Computers, Materials & Continua DOI:10.32604/cmc.2021.015421
Article

A Highly Efficient Algorithm for Phased-Array mmWave Massive MIMO Beamforming

Ayman Abdulhadi Althuwayb1, Fazirulhisyam Hashim2, Jiun Terng Liew2, Imran Khan3, Jeong Woo Lee4, Emmanuel Ampoma Affum5, Abdeldjalil Ouahabi*,6,7* and Sébastien Jacques8

1Department of Electrical Engineering, Jouf University, Sakaka, Aljouf, 72388, Kingdom of Saudi Arabia
2Department of Computer and Communication Systems Engineering, Faculty of Engineering, Universiti Putra Malaysia (UPM), Serdang, 43400, Malaysia
3Department of Electrical Engineering, University of Engineering and Technology Peshawar, Pakistan
4School of Electrical and Electronics Engineering, Chung-Ang University, Seoul, 06974, Korea
5Electrical and Electronic Department, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
6UMR 1253, iBrain, Université de Tours, INSERM, Tours, France
7LIMPAF Laboratory, Department of Computer Science, University of Bouira, Bouira, 10000, Algeria
8University of Tours (France), GREMAN UMR 7347, CNRS, INSA Centre Val-de-Loire, Tours, 37100, France
*Corresponding Author: Abdeldjalil Ouahabi. Email: ouahabi@univ-tours.fr
Received: 20 November 2020; Accepted: 30 December 2020

Abstract: With the rapid development of the mobile internet and the internet of things (IoT), the fifth generation (5G) mobile communication system is seeing explosive growth in data traffic. In addition, low-frequency spectrum resources are becoming increasingly scarce and there is now an urgent need to switch to higher frequency bands. Millimeter wave (mmWave) technology has several outstanding features—it is one of the most well-known 5G technologies and has the capacity to fulfil many of the requirements of future wireless networks. Importantly, it has an abundant resource spectrum, which can significantly increase the communication rate of a mobile communication system. As such, it is now considered a key technology for future mobile communications. MmWave communication technology also has a more open network architecture; it can deliver varied services and be applied in many scenarios. By contrast, traditional, all-digital precoding systems have the drawbacks of high computational complexity and higher power consumption. This paper examines the implementation of a new hybrid precoding system that significantly reduces both calculational complexity and energy consumption. The primary idea is to generate several sub-channels with equal gain by dividing the channel by the geometric mean decomposition (GMD). In this process, the objective function of the spectral efficiency is derived, then the basic tracking principle and least square (LS) techniques are deployed to design the proposed hybrid precoding. Simulation results show that the proposed algorithm significantly improves system performance and reduces computational complexity by more than 45% compared to traditional algorithms.

Keywords: 5G; mmWave; phased array; algorithm; antenna beamforming

1  Introduction

As the number of wireless devices continues to grow and wireless applications continue to expand, user demand for wireless network transmission rates continues to increase. The existing low frequency network (<3 GHz) is struggling to meet this increasing demand for speed. Addition, it is oriented towards greater bandwidth. High frequency resources are currently being studied and implemented [1] and the 60 GHz millimeter wave (mmWave) has aroused the interest of a large number of researchers. Many countries have opened unlicensed 60 GHz mmWave frequency bands for research and testing. For example, China’s open frequency band is 59~64 GHz, in the United States the band is 57~64 GHz, and Japan uses 59~66 GHz. Relevant standards have been established to promote the industrialization of 60 GHz mmWave applications [2–6]. Wireless HD (WiHD) is primarily used to achieve high-quality, high-definition, uncompressed video transmission indoors, while the IEEE802.15.3c standard is primarily used for high-quality indoor networks. Wireless personal area network (WPAN) applications and the IEEE802.11.ad standard provide high-quality wireless local area network (WLAN) applications [7].

