Hybrid Beamforming Structure Using Grouping with Reduced Number of Phase Shifters in Multi-User MISO

Abstract

This paper proposes a novel hybrid beamforming (HBF) structure for gain-aware grouping transmit antennas and users in multiuser multiple-input single-output (MU-MISO) systems. In the conventional HBF structure, all transmit antennas form a beam to each user. In this case, the gain of each antenna varies depending on the location of the base station and each user, and the transmit power after the digital beamformer is allocated to the antenna with the smallest gain. Signals transmitted from antennas with small gains are susceptible to noise and interference. Therefore, this paper proposes an HBF structure in which only the antenna with the highest gain forms the beam for each user. In the proposed scheme, the transmitting antennas are grouped and the beam is formed only by the group of antennas with the highest gain for each user. Simulation results show that the proposed scheme can reduce the number of phase shifters used on the transmit side compared to the conventional HBF scheme while maintaining sum-rate performance when the number of transmit antennas and users are the same. It was also shown that there is a trade-off between the reduction in the number of phase shifters used to form the beam and the improvement in performance as the number of transmit antennas increases. Furthermore, it is shown that when antenna selection is used, although there is a trade-off between the number of phase shifters and performance improvement, the number of phase shifters can be reduced while maintaining performance even when the number of transmit antennas increases.

1. Introduction

As mobile communication tools like smartphones and Internet of Things (IoT) devices proliferate, communication traffic is rapidly increasing. Therefore, the next-generation wireless communication system, 6G, which is expected to be in service around 2030, requires six major capabilities: ultra-high-speed communication, ultra-wide coverage, ultra-low power consumption, ultra-low latency, ultra-reliable communication, and ultra-multiple connections and sensing [1]. Massive multiple-input multiple-output (MIMO) technology plays a significant role in meeting these requirements, with beamforming at the heart of this technology [2,3].

Beamforming is a technology that provides directionality in a specific direction by adjusting the phase of signals transmitted from each antenna. This technology transmits a stronger signal in the target direction and improves communication efficiency. It also reduces interference by suppressing emission of radio waves in other directions. In the case of a linear array antenna, signals from adjacent antenna elements have different arrival times, resulting in a phase difference. In this case, by adjusting the phase of the signals to compensate for the phase difference, the phase of the signals from each antenna will be the same, and the radio waves will strengthen each other in a particular direction [4]. In the case of planar array antennas, as with linear array antennas, signals from adjacent antenna elements have different arrival times in the horizontal and vertical directions, resulting in phase differences. By adjusting the phase to compensate for the phase difference in both directions, a beam can be formed in any direction [5].

There are two types of beamforming: analog beamforming, which adjusts phase, and digital beamforming, which adjusts amplitude and phase [6]. Analog beamforming (ABF) makes it difficult to form multiple beams simultaneously because a single radio frequency (RF) signal is connected to all antennas via phase shifters to control the direction of the beams. This will be a limiting issue in next-generation communications, where diverse users and devices will access the system simultaneously. Also, ABF is limited by the resolution of the phase shifter used in practice, which reduces the accuracy of angular adjustment and makes it difficult to perform optimal beamforming [7], whereas digital beamforming (DBF) is mainly used because digital signal processing can control different phases and amplitudes for each antenna, allowing for better adjustment of the beam direction. However, DBF requires one RF chain per antenna, which consists of a filter, a digital-to-analog converter (DAC), an analog-to-digital converter (ADC), and an amplifier, making it expensive and making it consume more power [8]. DBF becomes impractical as the number of antennas increases. Hybrid beamforming (HBF), which combines ABF and DBF, is attracting attention as a practical solution because it reduces the number of RF chains compared to DBF, thus lowering cost and power consumption [9].

