou** according to the statistical mean values of the amplitudes of the $F_{opt}$ corresponding to the $H$ parameters of different channels, which is called statistical dynamic grou**;
• The fixed number of antennas are connected to each RF chain, which is similar to the partially-connected structure, but the difference is that the array elements with similar amplitudes are dynamically combined, which is called average dynamic grou**.

The main purpose of considering the grou** methods of 2 and 3 is that statistical dynamic grou** and average dynamic grou** are grou** methods with a fixed number of array elements, which can fix the working range of RF devices and further simplify system complexity and hardware cost.

In the hybrid precoding structure of this paper, there are two variables. One is the digital baseband precoding amplitude, and the other is the analog RF precoding phase. In this paper, we take the transmitting end as an example, and the objective function is expressed in Formula (7).

Considering that the analog radio frequency precoding is infinite precision or high precision, due to the number of data streams $N_S=1$, the phase information in $F_{RF}{S}_{WT}{F}_{BB}$ can be completely consistent with the optimal digital precoding vector $F_{opt}$, so the amplitude information of digital baseband precoding needs to be optimized in the objective function.

The objective function in Formula (7) is transformed into:

$$\min {\displaystyle \sum _{i=1}^{N_{RF}}{\displaystyle \sum _{j=1}^{n_i}{\Vert \left|F_{opt(i,j)}\right|-\left|F_i\right|\Vert }_F^2}}$$

(8)

where $n_i$ represents the number of sub-array antennas connected by the $i\mathrm{th}$ RF chain, $F_{opt(i,j)}$ represents the $j\mathrm{th}$ optimal digital precoding vector connected by the $i\mathrm{th}$ RF chain, and the amplitude error of each RF chain in the objective function is relatively independent, so the objective function can be transformed into $N_{RF}$ quadratic functions, then the corresponding optimal amplitude of each RF chain is

$$\left|F_i\right|=\frac{\left(\left|F_{opt(1)}\right|+\left|F_{opt(2)}\right|+\cdots +\left|F_{opt(n_i)}\right|\right)}{n_i}$$

(9)

where $\left|F_i\right|$ represents the amplitude of the $i\mathrm{th}$ RF chain.

When the 2nd and 3rd grou** methods are used, the statistical partition should give the information of the number of antennas connected to the RF chain, and the average grou** means the average allocation of the number of antennas.

Among them, there are 5 related algorithms in this paper, i.e., SNHBP-∞, SNHBP-H-∞, SNHBP-A-∞, SNHBP-Q-bit, and SNHBP-Q-bit-AltMin (bit represents the quantization number of the phase shifter), which represent: infinite precision optimal dynamic grou**, infinite precision statistical dynamic grou**, infinite precision average dynamic grou**, phase quantization optimal dynamic grou**, and phase optimization optimal dynamic grou**, respectively.

3.2. Low-Resolution Phase Shifter Precoding Algorithm Based on a Switching Network

The above algorithm flow is realized under the condition of a high-resolution analog phase shifter. If the quantized phase is considered, the following phase optimization algorithm can be obtained: When the corresponding amplitude approximate substitute value is determined, we will carry out the $B$ bit quantization of the phase in the $F_{RF}$, $F_{RF}(i,j)\in \left\{e^{j\frac{2\pi b}{2^B}}|b=0,1,2,\cdots ,(2^B-1)\right\}$, that is, the quantization phase under low precision conditions is obtained, and the gap between the quantization and the target vector $F_{opt}$ increases. However, in the above algorithm, we only use the amplitude control for the digital baseband precoding, and the phase control is not applied. Therefore, under the condition of a low precision phase shifter, the phase adjustment of baseband precoding can be used to reduce the performance loss caused by direct quantization and further improve the system performance.

In Figure 4, blue represents the target vector $l_i$, red represents the SNHBP-∞ after direct quantization vector $\overline{l_i}$, and green represents the deflection vector ${\overline{l_{i,}}}_{\theta }$ after adding phase information in $F_{BB}$.

