Hybrid Precoding for mmWave Massive Beamspace MIMO System with Limited Resolution Overlapped Phase Shifters Network

Ting DING, Jiandong ZHU, Jing YANG, Xingmeng JIANG, Chengcheng LIU

Summary :

Considering the non-convexity of hybrid precoding and the hardware constraints of practical systems, a hybrid precoding architecture, which combines limited-resolution overlapped phase shifter networks with lens array, is investigated. The analogy part is a beam selection network composed of overlapped low-resolution phase shifter networks. In particular, in the proposed hybrid precoding algorithm, the analog precoding improves array gain by utilizing the quantization beam alignment method, whereas the digital precoding schemes multiplexing gain by adopting a Wiener Filter precoding scheme with a minimum mean square error criterion. Finally, in the sparse scattering millimeter-wave channel for the uniform linear array, the proposed method is compared with the existing scheme by computer simulation by using the ideal channel state information and the non-ideal channel state information. It is concluded that the proposed scheme performs better in low signal-to-noise regions and can achieve a good compromise between system performance and hardware complexity.

• Publication: IEICE TRANSACTIONS on Electronics Vol.E107-C No.10 pp.355-363
• Publication Date: 2024/10/01
• Publicized: 2024/03/25
• Online ISSN: 1745-1353
• DOI: 10.1587/transele.2024MMP0004
• Type of Manuscript: Special Section PAPER (Special Section on Microwave and Millimeter-Wave Technologies)

1.  Introduction

The Massive Multiple-Input Multiple-Output (MIMO) technology in the millimeter-wave (mmWave) band has become one of the fundamental technologies of 5G and Beyond 5G wireless communication systems because it is capable of meeting the requirements of future mobile communication systems such as ultra-high transmission rate and ultra-low communication delays [1]-[3]. With such a large number of antennas, on the other hand, using individual radio frequency (RF) chains for every antenna will make systems more complex and result in an unacceptable amount of hardware and energy consumption [4].

Hardware costs, as well as power consumption, can both be reduced by using digital/analog hybrid precoding. There are two types of hybrid precoding architectures for millimeter-wave MIMO systems: fully and partially connected. For fully connected architectures, each RF chain can achieve the whole array gain of large antenna arrays. Still, it will lead to an increase in the number of phase shifters, which, to some extent, increases the hardware complexity and energy consumption of the system. The partially connected architecture is very different from the fully connected architecture. In partially connected architecture, each RF chain is connected to an antenna subarray through a phase shifter network composed of analog phase shifters; after adjusting the phase by the phase shifters network, it is mapped to the corresponding antenna subarray and is not mixed with the signals on other RF chains. However, with the number of analog phase shifters reducing in partially integrated hybrid architectures, the system’s power consumption and hardware complexity are significantly lower than in completely connected hybrid architectures. A suitable hybrid precoding optimization approach for mmWave massive MIMO must still be developed to minimize hardware cost and energy consumption.

Beamspace MIMO is also considered a practical approach for reducing the RF chains needed for massive mmWave MIMO systems. With beamspace MIMO, the power of each channel path is provided by an antenna element that employs a lens antenna array [5]-[7]. A typical spatial MIMO channel can be transformed into a beamspace channel, in which every element of the equivalence channel represents the gain of a specific beam. Due to the limitations of mmWave propagation scattering, a remarkable characteristic of the beamspace channel is its sparse structure [7]-[11]. Consequently, most of the beamspace channel gain can be collected by just a few channel elements. This can reduce energy consumption, leading to lower hardware costs and lower energy consumption. In previous studies, a beam selection algorithm based on the Maximum Magnitude (MM) of user data was proposed in [7], and an Interference-ware Beam Selection (IA-BS) algorithm in [8] is proposed to reduce the interference of different users in the same beam.

The above analysis is centered on the assumption that the resolution of the phase shifters is infinite. In a practical mmWave system, the phase shifters can not be set to an arbitrary angle, and the resolution of the phase shifters is usually minimal. The higher the resolution, the more expensive it is [12]-[16]. Therefore, the paper considers the non-convex characteristics of hybrid precoding and the hardware limitations of actual systems intending to reduce the hardware cost and energy consumption.

Following is a summary of the contributions in this paper:

• We proposed a hybrid precoding architecture combining limited-resolution overlapped phase shifter networks with a lens array.

• Based on the proposed structure, a two-stage hybrid scheme is proposed, where the analog precoding improves array gain by utilizing the quantization beam alignment method, and the digital precoding schemes multiplexing gain by adopting a Wiener Filter precoding scheme with a minimum mean square error (MMSE) criterion.