The application of the 60 GHz mmWave band presents significant challenges. Oxygen attenuation on the ground reaches 15 dB/km as a result of the oxygen attenuation window. In addition, reflection attenuation and other losses occur as part of the propagation process. As such, the link budget is very limited and if we are in a non-line-of-sight (NLOS) environment, the performance of the link will be severely affected [8]. Furthermore, the output power of the power amplifier in the 60 GHz mmWave frequency band is very limited. Therefore, exploitation of the short wavelength region and small antenna arrays in this frequency band, as well as the use of beamforming technology, can help improve link gain. In terms of implementation methods, beamforming technology can be divided into adaptive beamforming and fixed beamforming [9]. Although adaptive beamforming technology performs well, the complexity of its implementation is high. The implementation of codebook-based fixed beamforming technology is not very complex, but unfortunately it is not very efficient. In terms of hardware architecture, digital beamforming technology is generally used. Each antenna is configured with a radio frequency link, which sees high costs and power consumption [10]. MmWave generally adopts radio frequency beamforming technology and realizes the alignment of the transceiver through the radio frequency phase shifter to directly adjust the signal, reducing the cost [11,12]. MmWave beamforming technology has been the subject of numerous studies. The IEEE802.11.ad and IEEE802.15.3c standards both adopt codebook-based beamforming technology as part of the RF phase-shifting architecture and find an optimal beam through hierarchical searching [13–16].

In recent years, mmWave massive multiple-input multiple-output (massive-MIMO) technology has attracted the attention of the scientific community. This is primarily because mmWave communications present an abundant spectrum and hybrid massive-MIMO precoding technology provides a higher beam gain and compensates for the propagation defect of the high loss of mmWave communications [17]. However, at the same time, the scale of the hardware required and the complexity of encoding and decoding are both increasing and a new hybrid precoding scheme is needed to reduce system complexity.

MIMO multiplexing systems typically use singular value decomposition (SVD) to obtain several independent orthogonal spatial sub-channels for higher throughput. However, due to the large differences in the gain of each sub-channel in this method, it is necessary to use different codecs and modulation/demodulation methods to meet the bit error rate (BER) requirements, making the system very complex [18]. For this reason, the authors of [19] propose a scheme based on geometric mean decomposition (GMD). This scheme decomposes the Rayleigh fading channel under orthogonal frequency division multiplexing into several parallel sub-channels with equal channel gains. This helps to avoid complicated bit allocation and power loading processes and reduces the difficulty of system encoding and decoding, reducing overall system complexity [20]. In order to obtain a hybrid precoding scheme with the best frequency efficiency, the authors in [21] transformed the frequency efficiency optimization problem into a sparse approximation problem and optimized it using the orthogonal matching pursuit (OMP) algorithm to downlink the frequency efficiency of a single-user MIMO system [22]. The authors of [23] studied the frequency efficiency of a single-user MIMO system and designed a hybrid precoding scheme, proposing an algorithm to optimize the frequency efficiency of the system. However, they only considered the optimization of the algorithm and did not proceed in terms of the complexity of the coding and decoding of the system. After studying MIMO channel diagonalization, the authors of [24] used the GMD method to efficiently compensate for gain difference defects in SVD-weighted sub-channels. The authors of [25] examined the impact of the differential GMD precoder on the frequency efficiency of the system based on the assumption of a low feedback rate in single-user MIMOs, but the relationship between the frequency efficiency of the system and the number of BS antennas and the signal-to-noise ratio remains unclear and would be difficult to promote in the current communications environment.

Based on the current state of research on the spectral efficiency of the millimeter wave MIMO downlink system, the GMD channel processing method was applied to the processing of Saleh-Valenzuela mmWave channels. Compared to the traditional SVD algorithm, the complexity of the system can be effectively reduced and, for the single-user mobile downlink communication scenario, the hybrid GMD-based precoding scheme is proposed and implemented. This scheme can give good results, assuming that the system complexity can be reduced and the spectral efficiency can be improved.

2  System Modeling

2.1 Hybrid Precoding Model

Fig. 1 shows the proposed modeling of the mmWave Massive-MIMO system. To simplify the analysis, the system only takes into account the downlink transmission scenario of a single cell and a single user. The base station is configured with Nt antennas and NtRF radio frequency chains; the transmission data contains Ns data streams and satisfies the following in equation: Ns≤NtRF≤Nt. The user receiver is configured with Nr antennas and NrRF radio frequency chains to meet the following requirement: Ns≤NrRF≤Nr.