There are linear algorithms, such as zero-forcing (ZF), mean square error minimization (MMSE), and maximum ratio transmission (MRT), that determine the weights of beamforming. In HBF, these algorithms are used to determine the digital beamformer weight from the equivalent channel matrix multiplied by the analog beamformer weight matrix. The ZF-based weight uses the pseudo-inverse matrix of the equivalent channel as the weight of the digital beamformer. This algorithm can reduce user interference to zero, but it cannot increase the received power of the desired signal [10]. The MMSE-based weight is digital beamformer weights that minimize the mean square error between the transmitted and received signals using the equivalent channel. This algorithm can improve signal-to-interference plus noise ratio (SINR) because it takes into account the effects of channel interference and noise, but it is computationally more expensive than ZF and MRT [11]. The MRT-based weight uses the complex conjugate transpose of the equivalent channel as the digital beamformer weight. This algorithm can maximize signal-to-noise ratio (SNR), but user interference becomes non-negligible. This also results in a large variation in the power of the desired signal received by each user [12]. The objective of this paper is to increase the received power of the desired signal in a multi-user MISO system by grouping the transmit antennas and users based on the received strength of each transmit antenna, which varies for each user. Therefore, this paper employs digital beamformer weights based on MRT, which have large variations in received strength.

HBF research includes studies of HBF when channel state information (CSI) feedback from users is incomplete, studies of HBF in narrowband and wideband channels, and studies of HBF structure. In massive MIMO with a large number of antennas, channel estimation becomes more complex and the time to be fed back may exceed the channel coherent time. To solve this problem, ref. [13] proposed HBF with robustness of the difference between the estimated channel and the actual channel using deep learning. In [14], the impact of CSI imperfections and hardware impairments on beamforming in a multi-user system is investigated and shown to significantly reduce spectral efficiency. Beamforming in narrowband channels in previous studies assumes that the channel does not change beyond the coherent bandwidth, while beamforming in wideband channels assumes that the channel does not change beyond the sub-bandwidth. In beamforming on wideband channels, however, the channel is different for each sub-band, and it is necessary to determine the appropriate weights for each sub-band. In [15], a DBF that controls the phase and amplitude for each sub-band is proposed. On the other hand, HBF has the limitation of controlling the same phase for all sub-bands in the analog beamformer, and ref. [16] propose an iterative algorithm to alternatingly optimize the phase shifter-based wideband analog precoder and low-dimensional digital precoder and low-complexity non-iterative hybrid precoder. Fully-connected (FC) architectures and sub-connected (SC) architectures have been studied as typical HBF structures. The FC architecture has a structure in which each RF chain is connected to all antennas through phase shifters, while the SC architecture has a structure in which each RF chain is connected only to disjoint antenna subarrays [17]. In [18], the sum-rate performance of the FC architecture and SC architecture is compared in terms of simulation and analysis. In [7], the authors propose a structure that dynamically selects a non-fixed number of the transmit antennas connected to each RF chain in SC. In both papers, the FC architecture shows the best sum-rate performance among the HBF structures. Therefore, this paper focuses on the structure of the FC architecture and proposes a new HBF structure.

Antenna selection is a technique that determines the antenna elements to be used for beamforming and uses the appropriate antenna for the communication environment. This allows a limited number of RF chains to be allocated to the most appropriate antenna element instead of providing RF chains for all antenna elements, thus making efficient use of limited power. In addition, antenna selection based on channel conditions improves performance by allowing communication using antennas with better signal quality when the channel is dynamically changing [5].

In the conventional HBF structure (FC architecture), all transmit antennas form beams to each user. The gain of each antenna varies depending on the location of the base station and each user, resulting in the transmit power after digital beamformer being allocated to the antenna with the smallest gain [19,20]. Signals transmitted from antennas with small gains are susceptible to noise and interference when demodulated at the receiver side, making correct demodulation challenging.

This paper proposes an HBF structure that divides users and transmit antennas into multiple groups, considering the antenna gains obtained from channel estimation. In this method, the beam is formed only by the group of antennas with the highest gain for each user, ensuring that the desired signal received by each user has high power. Through simulations, this paper shows that the proposed method can reduce the number of phase shifters used on the transmit side compared to the conventional HBF scheme while maintaining sum-rate performance when the number of transmit antennas and users are the same. It was also shown that there is a trade-off between the reduction in the number of phase shifters used to form the beam and the improvement in performance as the number of transmit antennas increases. Furthermore, it is shown that when antenna selection is used, although there is a trade-off between the number of phase shifters and performance improvement, the number of phase shifters can be reduced while maintaining performance even when the number of antennas increases. Here, the power consumption at the transmit side decreases as the number of phase shifters to be used when transmitting radio waves decreases. In addition, the amount of heat generated when operating the phase shifters also decreases. As a result, the proposed method enables energy-efficient transmission of radio waves.