• The proposed scheme performs better in low signal noise regions. Moreover, the proposed scheme can achieve a good tradeoff between system performance and hardware complexity.

2.  Model for mmWave Massive MIMO Systems

There are a number of mmWave massive MIMO systems that are discussed in the paper. The systems include \(N\) transmit antennas and \(N_{RF}\) RF chains, which allows for \(K\) users with a single antenna.

2.1  Spatial MIMO: A Traditional Approach

Based on the conventional MIMO downlink system, the \(K\times 1\) received signal vectors can be represented as \(\mathbf{y}\) for the \(K\) users,

\[\begin{equation*} \mathbf{y}=\mathbf{H}^{H}\mathbf{P}\mathbf{s}+\mathbf{n}, \tag{1} \end{equation*}\]

where \(\mathbf{H} = \left [ \mathbf{h}_{1},\mathbf{h}_{2},\cdots ,\mathbf{h}_{K} \right ]\) represent the whole users’ channel matrix. Meanwhile, \(\mathbf{n}\) of the dimension of \(K\times 1\) is an AWGN vector, which has zero-mean and variance equal to \(\sigma ^{2}\). The matrices \(\mathbf{P}\in \mathbb{C}^{N\times K}\) and \(\mathbf{s}\) represent the precoding matrix and the transmitting signal vector.

2.2  Beamspace MIMO

It is possible to transfer from the spatial domain channel of a conventional system to the channel of the beamspace using a discrete lens array (DLA) [17]. The lens array contains N orthogonal directions (beams) array steering vectors, which cover the complete angular domain as [15], [17],

\[\begin{equation*} \mathbf{U}=\left [ \alpha \left ( \bar{\varphi}_{1} \right ) , \alpha \left ( \bar{\varphi}_{2} \right ) ,\cdots , \alpha \left ( \bar{\varphi}_{N} \right ) \right ]^{H}, \tag{2} \end{equation*}\]

where \(\bar{\varphi }_{n}=\frac{1}{N}\left ( n-\frac{N+1}{2} \right )\), for \(n=1,2,\cdots ,N\) represents the normalized spatial orientation. \(\alpha \left ( \varphi \right )\) represents the steer vector of the array. ULAs with \(N\) antennas are typically characterized as follows,

\[\begin{equation*} \alpha \left ( \varphi \right )=\frac{1}{\sqrt{N}}\left [ e^{-j2\pi \varphi m} \right ]_{m\in \tau \left ( N \right )}, \tag{3} \end{equation*}\]

where \(\tau \left ( N \right )=\left \{ l-\left ( N-1 \right )/2,l=0,1,\cdots ,N-1 \right \}\) is a set of symmetric numerical values which are evenly distributed around zero. \(\varphi\triangleq\frac{d}{\lambda }sin\theta\), where \(\theta\) refers to the directional component, and \(d\) is the space between adjacent antennas, which in mmWave frequencies is usually \(d=\lambda /2\).

Fig. 1  A hybrid precoding architecture for beamspace MIMO.

By utilizing DLA and incorporating minimal performance degradation into the design, it is possible to convert a conventional space channel into a beamspace channel. Channels in the mmWave band are sparse. Thus, the model captures this sparsity. Figure 2 illustrates a hybrid precoding architecture with finite-resolution phase shifters, where each RF chain is linked to each antenna. As a result, the particular architectural design can achieve a wide variety of gains with near-optimal performance. However, it is typically significantly more expensive and energy-intensive than other architectures. In beamspace MIMO, the system model is represented as follows,

\[\begin{equation*} \mathbf{\tilde{y}}=\mathbf{H}^{H}\mathbf{U}^{H}\mathbf{P}\mathbf{s}+\mathbf{n}=\mathbf{\tilde{H}}^{H}\mathbf{P}\mathbf{s}+\mathbf{n}, \tag{4} \end{equation*}\]

where \(\mathbf{\tilde{y}}\) represents the beamspace signal that was received by the receiver, \(\mathbf{\tilde{H}}=\left [\mathbf{\tilde{h}}_{1},\mathbf{\tilde{h}}_{2},\cdots ,\mathbf{\tilde{h}}_{K} \right ]\) represents the beamspace channel that was received by the receiver, and \(\mathbf{\tilde{h}}_{k}\) represents \(k\)-th user’s beamspace channel, we have

\[\begin{equation*} \mathbf{\tilde{H}}=\mathbf{U}\mathbf{H}=\left [ \mathbf{U}\mathbf{h}_{1},\mathbf{U}\mathbf{h}_{2},\cdots ,\mathbf{U}\mathbf{h}_{K} \right ]. \tag{5} \end{equation*}\]

