1. Introduction
Symmetry concept is utilized in almost all communications systems and cellular network analyses, which is then transformed into various conclusive outcomes. As the main direction of the development of information, communication in the fifth-generation (5G) will penetrate into various fields of future society. It aims to provide users with higher data rates and multiple connections for many Internet-of-Things (IoT) devices. Compared with fourth-generation (4G), the spectrum efficiency of 5G communication is increased by 5–15 times, and energy efficiency and cost efficiency are improved by a hundred times [
1]. Massive multiple-input multiple-output (MIMO) technology and millimeter wave (mmWave) communication technology plays an important role as the key technology of 5G. Massive MIMO technology uses multiple antenna units to transmit and receive transmission signals at both ends of the communication system [
2,
3,
4,
5,
6]. Through spatial multiplexing, beamforming, precoding, and other technologies, the space resources of the wireless channel are fully utilized to improve the system Stability, reducing the system’s bit error rate, simplifying signal processing and reducing system latency. Currently, 6, 15, 18, 28, 45, 60, and 72 GHz are typical candidate bands for 5G research in the industry. The 6–100 GHz frequency range is a typical millimeter-wave band [
7,
8]. Communication within this range can provide higher capacity for hotspot users and support user transmission rates above 10 GB/s [
9,
10,
11]. The millimeter-wave mainly propagates in space in the form of direct waves. The beam is narrow and has good directivity, but it is greatly affected by the environment, such as rainfall, sand, dust, etc., and the propagation distance is limited [
12,
13]. Since the millimeter-wave belongs to a very high frequency, the wavelength is small, and a large number of antennas can be packaged in a limited physical space. The combination of the array antenna and the precoding technology can effectively compensate for the transmission loss of the millimeter-wave, increase the transmission distance, and improve the stability of the system [
14].
The use of precoding techniques in massive MIMO systems can improve the spectral efficiency of the system, reduce the bit error rate, and simplify the complexity of the receiver. In the 4G communication system, traditional digital precoding technology has been widely used. In order to achieve the maximum theoretical advantage of massive MIMO technology [
15], traditional digital precoding technology needs to equip each transmitting antenna with a separate Radio Frequency (RF) link. However, in a millimeter-wave massive MIMO system, due to a large number of transmit antennas, if the traditional digital precoding technology is still used, then the system hardware cost will be too high and the system power consumption will be too large. Therefore, in millimeter-wave massive MIMO systems, hybrid precoding techniques are generally used, that is, precoding processing is divided into baseband precoding (digital precoding) processing and radio frequency precoding (analog precoding) processing [
16]. The solution uses only a small number of RF links to solve the high cost and high-power consumption of traditional digital precoding.
Hybrid precoding is used to process the multiplexed signal through the baseband precoder, send it to the RF precoder through the RF link for constant mode phase shift, and then send it to the transmitting antenna for transmission. The hybrid precoding architecture of the millimeter-wave massive MIMO system can be divided into a shared array architecture and a separate sub-array architecture [
17], in which each RF link of the shared array architecture is connected to all the transmit antennas, and the separated sub-arrays for each RF link of the architecture is only connected to a portion of the transmit antenna. The shared array architecture has complex hardware design and high-power consumption compared to the separate sub-array architecture, but it can fully exert the precoding performance and achieve better results. Therefore, the research in this paper is based on a shared array architecture.
In the hybrid precoding study of millimeter-wave massive MIMO systems, the literature [
18] uses the transmitter and receiver to jointly design the analog beamforming vectors under the multi-resolution codebook. Literature [
19] introduces beam space multiple-input multiple-output, and uses a discrete Fourier transform (DFT) beamforming vector to direct the transmitted signal to a subspace with progressively maximized received signal power. Literature [
20] converts the hybrid precoding problem into a sparse reconstruction problem based on the sparsity of the millimeter-wave channel. Under the condition of perfect known channel information, a low complexity hybrid precoding algorithm is proposed by using the concept of orthogonal matching pursuit. However, the design of RF precoding in [
20] requires that the candidate matrix be selected from the candidate matrix to multiply the largest product by the inner product of the residual, and the construction of the candidate matrix requires high-precision channel estimation. Upon completion, it is necessary to estimate the signal emission angle and angle of arrival, which will cause system delay and consume system resources. At the same time, the algorithm uses the least-squares method to perform matrix inversion when updating the baseband precoding. The complexity of matrix inversion increases with the number of base vectors. High dimensional matrix inversion will lead to longer calculation delay and higher power consumption. The authors in [
21] proposed a semidefinite relaxation algorithm by utilizing the idea of alternating minimization, which can provide substantial performance gains for phase shifters (PSs). However, while the PSs are studied substantially, there still exist an inevitable gap compared with the performance of PSs. The authors in [
22] proposed an analog PSs and spatial multiplexing by utilizing multiple RF chains connected to a fixed subset of antenna elements. This algorithm is based on analog beamforming and therefore, it lacks the ability to provide the required precoding performance for mmWave massive MIMO systems. In [
23], for single-stream single-user MIMO-OFDM systems, a hybrid precoding is proposed to maximize the received signal strength.