This paper is organized as follows. Section 2 describes the MRT hybrid precoding and the system model. In Section 3, we present our proposed method. In Section 4, we show the simulation results to demonstrate the effectiveness of our proposed scheme, and in Section 5, we conclude the paper.

2. System Model

This section describes MRT hybrid precoding and the reception model. Beamforming technology manipulates transmitted data to make radio waves more directional, forming a sharp beam. HBF combines ABF and DBF, where a digital beamformer and an analog beamformer process signals to form a beam. HBF remains practical because the number of RF chains can be reduced compared to DBF even as the number of antennas increases [21].

In this paper, we consider a downlink multi-user MISO system, where the number of base station (BS) antennas is $N_t$, the number of users is $N_u$, the number of RF chains at BS is $N_{rf}$, and the number of antennas per user is 1, as shown in Figure 1. Here, ${\mathit{h}}_{\mathit{k}}=[h_{k,1},h_{k,2},\cdots ,h_{k,N_t}]\in {\mathbb{C}}^{1\times N_t}$ denotes the channel vector from the BS to the k-th user. Rayleigh flat fading channels are considered as the channel models. Each entry of the channel vector is independent and identically distributed (i.i.d.) following $\mathcal{CN}(0,1)$, which represents a symmetric complex Gaussian distribution with zero mean and unit variance. We also assume perfect CSI is fed back from each user to the BS. The channel matrix $\mathit{H}$ is expressed as

$$\mathit{H}={[{\mathit{h}}_{\mathbf{1}},\cdots ,{\mathit{h}}_{{\mathit{N}}_{\mathit{u}}}]}^{\mathrm{T}}.$$

(1)

Different signals are transmitted to each user antenna, so that the data symbol vector $\mathit{S}$ is represented as

$$\mathit{S}={[s_1,\cdots ,s_{N_u}]}^{\mathrm{T}},$$

(2)

where signal power is normalized to $\mathbb{E}\left(\right|s_u{|}^2)=1$, and $s_u$ is the data symbol from the BS to the u-th user and $\mathbb{E}$ means the ensemble average. Multiplying the analog beamformer weight matrix ${\mathit{F}}_{\mathit{rf}}$ and the digital beamformer weight matrix ${\mathit{F}}_{\mathit{bb}}$ by the data symbol vector $\mathit{S}$, the transmitted signal vector $\mathit{X}$ is expressed as

$$\mathit{X}={\mathit{F}}_{\mathit{rf}}{\mathit{F}}_{\mathit{bb}}\mathit{S}.$$

(3)

Since the transmitted signal passes through the channel and is received with added noise, the received signal matrix $\mathit{Y}$ becomes

$$\begin{array}{ccc}\hfill \mathit{Y}& =& \mathit{H}\mathit{X}+\mathit{N}\hfill \\ & =& \mathit{H}{\mathit{F}}_{\mathit{rf}}{\mathit{F}}_{\mathit{bb}}\mathit{S}+\mathit{N},\hfill \end{array}$$

(4)

where ${\mathit{F}}_{\mathit{bb}}=[{\mathit{w}}_{\mathbf{1}},\cdots ,{\mathit{w}}_{{\mathit{N}}_{\mathit{u}}}]\in {\mathbb{C}}^{N_{rf}\times N_u}$, $\mathit{N}={[n_1,\cdots ,n_{N_u}]}^{\mathrm{T}}\in {\mathbb{C}}^{N_u\times 1}$ is the additive white Gaussian noise (AWGN) vector, and each element follows $\mathcal{CN}\left(0,{\sigma }^2\right)$. Therefore, the received signal at the k-th user is expressed as

$$\begin{array}{c}\hfill y_k=\left({\mathit{h}}_k{\mathit{F}}_{\mathit{rf}}\right){\mathit{w}}_k{s}_k+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^{N_u}\left({\mathit{h}}_k{\mathit{F}}_{\mathit{rf}}\right){\mathit{w}}_j{s}_j+n_k,\end{array}$$

(5)

where the first term is the desired signal, the second term is the inter-user interference with signal components for other users, the third term is noise. From (5), the SINR at the k-th user is

$${\mathrm{SINR}}_k={\displaystyle \frac{|\left({\mathit{h}}_k{\mathit{F}}_{\mathit{rf}}\right){\mathit{w}}_k{|}^2}{{\sum }_{\begin{array}{c}j=1\\ j\ne k\end{array}}^{N_u}{\left|\left({\mathit{h}}_k{\mathit{F}}_{\mathit{rf}}\right){\mathit{w}}_j\right|}^2+{\sigma }^2}}.$$