The hybrid precoding, which is performed at the base station, consists of digital precoding, PD∈CNtRF×Ns, and analog precoding, PA∈CNt×NtRF. The precoding of the corresponding base station is defined as follows: P=PAPD. Assuming that the base station sends a signal vector of s, then the hybrid precoding signal is expressed by (1).

x=Ps=PAPDs(1)

where P is the base station transmit power, which satisfies tr{PPH}≤Ns. Consequently, the signal vector received by the user is expressed by (2).

y=ρHPAPDs+n(2)

In the previous equations, (·)H is the complex conjugate transposition operation; ρ is the mean received power; H∈CNr×Nt is the transmission matrix of the mmWave channel; n∈CNr×1 is the noise vector, which obeys the complex Gaussian distribution, i.e., CN(0,σ2INr), where σ2 is the noise variance, INr is the unit dimensional matrix Nr×Nr. The analog precoding PA is implemented by a network of phase shifter, and respects the limit of the constant modulus, i.e., |{PA}i,j| =1/Nt, where {⋅}i,j is the element on the i-th row and j-th column of the matrix; |⋅| is the modulus of the complex numbers.

Figure 1: Proposed system model

2.2 Channel Model

The mmWave channel follows the Saleh Valenzuela channel model, and the channel transfer matrix H can be expressed by (3) [26].

H=NtNrL∑iLβiαr(Φir)αt(Φit)(3)

In (3), L is the number of (resolvable) channel paths, βi is the complex gain of the i-th path, and Φir and Φit are respectively the arrival angle (AoA) and departure angle (AoD) of the i-th path. αr(Φir) and αt(Φit) are response vectors of the antenna array of the user’s receiver corresponding to Φir and the base station transmitter corresponding to Φit.

For a simple uniform linear array (ULA) containing N elements, the array response vector is expressed as:

ϑULA(Φ)=1N[1,ejkdsin⁡(Φ),ej2kdsin⁡(Φ),…,ej(N−1)kdsin⁡(Φ)]T(4)

In (4), k=2π/λ, λ is the wavelength and d is the separation distance of the antenna. Due to the limited spatial scattering in mmWave propagation, the corresponding mmWave massive MIMO channel matrix is a lower rank matrix, so that a near-optimal system spectral efficiency can be achieved with a limited radio frequency (RF) chain [27].

3  Proposed Algorithm

3.1 Problem Description

The limited spatial scattering of the propagating mmWave considerably varies the singular value of the channel matrix H [28]. As shown in Fig. 2a, this results in a large difference in the signal-to-noise ratio (SNR) of the different sub-channels after power allocation.

Figure 2: Illustration of sub-channel gain. (a) traditional SVD precoding; (b) GMD precoding

In the same modulation/demodulation mode, the bit error rate (BER) of all sub-channels is determined by the sub-channel with the lowest fixed SNR. In order to ensure that all sub-channels maintain a similar bit error rate, the SVD-based precoding system requires careful bit allocation for each sub-channel, which will greatly increase the complexity of the system encoding/decoding process [29]. However, the H-channel adopts the GMD technique to effectively equalize the SNR of each sub-channel [24,25]. As shown in Fig. 2b, this avoids the complicated process of allocating bits and loading power into the sub-channels, and reduces the complexity of programming and decoding the system. The overall complexity of the system is thus reduced. On this basis, the design relies on the GMD hybrid precoding optimization matrix, to achieve the optimized value of spectral efficiency under the lower complexity of coding, decoding and modulation, and demodulation.

From (2), the spectral efficiency of the system is defined by (5) [30].

R=log⁡[det(I+ρNsσn2HPAPDPDHPAHHH)](5)

3.2 Optimization of Spectral Efficiency

From (5), we can see that for any arbitrary value having a rank of Ns and a singular value of σ1≥σ2≥…≥σNs. The complex channel matrix H can be expressed by (6).