Therefore, in view of the problems in [
20,
21,
22], this proposed an efficient algorithm for effective precoding in mmWave massive MIMO systems.
The novel contributions of this paper are as follows:
This study deploys bird swarm algorithm (BSA) to eliminate the need for the known candidate matrices in mmWave channel estimation as in the traditional algorithms.
The proposed algorithm uses the characteristics of BSA with global search optimal (GSO) value to search for the largest array response vector multiplied by the residual matrix, and uses the Banachiewicz–Schur (BS) block matrix generalized inverse to transform the high-dimensional matrix into a low-dimensional matrix, avoiding matrix inversion and reducing the amount of calculation.
It uses the results of each iteration to avoid matrix inversion and to simplify the computational complexity of the system.
The rest of the paper is organized as follows.
Section 2 described the system model.
Section 3 explains the proposed algorithm.
Section 4 gives the numerical simulation analysis while
Section 5 concludes the paper.
2. System Model
The shared array architecture of the mmWave massive MIMO system is shown in
Figure 1. The number of transmitting antennas at the transmitting end is
${N}_{t}$, the number of receiving antennas at the receiving end is
${N}_{r}$, the number of RF links at the transmitting end is
${N}_{t}^{RF}$, the number of RF links at the receiving end is
${N}_{r}^{RF}$, and the data streams of the transmitting end and the receiving end are both
${N}_{s}$.
In order to ensure multi-stream transmission, it is necessary to satisfy
${N}_{s}\le {N}_{t}^{RF}\le {N}_{t}$ and
${N}_{s}\le {N}_{r}^{RF}\le {N}_{r}$. Under this hardware structure, the signal is transmitted to the channel
$\mathit{H}\in {\u2102}^{{N}_{r}\times {N}_{t}}$ by the processing of the baseband precoder
${\mathit{F}}_{\mathrm{BB}}\in {\u2102}^{{N}_{t}^{RF}\times {N}_{s}}$ and the RF precoder
${\mathit{F}}_{\mathrm{RF}}\in {\u2102}^{{N}_{t}\times {N}_{t}^{RF}},$ and the transmitting signal of the transmitting end is:
where
$\mathit{S}={\left[{s}_{1},{s}_{2},\dots ,{s}_{{N}_{s}}\right]}^{T}$ is the data stream of the signal, and
$E\left[\mathit{S}{\mathit{S}}^{\mathit{H}}\right]=\frac{1}{{N}_{s}}{\mathit{I}}_{{N}_{s}}$;
$\mathit{x}={\left[{x}_{1},{x}_{2},\dots ,{x}_{{N}_{t}}\right]}^{T}$ is the transmitting signal for the transmitting end.
The received signal arriving at the receiving end antenna after channel transmission is:
where
$\mathit{n}\in {\u2102}^{{N}_{r}}$ represents a Gaussian white noise with zero mean and a covariance matrix of
${\sigma}^{2}{\mathit{I}}_{{N}_{r}}$,
$\mathit{y}={\left[{y}_{1},{y}_{2},\dots ,{y}_{{N}_{r}}\right]}^{T}$ is the signal received by the receiving antenna,
$\rho $ is the average received power,
$\mathit{H}$ is the channel matrix, and
$E\left[{\parallel \mathit{H}\parallel}_{\mathbf{F}}^{\mathbf{2}}\right]={N}_{t}{N}_{r}$.