In accordance with (5), the columns of \(\mathbf{\tilde{H}}\left (\tilde{\mathbf{h}}_{k}\right)\) correspond to \(N\) orthogonal beams whose orientation in space are \(\tilde{\varphi}_{1},\tilde{\varphi}_{2},\cdots,\tilde{\varphi}_{N}\), specifically. When propagating at mmWave frequencies, predominant scattering is quite limited. We use the Saleh-Valenzuela channel model [7], [8], [14], \(k\) user of \(\mathbf{h}\) can be summarized as follows:

\[\begin{equation*} \mathbf{h}_{k}=\sqrt{\frac{N}{L+1}}\sum_{l=0}^{L}\beta _{k}^{\left ( l \right )}\mathbf{a}\left ( \varphi _{k}^{\left ( l \right )},\theta _{k}^{\left ( l \right )} \right )=\sqrt{\frac{N}{L+1}}\sum_{i=0}^{L}\mathbf{c}_{i}, \tag{6} \end{equation*}\]

where \(\varphi _{k}^{(l)}\) and \(\theta _{k}^{(l)}\) represent the angel of depature (AOD) and angle of arrival (AOA) of the path for \(k\)-th user. \(\mathbf{a}\left ( \varphi _{k}^{(l)},\theta _{k}^{(l)} \right )\) is the steering vector for the array. \(\mathbf{c}_{0}=\beta _{k}^{(0)}\mathbf{a}\left ( \varphi _{k}^{(0)} \right )\) is the line-of-sight (LoS) for the \(k\)-th user with \(\beta _{k}^{(0)}\) denoting the complex gain and \(\varphi _{k}^{(0)}\) is spatial orientation, \(\mathbf{c}_{i}=\beta _{k}^{(l)}\mathbf{a}\left ( \varphi _{k}^{(l)} \right )\) for \(1\leq l\leq L\) is the \(l\)-th non-line-of-sight (NLoS) of the \(k\)-th user, and \(L\) is the whole amount of NLoS.

Fig. 2  Hybrid beamspace MIMO architecture based on phase shifters with low resolution.

2.3  The Selection of Beams

The number of dominant scatters in beamspace channel vectors limits the number of dominant scatters. Therefore, the amount of more prominent components in each beamspace channel vector is much smaller than \(N\) due to the limitation of the number of dominant scatters [8]. In terms of structure, it is notable that \(\mathbf{\tilde H}\) represents the directions of the different users sparsely. The number of RF chains can be reduced in the beam selection process without affecting performance significantly. For example, in the downlink, if there are \(K\) receivers equipped with singular antennas, the received signal is expressed as follows:

\[\begin{align} \mathbf{\tilde{y}}\approx \mathbf{\tilde{H}}_{r}^{H}\mathbf{P}_{r}\mathbf{s}+\mathbf{n}. \tag{7} \end{align}\]

Considering the limited scattering of the spatial channel matrix for beams, \(\mathbf{\tilde{H}}_{r}\in \mathbb{C}^{N_{RF}\times K}\) is the reduced dimension of the channel matrix, and selecting a reduced amount of main beams can decrease the MIMO size to \(\mathbf{\tilde{H}}_{r}\). \(\mathbf{P}_{r}\in \mathbb{C}^{N_{RF}\times K}\) is corresponds to the precoder which is dimensionally reduced.

3.  Design of a Hybrid Precoding Scheme

An optimal hybrid precoding scheme based on Quantized Beam Alignment (QBA) is presented in this section to maximize the achievable sum rate in an overlapped phase shifters network with a lens array.

3.1  Proposed Hybrid Precoding Architecture

As a solution to the high energy consumption issue, we have developed a hybrid precoding scheme that combines PSS overlapped with lens arrays to reduce its energy consumption. Figure 3 illustrates how a BS provides service to \(K\) single-antenna users with \(N\) components lens antenna arrays and \(N_{RF}\) RF chains. A received signal can be represented by

\[\begin{equation*} \mathbf{\tilde{y}}=\mathbf{\tilde{H}}^{H}\mathbf{\tilde{F}}_{RF}\mathbf{\tilde{F}}_{BB}\mathbf{s}+\mathbf{n}. \tag{8} \end{equation*}\]