(6)

From (6), the sum-rate at the k-th user is

$$\begin{array}{ccc}\hfill {\mathrm{R}}_k& =& {log}_2(1+{\mathrm{SINR}}_k)\hfill \\ & =& {log}_2(1+{\displaystyle \frac{|\left({\mathit{h}}_k{\mathit{F}}_{\mathit{rf}}\right){\mathit{w}}_k{|}^2}{{\sum }_{\begin{array}{c}j=1\\ j\ne k\end{array}}^{N_u}{\left|\left({\mathit{h}}_k{\mathit{F}}_{\mathit{rf}}\right){\mathit{w}}_j\right|}^2+{\sigma }^2}}).\hfill \end{array}$$

(7)

In this paper, we compare the sum of the expression (7) for all users as the sum-rate of each system.

3. Conventional and Proposed Methods

This section describes the differences in the HBF structure between the conventional and proposed methods and the algorithm for grouping users and transmit antennas.

3.1. HBF Structure of the Conventional Method

The HBF structure of the conventional method, when the number of transmit antennas is 4 and the number of users is 4, is shown in Figure 2. In the HBF structure of the conventional method, the signals of each RF chain are connected to all antennas through phase shifters, and all transmit antennas are used to form a beam to each user. The number of RF chains is limited by the number of data streams to be transmitted as a lower limit and the number of transmit antennas as an upper limit. As the number of RF chains increases, the signals connected to each individual RF chain can be processed separately in the digital domain, which allows for greater flexibility in beamforming, but it also creates the problem of more complex digital signal processing circuits [9,17]. This paper compares the conventional and proposed methods in a system where the number of RF chains equals the number of data streams and users. Here, in the HBF structure, when the number of RF chains is $N_{rf}$ and the number of transmit antennas is $N_t$ in Figure 2, the number of phase shifters to be operated on the transmit side in the conventional method is $N_{rf}{N}_t$. When the distribution of channel conditions between each transmitting antenna and the user is sparse in the structure shown in Figure 2, the signal in the RF chain is connected to the antenna with poor channel conditions, resulting in a low power of the desired signal being received [19,20]. In the conventional method, the analog beamformer weight matrix ${\mathit{F}}_{\mathit{rf}}$ is the phase component of each element of the complex conjugate transpose of the channel matrix $\mathit{H}$, where the analog beamformer weight matrix ${\mathit{F}}_{\mathit{rf}}$ is expressed as

$$\begin{array}{cc}\hfill {\mathit{F}}_{\mathit{rf}}& ={\displaystyle \frac{1}{\sqrt{N_t}}}\left[\begin{array}{cccc}{e}^{-j{\varphi }_{1,1}}& e^{-j{\varphi }_{2,1}}& e^{-j{\varphi }_{3,1}}& e^{-j{\varphi }_{4,1}}\\ e^{-j{\varphi }_{1,2}}& e^{-j{\varphi }_{2,2}}& e^{-j{\varphi }_{3,2}}& e^{-j{\varphi }_{4,2}}\\ e^{-j{\varphi }_{1,3}}& e^{-j{\varphi }_{2,3}}& e^{-j{\varphi }_{3,3}}& e^{-j{\varphi }_{4,3}}\\ e^{-j{\varphi }_{1,4}}& e^{-j{\varphi }_{2,4}}& e^{-j{\varphi }_{3,4}}& e^{-j{\varphi }_{4,4}}\end{array}\right]\in {\mathbb{C}}^{N_t\times N_{rf}},\hfill \end{array}$$

(8)

where ${\varphi }_{i,m}$ denotes the phase component of $h_{i,m}$. Using the MRT technique for the equivalent channel $\left(\mathit{H}{\mathit{F}}_{\mathit{rf}}\right)$, the unnormalized digital beamformer weight matrix $\mathit{D}$ is

$$\mathit{D}={\left(\mathit{H}{\mathit{F}}_{\mathit{rf}}\right)}^{\mathrm{H}},$$

(9)

where $\mathit{D}=[{\mathit{d}}_{\mathbf{1}},\cdots ,{\mathit{d}}_{{\mathit{N}}_{\mathit{u}}}]\in {\mathbb{C}}^{N_{rf}\times N_u}$. When the total transmit power is 1, the weight of the normalized digital beamformer ${\mathit{F}}_{\mathit{bb}}$ at the k-th user is [18]

$${\mathit{w}}_k=\sqrt{{\displaystyle \frac{1}{N_u}}}{\displaystyle \frac{{\mathit{d}}_{\mathit{k}}}{\parallel {\mathit{F}}_{\mathit{rf}}{\mathit{d}}_{\mathit{k}}{\parallel }_F}}.$$