The received signal is further processed by the RF combiner
${\mathit{W}}_{\mathrm{RF}}\in {\u2102}^{{N}_{r}\times {N}_{r}^{RF}}$ and the baseband combiner
${\mathit{W}}_{\mathrm{BB}}\in {\u2102}^{{N}_{r}^{RF}\times {N}_{s}}$, and the received signal received by the receiver is:
where
$\widehat{\mathit{y}}={\left[{\widehat{y}}_{1},{\widehat{y}}_{2},\dots ,{\widehat{y}}_{{N}_{s}}\right]}^{T}$ is the signal received at the receiving end.
${\mathit{F}}_{\mathrm{RF}}$ is implemented using the analog network so that it satisfies the elements
${\left({\mathit{F}}_{\mathrm{RF}}^{\left(i\right)}{\mathit{F}}_{\mathrm{RF}}^{\left(i\right)H}\right)}_{l,l}=\frac{1}{{N}_{t}}$, where
${(\xb7)}_{l,l}$ represent the
$l\mathrm{th}$ diagonal element of the matrix. Similarly,
${\left({\mathit{W}}_{\mathrm{RF}}^{\left(i\right)}{\mathit{W}}_{\mathrm{RF}}^{\left(i\right)H}\right)}_{l,l}=\frac{1}{{N}_{r}}$. The total power constraint at the transmitting end is
${\parallel {\mathit{F}}_{\mathrm{RF}}{\mathit{F}}_{\mathrm{BB}}\parallel =}_{\mathbf{F}}^{\mathbf{2}}={N}_{s}$.
Considering the high path loss of the mmWave channel, the sparse distribution in space, the close arrangement of the antenna array on the transceiver in the massive MIMO system, and the high correlation of the antenna elements, the traditional fading statistical channel model is not applicable. Therefore, the ray-tracing model is usually used for modeling. If the mmWave channel contains
${N}_{\mathrm{cl}}$ scattering clusters, with each cluster containing
${N}_{\mathrm{ray}}$ strip propagation paths, the channel
$\mathit{H}$ of the system can be described as:
where
${\mathit{\alpha}}_{i,l}$ represents the gain factor of the
$l\mathrm{th}$ propagation path in the
$i\mathrm{th}$ scattering cluster that follows a complex Gaussian distribution with zero mean and a variance of
${\sigma}_{{\alpha}_{i}}^{2}$, and it satisfies
${\sum}_{i=1}^{{N}_{\mathrm{cl}}}{\sigma}_{{\alpha}_{i}}^{2}=\gamma $, and
$\gamma $ must satisfy
$E\left[{\parallel \mathit{H}\parallel}_{\mathbf{F}}^{\mathbf{2}}\right]={N}_{t}{N}_{r}$ with
${\phi}_{il}^{r}$ as the angle of arrival (AoA). For the
$i\mathrm{th}$ scattering cluster,
${\phi}_{il}^{r}$ is randomly distributed on
${\phi}_{i}^{r}$ with
${\phi}_{il}^{t}$ as the angle of departure (AoD), and for the
$i\mathrm{th}$ scattering cluster,
${\phi}_{il}^{t}$ is randomly distributed on
${\phi}_{i}^{t}$. Generally, we choose the Laplacian distribution as a random distribution;
${\mathit{\alpha}}_{t}\left({\phi}_{il}^{t}\right)$ and
${\mathit{\alpha}}_{r}\left({\phi}_{il}^{r}\right)$ represent the array response vectors of the transmitter and receiver, respectively.