\(\mathbf{\tilde{F}}_{RF}=\left [ \mathbf{\tilde{f}}_{RF}^{\left ( 1 \right )},\mathbf{\tilde{f}}_{RF}^{\left ( 2\right )},\cdots ,\mathbf{\tilde{f}}_{RF}^{\left ( N_{RF} \right )} \right ]\in \mathbb{C}^{N\times N_{RF}}\) and \(\mathbf{\tilde{F}}_{BB}=\left [ \mathbf{\tilde{f}}_{BB}^{\left ( 1 \right )},\mathbf{\tilde{f}}_{BB}^{\left ( 2\right )},\cdots ,\mathbf{\tilde{f}}_{BB}^{\left (K \right )} \right ]\in \mathbb{C}^{N_{RF}\times K}\) are the analog beamformer and the digital precoder of beamspace, separately.

Fig. 3  Proposed overlapped phase shifters based precoding architecture with lens array.

4.  The Proposed Hybrid Approach

It is designed using \(\mathbf{\tilde{F}}_{RF}\) and \(\mathbf{\tilde{F}}_{BB}\) for the purpose of maximising the achievable sum rate \(R_{\rm{sum}}\). According to the following representation:

\[\begin{align} \begin{array}{@{}cc} \left ( \mathbf{\tilde{F}}_{RF}^{opt},\mathbf{\tilde{F}}_{BB}^{opt} \right )=\begin{matrix} \textbf{arg max}~~R_{\rm{sum}}, & \\ \mathbf{\tilde{F}}_{RF},\mathbf{\tilde{F}}_{BB} \end{matrix}\\ s.t. \mathbf{\tilde{F}}_{RF}\in \mathbf{\Gamma} ,\\ \left \| \mathbf{\tilde{F}}_{RF}\mathbf{\tilde{F}}_{BB} \right \|_{F}^{2}=P_{T}, \end{array} \tag{9} \end{align}\]

where \(\mathbf{\Gamma}\) denotes the set of every potential analog beam-former which satisfy the (9), the \(k\)-th column of \(\mathbf{\tilde{F}}_{BB}\) is \(\mathbf{\tilde{f}}_{BB}^{\left ( k\right )}\), meanwhile, \(R_{\rm{sum}}\) will be calculated as follows for a given channel realization,

\[\begin{align} R_{\rm{sum}}=\sum_{k=1}^{K}log_{2}\left ( 1+\gamma _{k} \right ), \tag{10} \end{align}\]

where \(\gamma _{k}\) refers to the ratio of signal to interference noise ratio (SINR) for the \(k\)th user as

\[\begin{align} \gamma _{k}=\frac{\left | \mathbf{\tilde{h}}_{k}^{H}\mathbf{\tilde{F}}_{RF}\mathbf{\tilde{f}}_{BB}^{(k)} \right |^{2}}{\sum_{{k}'\neq k}^{K}\left | \mathbf{\tilde{h}}_{k}^{H}\mathbf{\tilde{F}}_{RF}\mathbf{\tilde{f}}_{BB}^{({k}')} \right |^{2}+\sigma ^{2}}. \tag{11} \end{align}\]

In order to solve the problem, we first decouple \(\mathbf{\tilde{F}}_{RF}\) and \(\mathbf{\tilde{F}}_{BB}\) joint designs. Due to the limited number of possible values for \(\mathbf{\tilde{F}}_{BB}\), the optimization can be accomplished by trying all possible values for \(\mathbf{\tilde{F}}_{RF}\). Since there are a large number of \(N\), this will result in a level of complexity that cannot be afforded. Therefore, we come up with a more intelligent way to reduce complexity. It is significant to note that as a result of the specific structure of the selected network, the \(\mathbf{\tilde{F}}_{RF}\) matrix does not have the same design as the fully connected subarray matrix but becomes a diagonalization matrix with a unique structure as follows,

\[\begin{equation*} \begin{aligned} \mathbf{\tilde{F}}_{\rm{RF}}&=\begin{bmatrix} \mathbf{\tilde{F}}_{\rm{RF}}^{\left ( 1\right )}, & \mathbf{\tilde{F}}_{\rm{RF}}^{\left ( 2\right )}, & \cdots, & \mathbf{\tilde{F}}_{\rm{RF}}^{\left ( N_{RF}\right )} \end{bmatrix}\\ &=\begin{bmatrix} \mathbf{\tilde{f}_{1}}\left ( 1 \right ) & & & \\ \vdots & \mathbf{\tilde{f}_{2}}\left ( 1 \right ) & \ddots & \\ \mathbf{\tilde{f}_{1}}\left ( M_{s} \right ) & \vdots & & \mathbf{\tilde{f}_{N_{\rm{RF}}}}\left ( 1 \right )\\ & \mathbf{\tilde{f}_{2}}\left ( M_{s} \right ) & \ddots & \vdots \\ & & & \mathbf{\tilde{f}_{N_{\rm{RF}}}}\left ( M_{s} \right ) \end{bmatrix} \end{aligned}, \tag{12} \end{equation*}\]