We deduce with the quantization bit $B=2$. That is, the quantized phase is four discrete phase values of $0,\pi /2,\pi ,3\pi /2$. In the interval formed by each quantization phase $0\le \theta <\pi /2$. Taking 1/2 of the quantized interval as the dividing line, the quantized interval is divided into the upper and lower half, the quantized phase of the upper half is the left boundary of the quantized interval, and the quantized phase of the lower half is the right boundary of the quantized interval. In the dynamic grou** structure, we assume that the number of antennas connected to the first RF chain is $n_1$. Among the phases corresponding to the $n_1$ antennas, $n_{1,up}$ antennas are located in the upper half of each quantized interval, and $n_{1,down}$ antennas are located in the lower half of each quantized interval, which meets $n_{1,up}+n_{1,down}=n_1$. Assuming that the phase of the digital baseband precoding part of the RF chain is $e^{j\theta }$, as shown in the Figure 5, after adding the phase of the digital baseband precoding, the vector after the original quantized phase will be deflected, then the error change after the deflection, and the error change of the upper half is: ${\left(\Delta l_i\right)}^2=l_i^2+{{\overline{l}}_i}^2-2l_i{\overline{l}}_i\cos ({\theta }_i+\theta )$ and the error change of the lower half is ${\left(\Delta l_i\right)}^2=l_i^2+{{\overline{l}}_i}^2-2l_i{\overline{l}}_i\cos ({\theta }_i-\theta )$, then the error of the first RF chain after adding phase control is

$$\begin{array}{cc}\Delta & ={\displaystyle \sum _{i=1}^{n_1}{\left(\Delta l_i\right)}^2=}{\displaystyle \sum _{i=1}^{n_{1,up}}{\left(\Delta l_{i,up}\right)}^2+{\displaystyle \sum _{i=1}^{n_{1,down}}{\left(\Delta l_{i,down}\right)}^2}}\hfill \\ & ={\displaystyle \sum _{i=1}^{n_{1,up}}\left(l_{i,up}^2+{\overline{l}}_{i,up}^2-2l_{i,up}{\overline{l}}_{i,up}\cos ({\theta }_{i,up}+\theta )\right)}\hfill \\ & +{\displaystyle \sum _{i=1}^{n_{1,down}}{\left(\left(l_{i,down}^2+{\overline{l}}_{i,down}^2-2l_{i,down}{\overline{l}}_{i,down}\cos ({\theta }_{i,down}-\theta )\right)\right)}^2}\hfill \end{array}$$

(10)

Our goal is $\min \Delta$, since there is only one variable $\theta$ of $\Delta$, the objective function can be transformed into

$$\max Y={\displaystyle \sum _{i=1}^{n_{1,up}}\left(l_{i,up}{\overline{l}}_{i,up}\cos ({\theta }_{i,up}+\theta )\right)}+{\displaystyle \sum _{i=1}^{n_{1,down}}\left(l_{i,down}{\overline{l}}_{i,down}\cos ({\theta }_{i,down}-\theta )\right)}$$

(11)

$$\begin{array}{cc}{Y}^{\prime }& ={\displaystyle \sum _{i=1}^{n_{1,up}}\left(-l_{i,up}{\overline{l}}_{i,up}\sin ({\theta }_{i,up}+\theta )\right)}+{\displaystyle \sum _{i=1}^{n_{1,down}}\left(l_{i,down}{\overline{l}}_{i,down}\sin ({\theta }_{i,down}-\theta )\right)}\hfill \\ & ={\displaystyle \sum _{i=1}^{n_{1,up}}\left(-l_{i,up}{\overline{l}}_{i,up}\left(\sin {\theta }_{i,up}\cos \theta +\cos {\theta }_{i,up}\sin \theta \right)\right)}\hfill \\ & +{\displaystyle \sum _{i=1}^{n_{1,down}}\left(l_{i,down}{\overline{l}}_{i,down}\left(\sin {\theta }_{i,down}\cos \theta -\cos {\theta }_{i,down}\sin \theta \right)\right)}\hfill \\ & =P\cos \theta +Q\sin \theta \hfill \end{array}$$