The types of antenna arrays can be combined into various configurations depending on the arrangement of the antenna in the array. In mmWave massive MIMO systems, a uniform antenna array is generally selected to design the antennas at both ends of the transceiver. Common uniform antenna arrays have a uniform linear array (ULA) and uniform planar array (UPA). In a ULA, either the elevation or the azimuth perspective is considered, since it is a one-dimensional antenna array. The UPA is a two-dimensional antenna array, and this type of antenna arrays is preferable for mmWaves since it can accommodate more antenna elements within a small area at both the user equipment and the base station (BS). It also facilitates beamforming in an extra dimension, which results in the 3D-beamforming. For the convenience of analysis, a ULA is used in this paper. For a ULA, assuming that there are
$N$ antennas on the y-axis, the array response vector can be expressed as:
where
$\phi \in \left[0,2\pi \right]$,
$k=\frac{2\pi}{\lambda}$,
$\lambda $ is the wavelength of the signal, and
$d$ is the spacing between antenna elements. In an actual system, channel state information (CSI) can be known by channel estimation. In order to focus only on precoding research, assuming that the transceiver knows the CSI, the spectral efficiency of the system is:
where
${\mathit{R}}_{\mathrm{n}}={\sigma}_{n}^{2}{\mathit{W}}_{\mathrm{BB}}^{H}{\mathit{W}}_{\mathrm{RF}}^{H}{\mathit{W}}_{\mathrm{RF}}{\mathit{W}}_{\mathrm{BB}}$ is the noise covariance matrix processed by the receiver and
${\mathit{I}}_{{N}_{s}}$ is the identity matrix of the noise. Reference [
9] approximates the spectral efficiency to minimize the Euclidian distance of the hybrid precoding matrix and the all-digital precoding matrix, whereby the precoding design problem can be written as:
where
${\mathit{F}}_{\mathrm{opt}}$ is an all-digital precoding matrix, which is the first
${N}_{\mathbf{s}}$ column of the right singular matrix of the channel matrix
$\mathit{H}$. Then, this precoding design problem can be expressed as finding the projection of
${\mathit{F}}_{\mathrm{opt}}$ on the subspace formed by the hybrid precoder
${\mathit{F}}_{\mathrm{RF}}{\mathit{F}}_{\mathrm{BB}}$ set under the condition of
${\mathit{F}}_{\mathrm{RF}}\in {F}_{\mathrm{RF}}$. Similarly, the design method of the combiner at the receiving end is similar.
4. Numerical Simulation Analysis
In order to verify the performance of the proposed hybrid precoding Algorithm 1, this section gives the simulation results of full-digital precoding, analog precoding, and OMP-based hybrid precoding, and compares them with the proposed Algorithm 1 in mmWave massive MIMO systems. The simulation parameters are shown in
Table 2. We used MATLAB R2017a for simulations, while the results are averaged over 1000 random channel implementations.
Figure 2 shows the difference in spectral efficiency (SE) with SNR for different precoding algorithms with
${N}_{t}=64$ antennas at the transmitter,
${N}_{r}=16$ antennas at the receiver, and
${N}_{\mathrm{RF}}^{\mathrm{t}}={N}_{\mathrm{RF}}^{\mathrm{r}}={N}_{s}=\left[2,3\right]$. It can be seen from
Figure 2a,b that as the SNR increases, the spectral efficiency of the different precoding algorithms in improved to different degrees, and as the data streams increase, the spectral efficiency of different precoding will be improved to varying degrees. For digital data streams, the full-digital precoding algorithm performs best because it is optimal precoding, and all precoding is aimed at approaching it. The proposed BSAMIBOMP algorithm performs better than the traditional OMP-based hybrid precoding algorithm as it eliminates the matrix inversion operation and candidate matrix requirement. Also, the proposed algorithm outperforms traditional hybrid OMP precoding as it utilizes the BSA algorithm to search for the global optimal solution, while the traditional OMP-based precoding algorithm uses the candidate matrix to select the column with the highest correlation. However, there is an interval between the angles of each column in the candidate matrix, so the selected array response vector is not necessarily the global optimal solution. The analog precoding performance is the worst because the analog precoding is constant modulus; only the phase characteristics are utilized, and the amplitude characteristics are not utilized. Therefore, the proposed algorithm can achieve better results under
${N}_{\mathrm{RF}}^{\mathrm{t}}={N}_{\mathrm{RF}}^{\mathrm{r}}={N}_{s}$.
Figure 3a,b shows the spectral efficiency versus SNR for different precoding algorithms with
${N}_{t}=64$ antennas at the transmitter,
${N}_{r}=16$ antennas at the receiver,
${N}_{\mathrm{RF}}^{\mathrm{t}}={N}_{\mathrm{RF}}^{\mathrm{r}}=4$, and
${N}_{s}=3$ and 4, respectively. As can be seen from
Figure 3a,b, as the SNR and data streams increase, the spectral efficiency of different algorithms improves to different degrees. Similarly, for all data streams, the performance of the fully-digital precoding algorithm is the best. The performance of the proposed BSAMIBOMP algorithm is close to the full-digital algorithm and better than the traditional OMP-based hybrid precoding algorithm. The analog precoding performance is the worst of all precoding schemes. Comparing
Figure 2 and
Figure 3, it can be found that the effect is better when
${N}_{\mathrm{RF}}^{\mathrm{t}}={N}_{\mathrm{RF}}^{\mathrm{r}}\ge {N}_{s}$, because in this case, the dimensions of baseband precoding
${F}_{\mathrm{BB}}$ and RF precoding
${F}_{\mathrm{RF}}$ are higher, and the precoding matrix contains more information, so the effect is better. Therefore, the proposed algorithm can achieve better results for
${N}_{\mathrm{RF}}^{\mathrm{t}}={N}_{\mathrm{RF}}^{\mathrm{r}}\ge {N}_{s}$.