(10)

From (8)–(10), the coefficient of the first term in (5) is

$$\begin{array}{cc}\hfill G_k& =\sqrt{{\displaystyle \frac{1}{N_u{N}_t}}}{\displaystyle \frac{1}{\parallel {\mathit{F}}_{\mathit{rf}}{\mathit{d}}_{\mathit{k}}{\parallel }_F}}({\left({\mathit{h}}_{\mathit{k}}{\mathit{f}}_{\mathit{k}}\right)}^2+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^{N_{rf}}{|{\mathit{h}}_{\mathit{k}}{\mathit{f}}_{\mathit{j}}|}^2)\hfill \\ \hfill & =\sqrt{{\displaystyle \frac{1}{N_u{N}_t}}}{\displaystyle \frac{1}{\parallel {\mathit{F}}_{\mathit{rf}}{\mathit{d}}_{\mathit{k}}{\parallel }_F}}\hfill \\ \hfill & ·\left({\left(\sum _{\begin{array}{c}i=1\end{array}}^{N_t}{h}_{k,i}{f}_{i,k}\right)}^2+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^{N_{rf}}{\left|\sum _{\begin{array}{c}l=1\end{array}}^{N_t}{h}_{k,l}{f}_{l,j}\right|}^2\right),\hfill \end{array}$$

(11)

where ${\mathit{F}}_{\mathit{rf}}=[{\mathit{f}}_{\mathbf{1}},\cdots ,{\mathit{f}}_{{\mathit{N}}_{\mathit{rf}}}]\in {\mathbb{C}}^{N_t\times N_{rf}}$ with ${\mathit{f}}_k={[f_{1,k},f_{2,k},\cdots ,f_{N_t,k}]}^{\mathrm{T}}\in {\mathbb{C}}^{N_t\times 1}$. In (11), the first term $\left(h_{k,i}{f}_{i,k}\right)$ is the inner product of conjugate quantity, while the second term $\left(h_{k,l}{f}_{l,j}\right)$ is not, that is, while the first term is real and analytically larger than the second term.

3.2. HBF Structure of the Proposed Method

The HBF structure of the proposed method, when the number of transmit antennas is 4 and the number of users is 4, is shown in Figure 3. In the HBF structure of the proposed method, the signal of each RF chain is connected through phase shifters to only the group of antennas with the highest gain for each user, forming a beam to each user. The antenna to which the signal of each RF chain is connected is determined by considering the gain of the antenna obtained from the estimated channel. Here, in the HBF structure, when the number of RF chains is $N_{rf}$ and the number of transmit antennas is $N_t$ in Figure 3, the number of phase shifters to be operated on the transmit side in the proposed method is $\left(N_{rf}{N}_t\right)/2$. This means that in Figure 2 and Figure 3, the number of phase shifters operated in the proposed method can be half the number operated in the conventional method. Compared to the conventional method, the proposed method is expected to improve power efficiency because the number of antennas connected to one RF chain is reduced, and the number of phase shifters used is also reduced. In the structure shown in Figure 3, when the distribution of the channel state between each transmitting antenna and user is sparse, the proposed method groups the transmit antennas and users to determine the transmit antenna to connect the RF chain signal according to the channel state; as a result, the transmit power is not allocated to the antennas with low gain and the power of the desired signal being received is expected to be larger than that of the conventional method. In the proposed method, the analog beamformer weight matrix ${\mathit{F}}_{\mathit{rf}}$ is the phase component of each element of the complex conjugate transpose of the channel matrix $\mathit{H}$, where the analog beamformer weight matrix ${\mathit{F}}_{\mathit{rf}}$ is expressed as