(12)

where

$$\begin{array}{c}P\end{array}={\displaystyle \sum _{i=1}^{n_{1,up}}\left(-l_{i,up}{\overline{l}}_{i,up}\sin {\theta }_{i,up}\right)}+{\displaystyle \sum _{i=1}^{n_{1,down}}\left(l_i{\overline{l}}_i\sin {\theta }_{i,down}\right)}$$

$$\begin{array}{c}Q\end{array}={\displaystyle \sum _{i=1}^{n_{1,up}}\left(-l_{i,up}{\overline{l}}_{i,up}\cos {\theta }_{i,up}\right)}+{\displaystyle \sum _{i=1}^{n_{1,down}}\left(l_{i,down}{\overline{l}}_{i,down}\left(-\cos {\theta }_{i,down}\right)\right)}$$

$$Y^{\prime }=P\cos \theta +Q\sin \theta =\sqrt{P^2+Q^2}\sin \left(\phi +\theta \right),\tan \phi =\frac{P}{Q}$$

(13)

Then the extreme point is $\theta =-\phi$. On the premise that the dynamic grou** in this paper is the best grou** method, the maximum value of $Y$ can be determined by considering the extreme point and the adjustment range of the vector phase after quantization. After the determination of $\theta$, we optimize the amplitude information of the corresponding digital beamforming, and the objective function is still

$$\begin{array}{cc}\Delta & ={\displaystyle \sum _{i=1}^{n_1}{\left(\Delta l_i\right)}^2=}{\displaystyle \sum _{i=1}^{n_{1,up}}{\left(\Delta l_{i,up}\right)}^2+{\displaystyle \sum _{i=1}^{n_{1,down}}{\left(\Delta l_{i,down}\right)}^2}}\hfill \\ & ={\displaystyle \sum _{i=1}^{n_{1,up}}\left(l_{i,up}^2+{\overline{l}}_{i,up}^2-2l_{i,up}{\overline{l}}_{i,up}\cos ({\theta }_{i,up}+\theta )\right)}\hfill \\ & +{\displaystyle \sum _{i=1}^{n_{1,down}}{\left(\left(l_{i,down}^2+{\overline{l}}_{i,down}^2-2l_{i,down}{\overline{l}}_{i,down}\cos ({\theta }_{i,down}-\theta )\right)\right)}^2}\hfill \end{array}$$

(14)

If the variable here is ${\overline{l}}_i$, then the objective function is transformed into $\min \Delta$, and we can transform the objective function into

$$\min \mathrm{Z}={\displaystyle \sum _{i=1}^{n_{1,up}}\left({\overline{l}}_{i,up}^2-2l_{i,up}{\overline{l}}_{i,up}\cos ({\theta }_{i,up}+\theta )\right)}+{\displaystyle \sum _{i=1}^{n_{1,dowm}}\left({\overline{l}}_{i,down}^2-2l_{i,down}{\overline{l}}_{i,down}\cos ({\theta }_{i,down}-\theta )\right)}$$

(15)

${\mathrm{Z}}^{\prime }={\displaystyle \sum _{i=1}^{n_{1,up}}\left(2{\overline{l}}_{i,up}^{}-2l_{i,up}\cos ({\theta }_{i,up}+\theta )\right)}+{\displaystyle \sum _{i=1}^{n_{1,dowm}}\left(2{\overline{l}}_{i,down}^{}-2l_{i,down}\cos ({\theta }_{i,down}-\theta )\right)}$, when the derivative is 0, ${\displaystyle \sum _{i=1}^{n_1}\left({\overline{l}}_{i,up}^{}\right)}={\displaystyle \sum _{i=1}^{n_{1,up}}\left(l_{i,up}\cos ({\theta }_{i,up}+\theta )\right)}+{\displaystyle \sum _{i=1}^{n_{1,dowm}}\left(l_{i,down}\cos ({\theta }_{i,down}-\theta )\right)}$ then the amplitude value of the antenna sub-array when the objective function takes the minimum value is:

$${\overline{l}}_i=\frac{{\displaystyle \sum _{i=1}^{n_{1,up}}\left(l_{i,up}\cos ({\theta }_{i,up}+\theta )\right)}+{\displaystyle \sum _{i=1}^{n_{1,dowm}}\left(l_{i,down}\cos ({\theta }_{i,down}-\theta )\right)}}{n_1}$$

(16)

The amplitude of the digital beamforming is updated after optimizing the digital phase.