H=GGMDRGMDQGMDH=[G1G2][R1∗0R2][Q1HQ2H](6)

In (6), GGMD=[G1G2]∈CNr×Nt, G1∈CNr×Ns is a positive semi-definite matrix containing the unitary matrix GGMD containing the Ns column vector from the left; QGMDH=[Q1HQ2H] is a unitary matrix, Q1∈CNt×Ns is a positive semi-definite matrix containing the Ns column vector from the left of the unitary matrix QGMD; RGMD=[R1∗0R2]∈CNr×Nt is a diagonal matrix with singular values of matrix H. R1∈CNs×Ns is an upper triangular matrix with the same diagonal elements, and its diagonal elements contain the geometric mean of the first Ns singular values of matrix H. That is, ∀i, where ri,i=(σ1,σ2,…,σNs)1Ns=r¯ holds, ri,i is equivalent to the parallel subchannel gain after GMD processing, σ1,σ2,…,σNs is the singular value of H. R2∈C(Nr−Ns)×(Nt−Ns) is a triangular matrix, and its diagonal elements are the singular values of matrix H after Ns, and the corresponding matrix elements are arbitrary values. The * denotes any random possible value within the matrix set. As there are only Ns transmission data streams, only the geometric mean of the first Ns singular values is taken into account here.

From (6), the H-channel matrix will be processed by GMD, and the processed efficiency can be expressed by (7).

R=log⁡[det(I+ρNsσn2(RGMD)2QGMDHPAPDPDHPAHQGMD)](7)

Assuming that P=PAPD is very close to the optimal unit matrix Qres=Q1, the following two approximate conclusions can be obtained [6]:

1.    The matrix INs−Q1HPAPDPDHPAHQ1 has a small eigenvalue. It can thus be expressed in a way equivalent to Q1HPAPD≈INs in precoding mmWave.

2.    Moreover, since the singular value of Q2HPAPD is low, it can be equivalent to Q1HPAPD≈0, resulting in the transformation formula defined by (8).

QGMDHPAPDPDHPAHQGMD=[Q1HPAPDPDHPAHQ1Q1HPAPDPDHPAHQ2Q2HPAPDPDHPAHQ1Q2HPAPDPDHPAHQ2]=[Q11Q12Q21Q22](8)

Substituting Eq. (8) into Eq. (7), we obtain Eq. (9).

R=log⁡[det(I+ρNsσn2[R12∗0R22][Q11Q12Q21Q22])]≈log⁡[det(INs+ρNsσn2R12Q1HPAPDPDHPAHQ1)]=log⁡[|INs+ρNsσn2R12|]−(Ns−‖Q1HPAPD‖F2)(9)

In (9), the first term can be obtained by setting Qres=R1, and the second term is the squared chord distance between two points Qres=R1 and (PAPD) on the distance of the Grassmann manifold coconut ‖Qres−QAQD‖F. Therefore, the problem of maximizing the spectral efficiency can be converted into that of minimizing ‖Qres−QAQD‖F.

Assuming that Qres is the optimal precoding based on GMD, and that QA and QD are respectively the corresponding analog precoding and digital precoding, the optimized objective function after transformation is defined by (10).

tr(QAQDQDHQAH)≤Ns(10)

3.3 Conversion Optimization Objective Function

To reduce the complexity of the GMD transformation and obtain the optimized solution of Eq. (10), the optimization problem can be transformed by Lemma 1.

Lemma 1: Suppose the SVD of the channel matrix H is defined by (11).

The singular values of the channel matrix are arranged in descending order, then there is a unit matrix SR∈CNs×Ns, SL∈CNs×Ns such that Q1=V1SR; G1=U1SL; R1=SLT∑1SR.

H=U∑VH=[U1U2][∑100∑2][V1HV2H](11)

Lines of evidence: Calculate the geometric mean of the channel matrix r¯=(σ1,σ2,…,σNs)1Ns, define the auxiliary matrix by (12).

R1:={M(i)}HR1M(i),Q1:=Q1M(i),G1:=G1M(i)(12)

In the previous equation, “:=” means is defined as, and M(i) is the corresponding permutation matrix. Let R1=∑1, Q1=V1, G1=U1, and define 2 replacements from INs containing 4 elements {NL(i)}i,i, {NL(i)}i+1,i {NL(i)}i,i+1, {NL(i)}i+1,i+1sub-matrix generation. The four elements of the matrices NL(i), NR(i) have the following two-dimensional matrix form (see (13)).