$$\begin{array}{cc}\hfill {\mathit{F}}_{\mathit{rf}}& ={\displaystyle \frac{1}{\sqrt{N_c}}}\left[\begin{array}{cccc}{e}^{-j{\varphi }_{1,1}}& 0& e^{-j{\varphi }_{3,1}}& 0\\ 0& e^{-j{\varphi }_{2,2}}& 0& e^{-j{\varphi }_{4,2}}\\ e^{-j{\varphi }_{1,3}}& 0& e^{-j{\varphi }_{3,3}}& 0\\ 0& e^{-j{\varphi }_{2,4}}& 0& e^{-j{\varphi }_{4,4}}\end{array}\right]\in {\mathbb{C}}^{N_t\times N_{rf}},\hfill \end{array}$$

(12)

where $N_c$ is the number of antennas to which the signal of one RF chain is connected in the proposed method. In (12), the first and third RF chains are connected to the first and third transmit antennas, and the second and fourth RF chains are connected to the second and fourth transmit antennas in the structure. As in (9), using the MRT technique for the equivalent channel $\left(\mathit{H}{\mathit{F}}_{\mathit{rf}}\right)$, the coefficient of the first term in (5) is

$$\begin{array}{cc}\hfill G_k& =\sqrt{{\displaystyle \frac{1}{N_u{N}_c}}}{\displaystyle \frac{1}{\parallel {\mathit{F}}_{\mathit{rf}}{\mathit{d}}_{\mathit{k}}{\parallel }_F}}({\left({\mathit{h}}_{\mathit{k}}{\mathit{f}}_{\mathit{k}}\right)}^2+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^{N_{rf}}{|{\mathit{h}}_{\mathit{k}}{\mathit{f}}_{\mathit{j}}|}^2)\hfill \\ \hfill & =\sqrt{{\displaystyle \frac{1}{N_u{N}_c}}}{\displaystyle \frac{1}{\parallel {\mathit{F}}_{\mathit{rf}}{\mathit{d}}_{\mathit{k}}{\parallel }_F}}\hfill \\ \hfill & ·\left({\left(\sum _{\begin{array}{c}i=1\end{array}}^{N_t}{h}_{k,i}{f}_{i,k}\right)}^2+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^{N_{rf}}{\left|\sum _{\begin{array}{c}l=1\end{array}}^{N_t}{h}_{k,l}{f}_{l,j}\right|}^2\right).\hfill \end{array}$$

(13)

In (13), as in (11), the first term $\left(h_{k,i}{f}_{i,k}\right)$ is the inner product of conjugate quantity, while the second term $\left(h_{k,l}{f}_{l,j}\right)$ is not. Therefore, we propose a combination such that $\left({\mathit{h}}_{\mathit{k}}{\mathit{f}}_{\mathit{k}}\right)$ for each user is large in the grouping of users and antennas in the proposed method.

3.3. Grouping of Users and Transmitting Antennas in the Proposed Method

The flowchart that divides the transmit antennas and users into multiple groups considering the antenna gains obtained from the channel estimation is shown in Figure 4. First, from the estimated channel matrix, the matrix ${\mathit{G}}_{\tilde{\mathbf{1}}}$ of the sum of the gains for all antennas for each user when transmitting using all antennas is calculated. The gain ${\mathit{G}}_{\tilde{\mathbf{1}}}$ for the k-th user is calculated as

$$\begin{array}{c}\hfill {g_{\tilde{1}}}_{\left(k\right)}=\sum _{\begin{array}{c}i=1\end{array}}^{N_t}\left|h_{k,i}\right|,\end{array}$$

(14)

where ${\mathit{G}}_{\tilde{\mathbf{1}}}=[{g_{\tilde{1}}}_{\left(1\right)},{g_{\tilde{1}}}_{\left(2\right)},\cdots ,{g_{\tilde{1}}}_{\left(N_u\right)}]\in {\mathbb{C}}^{1\times N_u}$. Also, the matrix ${\mathit{G}}_{\tilde{\mathbf{2}}}$ of gain for each user in all combinations that divide all antennas into $N_t/2$ antennas is calculated. Each gain for the k-th user is calculated as

$$\begin{array}{c}\hfill {g_{\tilde{2}}}_{(k,l)}=|h_{k,m}|+|h_{k,n}|,\end{array}$$