where \(\mathbf{\tilde{f_{i}}}\left ( n \right )\left ( i=1,2,\cdots ,N_{\rm{RF}},n=1,2,\cdots ,M_{s} \right )\) is the precoding matrix \(\mathbf{\tilde{F}}_{RF}\) of nonzero weight, and the magnitude is

\[\begin{align} \mathbf{\tilde{f_{i}}}\left ( n \right )\mathbf{\tilde{f_{i}}}\left ( n \right )^{H}=\frac{1}{M_{s}}. \tag{13} \end{align}\]

The matrix structure for analog precoding varies depending on the array structure. A block diagonal matrix is formed by \(\mathbf{\tilde{F}}_{RF}\), which breaks the standard Eigenvalue problem structure. Therefore, Eigenvalue Decomposition (EVD) cannot be applied.

Taking full advantage of a high-capacity massive MIMO system and reducing the hardware limitation, it is suggested to use economical and practical phase shifters to couple \(K\) RF Chains outputs with \(N_{RF}\) transmitting antennas, beam alignment (BA) is used to maximize the gain of the array antenna, as a result, in the RF domain, phase control is obtained by taking the phase of the conjugate transpose from a base station and applying it to the aggregated downlinks of multiple users to get phase control. This is to align the phase of the channel elements, ensuring that the array’s high gain results from too many antennas. The term beam alignment has been used in some literature to describe the process of aiming the beam toward the intended target [18], [19]. Nevertheless, the beam alignment precoding presented in this manuscript differs from Beam alignment in that the alignment gain is not the direction of the selected beam. Inspired by this idea, for MIMO systems with overlapped phase shifters, the RF precoding matrix of the system is formulated as follows,

\[\begin{align} \mathbf{\tilde{F}}_{\rm{RF}}\left ( s,t\right )=\frac{1}{\sqrt{M_{s}}}e^{j\times angle\left ( \mathbf{\tilde{H}}^{H}\left ( s,t\right )\right )}, \tag{14} \end{align}\]

where \(\left ( s,t \right )\) and \(angle\left (\mathbf{\tilde{H}}^{H}\left ( s,t \right ) \right )\) denote the element and the phase of the element in the \(s\)-th row while \(t\)-th column of the matrix, respectively. \(s=1,\cdots ,M_{s}\), \(t=1+\left ( M_{s}-\Delta M \right )\left ( s-1 \right ),\cdots ,s M_{s}-\Delta M\left ( s-1 \right )\), and the other elements in \(\mathbf{\tilde{F}}_{\rm{RF}}\) are all zero. Each RF chain is connected to a subarray in the overlapped subarrays while allowing these subarrays to overlap. Since the antennas in the overlapped area are connected to more RF chains, \(N_{RF} \times M_{s}\) phase shifters are required. \(M_{s}\) denotes the number of phase shifters connected to an RF chain, that is, the number of precoded antennas in each subarray, \(\Delta M\) represents the number of overlapped subarrays. When \(\Delta M = M_{s} =N /N_{RF}\), the system is a non-overlapped subarray, when \(\Delta M = M_{s} =N\), the system is a fully connected structure, when \(0\le \Delta M\le M_{s}\), the system is an overlapped subarray. Index vectors are set as follows,

\[\begin{align} x_{n}\left [ j\right ]=\left\{\begin{matrix} 1, &j\subseteq \Theta _{n} \\ 0, &j\subseteq \bar{\Theta }_{n} \end{matrix}\right., \tag{15} \end{align}\]

where the \(n\)th index set \(\Theta _{n}, n=1,\cdots ,N_{RF}\) denotes the range of \(1+(n-1)\Delta M_{s}:M_{s}+(n-1)\Delta M_{s}\) and its complement \(\bar{\Theta }_{n}\) follows \(\bar{\Theta }_{n}\bigcup \Theta_{n}=\varsigma\), where \(\varsigma\) is a universal set of \(1:N\). Therefore, the formula (14) can also be expressed as

\[\begin{align} \mathbf{\tilde{F}}_{\rm{RF}}^{\left ( n\right )}=\frac{1}{\sqrt{M_{s}}}e^{j\times angle\left ( \mathbf{\tilde{H}}^{H}\right )}\odot \mathbf{x}_{n}. \tag{16} \end{align}\]