In the third part, three dynamic grou** schemes of hybrid precoding based on a switching network are introduced, and the infinite precision optimal dynamic grou** algorithm and phase optimization optimal dynamic grou** algorithm are derived.

4. Simulation Experiment and Experimental Results

In this part, we will present simulation results to demonstrate the effectiveness of our proposed algorithm. In the MIMO system, the transmit and receive antenna arrays are all uniform square arrays (USPAs), the channel parameters are set to $N_{cl}=5$ clusters, $N_{ray}=10$ and the average power of each cluster is ${\sigma }_{\alpha ,i}^2=1$. The azimuth and elevation AODs and AOAs follow the Laplacian with uniformly distributed mean angles over $[0,2\pi )$ and the angular spread of 10 degrees [24], the transmit power ${\mathrm{P}}_T=1$. The antenna elements in the USPA are separated by a half wavelength distance and all simulation results are averaged over 1000 channel realizations, except for Figure 6a.

In the experimental results shown in Figure 5a–f, all the hybrid precoding algorithms increase with the increase of the number of RF chains. In the hybrid precoding structure based on the switching network proposed in this paper, under the condition of the high-resolution phase shifter, when the number of RF chains is 2, the difference of the spectral efficiency between the SNHBP and the full digital precoding is about 0.1 dB, which is similar to the spectral efficiency of a full digital precoding. Among the three dynamic grou** methods, the optimal dynamic grou** is better than the statistical dynamic grou**, which is better than the average dynamic grou**, but there is no significant difference between the three. Therefore, under certain application conditions, the statistical dynamic grou** and the average dynamic grou** can be used to further simplify the hardware parameter selection of the system. In addition, in the experimental results for low-resolution phase shifters, the spectral efficiency of the phase optimization algorithm is slightly improved compared to that of direct quantization, but in the future millimeter wave massive MIMO array, the number of antennas will be enormous, so it will also bring a significant performance gain. Moreover, using the structure of this paper, when the number of quantization bits of the phase shifter is 3, the spectral efficiency of the system is obviously better than that of the partially-connected structure.

In the simulation experiment shown in Figure 6a,b, the relationship between the number of RF chains and the spectral efficiency is analyzed under the condition of SNR = 0. The main difference between the two results is that the number of the selected channel parameters is different. The number of channels $H$ in the experiment in Figure 6a is 10, and the number of channels $H$ in the experiment in Figure 6b is 1000. When the channel parameters are small, the effect of statistical dynamic grou** is poor, which proves that a statistical dynamic grou** can obtain appropriate statistical grou** data after statistics based on a large number of channel parameters. Relatively speaking, under the same experimental conditions, the spectral efficiency of the fully-connected structure is high, and the spectral efficiency of the partially-connected structure is low. The structure proposed in this paper can further approximate the spectral efficiency of the full digital precoding on the basis of the partially-connected structure, but the complexity of the hardware structure and algorithm is much lower than that of the fully-connected structure.

5. Conclusions

The hybrid precoding structure based on a switching network can effectively reduce the number of RF chains in massive MIMO antenna systems, especially when the antenna scale is large. The spectral efficiency of the SNHBP structure can approach the performance of the full digital precoding structure, but the number of RF chains used by the SNHBP structure is smaller than that of full digital precoding structure, and the number of phase shifters used by the SNHBP structure is smaller than that of the fully-connected precoding structure, which reduces the hardware complexity of hybrid precoding structure. The switched network is used in the SNHBP, which is more dynamic and optimized than the partially-connected structure, so the spectral efficiency is greatly improved compared with the partially-connected structure. With the phase optimization optimal dynamic grou** algorithm, the SNHBP structure can still obtain better spectral efficiency than the partially-connected structure under the condition that the phase shifter quantization bit is 3. The three dynamic grou** schemes can achieve similar spectrum efficiency of full digital precoding and can be selected in different application environments. Further simplifying the hardware requirements of the system is conducive to the engineering implementation of the SNHBP structure in the hybrid precoding field.