(15)

where ${\mathit{G}}_{\tilde{\mathbf{2}}}={[{{\mathit{g}}_{\tilde{\mathbf{2}}}}_{\left(1\right)},{{\mathit{g}}_{\tilde{\mathbf{2}}}}_{\left(2\right)},\cdots ,{{\mathit{g}}_{\tilde{\mathbf{2}}}}_{\left(N_u\right)}]}^{\mathrm{T}}\in {\mathbb{C}}^{N_u\times N_{com}}$ with ${{\mathit{g}}_{\tilde{\mathbf{2}}}}_{\left(k\right)}=[{g_{\tilde{2}}}_{(k,1)},{g_{\tilde{2}}}_{(k,2)},\cdots ,{g_{\tilde{2}}}_{(k,N_{com})}]\in {\mathbb{C}}^{N_u\times N_{com}}$. $N_{com}$ is the number of all combinations that divide the transmitting antenna into $N_t/2$. (15) denotes the gain at the combination of the m-th and n-th transmit antennas and the gain of the l-th combination. Next, we select the combination that maximizes the gain ${\mathit{G}}_{\tilde{\mathbf{2}}}$ of the user with the smallest gain ${\mathit{G}}_{\tilde{\mathbf{1}}}$. The other group is the combination of the antennas that has not been selected. Finally, we determine the group of antennas with the larger gain in ${\mathit{G}}_{\tilde{\mathbf{2}}}$ of the two groups of antenna combinations, starting from the user with the second-smallest gain ${\mathit{G}}_{\tilde{\mathbf{1}}}$, in decreasing order. In this case, the number of users in each group should be $N_u/2$.

4. Comparison

In this section, we compare the performance of the conventional and the proposed methods.

4.1. Simulation Parameters

The simulation parameters are listed in

Table 1. Simulation parameters.

Parameters	Values
Number of BS antennas $N_t$	4
Number of users $N_u$	4
Number of antennas per users $N_u$	1
Channel model $\mathit{H}$	Rayleigh flat fading

. From (14) and (15), the proposed method uses each element of the channel matrix $\mathit{H}$ in grouping users and transmitting antennas. Therefore, in this paper, the channel matrix is assumed to be Rayleigh flat fading in order to compare the proposed method with the conventional method in the system model where each element of the channel matrix is independent and identically distributed (i.i.d.). In addition, in order to eliminate the difference in sum-rate performance of each user due to noise distribution in (7), the noise configuration is assumed to be a Gaussian distribution.

The analog beamformer weight matrix ${\mathit{F}}_{\mathit{rf}}$ is the phase components of each element of the complex conjugate transpose of the channel matrix $\mathit{H}$, and the digital beamformer weight matrix ${\mathit{F}}_{\mathit{bb}}$ is the complex conjugate transpose of the equivalent channel $\left(\mathit{H}{\mathit{F}}_{\mathit{rf}}\right)$. Simulations are compared in two cases. #1 (case 1) is the case where $N_t$ transmit antennas are used without antenna selection. #2 (case 2) is the case where $N_t$ transmit antennas are used by selecting the top $N_t$ total gain ${\mathit{G}}_{\tilde{\mathbf{3}}}$ from $2N_t$ antennas. The matrix ${\mathit{G}}_{\tilde{\mathbf{3}}}$ is the sum of the gains of each antenna for all users. The gain ${\mathit{G}}_{\tilde{\mathbf{3}}}$ for the k-th antenna is calculated as

$$\begin{array}{c}\hfill {g_{\tilde{3}}}_{\left(k\right)}=\sum _{\begin{array}{c}i=1\end{array}}^{N_u}\left|h_{i,k}\right|,\end{array}$$

(16)

where ${\mathit{G}}_{\tilde{\mathbf{3}}}=[{g_{\tilde{3}}}_{\left(1\right)},{g_{\tilde{3}}}_{\left(2\right)},\cdots ,{g_{\tilde{3}}}_{\left(N_t\right)}]\in {\mathbb{C}}^{1\times 2N_t}$. In case 2, the antenna selection reduces the number of antennas used for transmission, thus reducing power consumption during transmission and enabling more efficient radio transmission compared to the case with $2N_t$ transmit antennas. In addition, by selecting antennas with good channel conditions, the quality of the received signal is improved [22].