Figure 4a,b shows the spectral efficiency with the number of RF links for different precoding algorithms with
${N}_{t}=64$ antennas at the transmitter,
${N}_{r}=16$ antennas at the receiver,
${N}_{s}=1$,3, and
$SNR=0\text{}\mathrm{dB}$. As can be seen from
Figure 4a,b, since full-digital precoding is precoded only at the baseband, the analog precoding is precoded only at the radio frequency, so they are not affected by the change in the number of RF links. With the increase of the number of RF links, the proposed algorithm has better spectral efficiency than the traditional OMP-based hybrid precoding scheme. When
${N}_{s}=1$, the proposed algorithm and the traditional OMP-based hybrid precoding algorithm have approximately similar performances (
Figure 4a). When
${N}_{s}=3$, the proposed algorithm shows a better performance than the traditional OMP-based hybrid precoding algorithm (
Figure 4b). Therefore, the proposed algorithm performance gets better when the number of RF links and the number of data streams are relatively large. However, if the difference is too large, then the meaning of hybrid precoding is lost. So, the number of RF links is generally twice that of the data streams for better performance. Therefore, the proposed algorithm can achieve better results for different number of RF link.
Figure 5 and
Figure 6 show the different BERs with
${N}_{t}=64$ antennas at the transmitter end,
${N}_{r}=16$ antennas at the receiving end,
${N}_{\mathrm{RF}}^{\mathrm{t}}={N}_{\mathrm{RF}}^{\mathrm{r}}=4$, and
${N}_{s}$ at 1 and 3, respectively. As can be seen from
Figure 5 and
Figure 6, the BER of the full-digital precoding, the proposed algorithm, and the traditional OMP-based hybrid precoding algorithms decrease with the increase of SNR, while the analog precoding BER remains unchanged. This is because the analog precoder selects the
${N}_{s}$ column array response vector with the largest channel gain, and its selection is independent of the SNR. Comparing
Figure 5 with
Figure 6, it can be found that when
${N}_{s}=1$, the full-digital precoding algorithm, the proposed BSAMIBOMP, and the traditional OMP-based hybrid precoding algorithm achieved the best BER performance, that is, with no error rate. When
${N}_{s}=3$, the full-digital precoding algorithm can be optimized at 5 dB, while the OMP-based hybrid precoding and the proposed algorithm tend to be stable after 5 dB SNR, and a closer observation can be found in this paper. The proposed algorithm shows better BER performance than the traditional OMP-based hybrid precoding, and also its performance is closer to the full-digital precoding scheme. We also conclude that the proposed algorithm is especially suitable when there is a difference between the number of RF links and the number of data streams. From the above results, it is clear that the proposed algorithm also shows better performance in terms of BER.
Figure 7 compares the computational complexity of the proposed BSAMIBOMP algorithm with another existing algorithm under a different number of RF chains. As can be seen from
Figure 7, when the number of RF chains increases, the number of multiplications and additions required for all the algorithms increases. It is also shown in the results that the proposed algorithm requires a smaller number of multiplications and additions than the OMP precoding [
15] and the SDRAltMin precoding [
16] for the same number of RF chains and operating conditions, which makes effective for mmWave systems. This means that the proposed algorithm requires a lesser amount of energy to operate the systems and also results in reduced hardware complexity. On the other hand, the conventional OMP hybrid precoding scheme requires a greater number of complex multiplications and additions, which makes it unsuitable for mmWave communications systems hardware.
Figure 8 shows the energy efficiency analysis under different number of RF chains for different precoding schemes. As it can be seen, the energy efficiency of all the precoding schemes decreases with an increasing number of RF chains. Moreover, the proposed BSAMIBOMP precoding algorithm shows better energy efficiency than the existing OMP hybrid precoding [
15] and SDRAltMin [
16] precoding under the same number of RF chains and operating conditions, which makes it more effective for mmWave systems. Therefore, the proposed algorithm has a close performance to Fully-Digital precoding and an overall better performance than other competing alternatives.