From Eq. (16), the RF precoding \(\mathbf{\tilde{F}}_{\rm{RF}}\) differs only in the phase of the assumed continuous value. However, in practical applications, due to the practical limitations of variable phase shifters, the phase of each input is often heavily quantized [20]. Hence, it is necessary to test the precoding scheme in practice, where all the phases of \(\mathbf{\tilde{F}}_{\rm{RF}}\) are quantized to the precision of \(B\) bits, quantized to the nearest neighboring phase according to the nearest Euclidean distance. The quantized phase in \(\mathbf{\tilde{F}}_{\rm{RF}}\) is represented as

\[\begin{align} \hat{\varphi }=\left ( 2\pi \hat{n}\right )/2^{B}, \tag{17} \end{align}\]

where \(\hat{n}\) can be chosen according to the following formula,

\[\begin{align} \hat{n}=\begin{matrix} \begin{matrix} \mathbf{arg min}\\ n\in \left ( 0,\cdots ,2^{B}-1\right ) \end{matrix} \end{matrix}\left | \varphi -\frac{2\pi n}{2^{B}}\right |, \tag{18} \end{align}\]

where \(\varphi\) is the unquantized phase obtained by (16).

Then we use the classic Wiener Filter (WF) precoding algorithm [21], [22], based on the following calculation,

\[\begin{align} & \mathbf{G}=\left ( \mathbf{\tilde{H}}_{eq}\left ( \mathbf{\tilde{H}}_{eq}\right )^{H}+\zeta \mathbf{I}\right )^{-1}\mathbf{\tilde{H}}_{eq}, \tag{19} \\ &\zeta =\sigma ^{2}K/P_{T}, \tag{20} \\ & \mathbf{\tilde{F}}_{\rm{BB}}=\beta \mathbf{G}, \tag{21} \\ & \beta =\sqrt{\frac{P_{T}}{tr\left ( \mathbf{G}\mathbf{\Lambda} \mathbf{G}^{H}\right )}}, \tag{22} \end{align}\]

where \(\mathbf{\tilde{H}}_{eq}=\mathbf{\tilde{H}}\mathbf{\tilde{F}}_{RF}\) represents the equivalent channel, \(\mathbf{\Lambda}=E\left [ \mathbf{s}\mathbf{s}\right ]^{H}\) represents the matrix of diagonal correlations of the data \(\mathbf{s}\). Finally, \(\mathbf{\tilde{F}}_{RF}\) and \(\mathbf{\tilde{F}}_{BB}\) are brought into Eqs. (11) and (12) to obtain \(\mathbf{\tilde{R}}_{sum}\). The hybrid precoding scheme proposed in the paper is called Quantized Beam Alignment (QBA) hybrid precoding, and the detailed flow of the algorithm is shown in Algorithm 1.

However, in the practical mmWave MIMO system, obtaining the complete Channel State Information (CSI) is very challenging [23]-[28]. Hybrid precoding systems that use phase shifters networks usually have fewer RF chains than antennas. As a result, CSI for a hybrid precoding architecture is more complex than CSI for fully digital precoding systems. This is a complex issue, and there are two typical approaches to solving it. Firstly, the approach consists of two steps for channel estimation. The initial step involves training the beam using analog beamforming techniques to calculate \(\mathbf{A}\) [30]. Secondly, the procedure requires analysis of an efficient channel matrix \(\mathbf{HA}\) by classical methods, such as minimum mean square error. Instead of estimating the MIMO channel matrix \(\mathbf{HA}\), it is also possible to exploit mmWave’s low-rank MIMO channel to get a complete channel \(\mathbf{H}\) matrix with minimal pilot overhead by utilizing the channel matrix \(\mathbf{H}\) in mmWave. For example, in [31], a channel estimation scheme employing adaptive Compressed Sensing (CS) is proposed. The complete channel estimation process combines multiple subproblems, each taking a single channel path into account. Using an Orthogonal Matching Pursuit (OMP) algorithm, it determines the direction of each way based on a coarsely oriented grid. Refine the path direction further with a narrower directional grid. After that, use a more limited directional grid and further refine the path direction.