4.2. Performance Comparison

Figure 5 shows the sum-rate characteristics of the proposed method and the conventional method when the number of transmit antennas is 4 and the number of users is 4. Comparing the conventional and the proposed method, there is almost no difference at low SNR because the noise power is dominant, but at high SNR, the sum-rate of the proposed method becomes larger than that of the conventional method. In particular, the sum-rate of the proposed method improves by about 14.5% for case 1 and by about 15.2% for case 2 when the SNR is 30 dB compared to the conventional method. Comparing case 1 and case 2, the sum-rate of case 2 is larger for both conventional and proposed methods. Thus, antenna selection is also considered effective in the proposed method.

Figure 6 and Figure 7 show the sum-rate when the number of users is set to 4 and the number of transmit antennas is increased. Figure 6 shows the case without antenna selection, and Figure 7 shows the case with antenna selection. Figure 6 shows that the sum-rate of the proposed method has the same performance as the conventional method when $N_t$ is 8 with SNR of 0 dB, and when $N_t$ is 20 with SNR of 30 dB. Similarly, Figure 7 shows that the sum-rate of the proposed method has the same performance as the conventional method when $N_t$ is 8 with SNR of 0 dB, and when $N_t$ is 22 with SNR of 30 dB. This means that when the number of transmit antennas is small, the number of phase shifters can be reduced while maintaining better performance compared to the conventional method. However, as the number of transmit antennas increases, the grouping algorithm will not optimize the transmit antennas for users who are late in the order of determining the group of antennas, resulting in gradually inferior performance compared to the conventional method. Also, Figure 6 and Figure 7 show that the difference between the sum-rate performance with antenna selection and without antenna selection increases as the number of transmit antennas increases for both the conventional and proposed methods. This is because the distribution of the gain of each transmit antenna becomes more diverse when the number of transmit antennas increases, and therefore, the antenna with higher gain can be selected for each user.

Figure 7 shows the sum-rate of the proposed method and the conventional method with antenna selection from $2N_t$ antennas, whereas Figure 8, Figure 9 and Figure 10 show the sum-rate of the proposed method and the conventional method with antenna selection from 24, 100, and 1000 antennas. In Figure 6 and Figure 7, as the number of antennas increases, the sum-rate of the proposed method becomes inferior to that of the conventional method, and when the number of transmit antennas is 24 and SNR is 0 dB, the proposed method is the most inferior to the conventional method. On the other hand, in Figure 8, Figure 9 and Figure 10, when the number of transmit antennas is 24 and SNR is 0 dB, the proposed method is 9.72% lower than the conventional method when antenna selection is made from 24 antennas, 7.13% lower than the conventional method when antenna selection is made from 100 antennas, 6.26% lower than the conventional method when antenna selection is made from 1000 antennas. These results show that, in the case with antenna selection, the sum-rate of the proposed method can be maintained to some extent compared to the conventional method even when the number of transmit antennas increases as the number of antennas for antenna selection increases. In Figure 6 and Figure 7, as the number of transmit antennas increased, there was a problem, that is, transmit antennas were not optimized for users who were slow in selecting groups of antennas in order. On the other hand, in Figure 8, Figure 9 and Figure 10, as the number of antennas used for antenna selection increases, the distribution of the gain of each antenna becomes more diverse, and the average gain of all the transmit antennas used can be higher. Therefore, a user who selects a group of antennas in late order can select a group of antennas with higher gain, and the proposed method can maintain the performance of the conventional method even when the number of transmit antennas increases. These results suggest that the proposed method will be more effective when the number of antennas used for antenna selection is large.

5. Conclusions

The purpose of this paper is to reduce the number of phase shifters while maintaining sum-rate performance compared to the conventional HBF scheme, which has the problem that the transmit power after the digital beamformer is allocated to the antenna with small gain when the beam is formed at the transmitter side. Therefore, we proposed a structure of HBF with grouping of users and antennas considering the gain of the antennas. Simulation results show that the proposed method improves the sum-rate compared to the conventional method when the number of transmit antennas and users are the same, indicating the effectiveness of the proposed method. It was also shown that there is a trade-off between the reduction in the number of phase shifters used to form the beam and the improvement in performance as the number of transmit antennas increases. Furthermore, when antenna selection is used to select antennas with good channel conditions as transmit antennas, the number of phase shifters can be reduced while maintaining performance even when the number of antennas increases, although there is a tradeoff between the number of phase shifters and performance improvement, indicating the effectiveness of the proposed method.