The paper utilizes the CSI under the optimal channel state and CSI obtained by the OMP algorithm. The proposed scheme is validated by using the channel estimation of the OMP algorithm.

5.  Energy Efficiency

It is generally considered that energy efficiency (EE) refers to the ratio between the achievable sum-rate \(R_{\rm{sum}}\) and the total energy consumption [8], [32]. EE is expressed based on the following equation,

\[\begin{align} EE=\frac{R_{\rm{sum}}}{P_{\rm{T}}+P_{BB}+P_{\rm{H}}}(\rm bps/Hz/W). \tag{23} \end{align}\]

The transmission power is called \(P_{T}\), while the baseband’s power consumption is called \(P_{BB}\), and the energy consumed by the hardware architecture is called \(P_{H}\).

The spatial MIMO is usually achieved by linking one antenna to every RF chain. So we have

\[\begin{align} P_{\rm{H}}=NP_{\rm{RF}}, \tag{24} \end{align}\]

where \(P_{\rm{RF}}\) represents energy consumption from each RF chain. RF chains usually depend on the number of antennas, which is large for mmWave massive MIMO.

A beamspace MIMO system is shown in Fig. 1, in which the selecting network consists of switches having equal numbers as RF chains. In the structure, we have

\[\begin{align} P_{\rm{H}}=N_{\rm{RF}}P_{\rm{RF}}+N_{\rm{RF}} P_{\rm{SW}}. \tag{25} \end{align}\]

\(P_{\rm{SW}}\) represents a switch’s energy consumption, which is much smaller than \(P_{\rm{PS}}\).

Figure 2 illustrates a hybrid precoding architecture with completely connected components. In the architecture, each RF chain on the BS is linked to each antenna by phase shifters, where

\[\begin{align} P_{\rm{H}}=N_{\rm{RF}}P_{\rm{RF}}+N_{\rm{RF}}N P_{\rm{PS}}+NP_{\rm{COM}}. \tag{26} \end{align}\]

The power consumption of power combiner, phase shifter and RF chains are represented by \(P_{\rm{PS}}\), \(P_{\rm{COM}}\), and \(P_{\rm{RF}}\), respectively.

6.  Simulation Results

The simulation parameters are summarized in Table 1.

Table 1  List of parameters

In accordance with (6), the following steps are taken in generating mmWave MIMO channels:

• NLoS consists of two components and LoS consists of one component;

• \(\beta _{k}^{\left ( 0 \right )}\sim \mathcal{CN} \left ( 0,1 \right )\), \(\beta _{k}^{\left ( l \right )}\sim \mathcal{CN} \left ( 0,10^{-1} \right )\) for \(l=1,2\);

• Bath \(\varphi _{k}^{\left ( 0 \right )}\) and \(\varphi _{k}^{\left ( l \right )}\) follow the principle of i.i.d. uniform distribution, which has a range of \(\left [ -\frac{1}{2},\frac{1}{2} \right ]\).

The remaining simulation parameters are listed in Table 1. There are four algorithms analyzed in this paper for their achievable sum-rate performance: (1) the fully digital precoding algorithm; (2) the MM-BS antenna selection algorithm (MM-BS) [7]; (3) the hybrid precoding algorithm using interference-sensing antenna selection algorithm (IA-BS) [8], and (4) the proposed QBA hybrid precoding algorithm.

We propose a hybrid precoding algorithm that utilizes QBA in this paper, as shown in Fig. 4. As soon as \(\Delta = 8\) (i.e., there are eight overlapped phase shifters between adjacent subarrays), the system architecture is fully connected, with eight overlapped elements between two adjacent subarrays. Regarding signal-to-noise ratios (SNR) (less than 27 dB), the beam selection algorithm performs better than the MM-BS and IA-BS algorithms and is closer to fully digital precoding. It is also possible to infer that the IA-BS algorithm is very similar to the MM-BS algorithm of two beams per user.

Fig. 4  Sum-rate comparisons (perfect CSI).

As shown in Fig. 5, in the low SNR region (less than 25 dB), the proposed algorithm outperforms MM-BS and IA-BS when \(\Delta = 7\) (i.e., the number of overlapped elements between adjacent subarrays is 7) and is closer to fully digital precoding performance. In particular, the fundamental cause is that traditional beam selection algorithms are not integrated with baseband digital precoding algorithms. Furthermore, as the number of phase shifters overlapped decreases, the system’s hardware complexity decreases, which results in reduced performance for the hybrid precoding algorithm based on QBA. The reduction in array gain is primarily due to the decrease in the number of overlapped elements.

Fig. 5  Sum-rate comparisons (perfect CSI).

According to Fig. 6, the proposed QBA hybrid precoding algorithm can be used in low signal-to-noise ratio (SNR) situations with \(\Delta M= 6\) (i.e., the number of overlapped elements between adjacent subarrays is 6). Compared to the MM-BS and IA-BS beam selection algorithms, the proposed QBA algorithm performs better and is closer to the fully digital precoding performance. The QBA based hybrid precoding algorithm outperforms the MM-BS and IA-BS beam selection algorithms in low SNR. The main reason is that traditional beam selection algorithms are designed differently than baseband digital precoding algorithms. In the low SNR region, the sum rate of the MM-BS algorithm and IA-BS algorithm are lower than that of the hybrid precoding algorithm based on QBA because of the power allocation among multiple users and the aggravation of inter-user interference. In the proposed QBA algorithm, from formula (14), a radio frequency precoding matrix is designed to obtain the maximum array gain by extracting a portion of the downlink channel matrix phase information from the transmitter to the user. In particular, phase control is achieved by removing the phase of a conjugate transposition from a base station to an aggregate downlink channel for multiple users to align the phase of the channel element; thus, the significant array gain provided by multiple antennas can be obtained. With the increase in SNR, the proposed algorithm’s performance decreases because of quantization. In practical applications, the low SNR region usually corresponds to the situation where the channel conditions, such as the cell edge of the mobile communication system, are not ideal; that is, with the increase of SNR, the superiority of the proposed algorithm decreases. Thus, the hybrid precoding technique presented in this paper is more suitable for the cell edge of a mobile communication system. Moreover, Figs. 4, 5, and 6 show that as the number of overlapped subarrays increases, that is, the hardware complexity of the system increases and the system’s sum-rate increases; thus, as we understand it, the overlapped system can achieve a compromise between hardware complexity and system performance.

Fig. 6  Sum-rate comparisons (perfect CSI).

From Fig. 7, the proposed QBA hybrid precoding scheme (when \(\Delta M = 8\), i.e., the number of overlapped phase shifters between adjacent subarrays is 8) has slightly lower (averages less than one bps/Hz/W) energy efficiency than the traditional beam selection algorithm when the user number is \(K\leq 32\).

Fig. 7  Energy efficiency comparisons (perfect CSI).

Simultaneously, as shown in Figs. 8, 9, and 10, the system’s hardware complexity decreases as the number of phase shifters overlapping in the subarray decreases. The main reason is that the hybrid precoding scheme with QBA performs worse. The array gain decreases due to decreased overlapped and practical elements. Based on nonideal CSI (such as the CSI estimated by the OMP algorithm), the hybrid precoding method based on the QBA presented in this paper may achieve a favorable balance between hardware complexity and sum-rate performance, depending on how many phase shifters overlap. At the same time, the results show that the hybrid precoding algorithm has high robustness under non-ideal channel conditions.

Fig. 8  Sum-rate comparisons (OMP CSI).

Fig. 9  Sum-rate comparisons (OMP CSI).

Fig. 10  Sum-rate comparisons (OMP CSI).

7.  Conclusion

The paper proposed a hybrid precoding architecture combining limited-resolution overlapped phase shifter networks with a lens array. Based on the proposed structure, a two-stage hybrid scheme is proposed, where the analog precoding improves array gain by utilizing the quantization beam alignment method, and the digital precoding schemes multiplexing gain by adopting a Wiener Filter precoding scheme with a minimum mean square error criterion. According to simulations, the developed QBA-based hybrid precoding algorithm has better performance under low SNR in terms of both ideal CSI and non-ideal CSI. Moreover, the proposed scheme can achieve a good tradeoff between system performance and complexity. Furthermore, 5G and 6G technology can be a crucial enabler for mmWave communications [33]. We expect mmWave beamspace MIMO systems to have great potential in future mobile communications.

Acknowledgments

This work was Sponsored by Natural Science Foundation of Henan Province No.222300420379.

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Authors

Ting DING
  Henan Province Engineering Research Center of High-speed Railway Operation and Maintenance Engineering

Jing YANG
  National Digital Switching System Engineering and Technological Research Center

Xingmeng JIANG
  Henan Province Engineering Research Center of High-speed Railway Operation and Maintenance Engineering

Chengcheng LIU
  National Digital Switching System Engineering and Technological Research Center

Keyword

massive MIMO,  mmWave,  hybrid precoding,  lens array,  phase shifters,  energy efficiency