θ¨L(i)=1r¯[cri,isri+1,i+1−sri+1,i+1cri,i],

θ..L(i)=1r¯[ cri,isri+1,i+1−sri+1,i+1cri,i ],θ..R(i)=[ c−ssc ](13)

where ri,i=ri+1,i+1=r¯, when c=1, s=0, then ri,i≠ri+1,i+1, therefore:

c=r¯2−ri+1,i+12ri,i2−ri+1,i+12,s=1−c2(14)

Update R1, Q1 and G1, then:

R1:=NL(i)R1NR(i),Q1:=Q1{NL(i)}T,G1:=G1NR(i)(15)

Further:

R1i+1=θ¨L(i)[ri,i00ri+1,i+1]θ¨R(i)=[r¯∗0ri,iri+1,i+1r¯](16)

This shows that through NL(i)R1NR(i), we can make ri,i=r¯, and keep the other elements unchanged. Substitute Ri for R1 in the calculation in turn, where 1≤i≤Ns−1.

Eq. (15) shows that M(i), NL(i) and NR(i) are all unitary matrices, and Eq. (16) shows that only the value of diagonal elements is changed through GMD transformation, and other elements remain unchanged. Therefore, the matrix SL=M(1)NR(1)M(2)NR(2)…M(Ns)NR(Ns) and SR=M(1)(NL(1))TM(2)(NL(2))T…M(Ns)(NL(Ns))T, substituting into Eq. (15), we get R1=SLT∑1SR. Similarly, Q1=V1SR; G1=U1SL. Hence, Lemma 1 is proved.

According to Lemma 1, the right side of Eq. (10) can be rewritten as (17).

‖Qres−QAQD‖F=‖R1−QAQD‖F=‖V1SR−QAQD‖F=‖V1−QAQD{SR}H‖F(17)

The condition that the third step of the Eq. (17) holds is because the F-norm remains unchanged under the matrix rotation, and there is a QD~=QD{SR}H.

Substituting Eq. (17) into Eq. (10), we can get (18).

tr(QAQD~QDHQ~AH)≤Ns(18)

So far, the objective function (i.e., Eq (10)) has been effectively transformed.

3.4 Optimize Objective Function Solution

From (18), although QA and QD are coupled to each other, because the elements of QA have constant modulus constraints, QA and QD can be designed separately by decoupling, i.e., when designing QA, QD remains unchanged [21]. The reverse is also true because the constant modulus elements of αt(Φit) satisfy the constraint of |{QA}i,j|=1Nt, and the column vector of V1 forms the orthogonal base of the channel row vector space, so that V1 can be considered as a linear combination of αt(Φit). By approximating V1 by QA and QD, the optimal solution of QA and QD can be obtained. For this reason, it is necessary to reasonably find the “optimal” NtRF array response vector from the αt(Φit) to form the column QA, so that the design problem of analog precoding can be converted (see (19)).

tr(AtTQD~DDHTHAAH)≤Ns(19)

In (19), At=[αt(Φ1t),αt(Φ2t),…,αt(ΦLt)] is a matrix of order Nt×L and sparse constrained ‖diag(TTH)‖0=NtRF indicates that T is impossible because there are non-zero rows with more than NtRF. When T has only NtRF non-zero rows, NtRF non-zero columns in At can be effectively selected as the analog precoding matrix.

Eq. (19) is a sparse reconstruction problem, which can be solved by tracking the bases [6]. When the analog precoding matrix QA is determined, the digital precoding design problem can be converted into a minimizing problem of the F-norm (see (20)).

s.t.tr(QAQD~Q~DHQAH)≤Ns(20)

The optimal solution of Eq. (20) is presented in the form of an LS method (see (21)) [26].

QD~={QA}†V1(21)

Furthermore:

QD=QD~SR={QA}†V1SR(22)

According to the derivation of the LS matrix:

QD=(QAHQA)−1QAHV1(23)

According to the above analysis, the problem of selecting the optimal analog precoding matrix is a sparse reconstruction problem, which can be solved by the basic tracking principle. After obtaining the analog precoding, the LS method is used to optimize the solution method in order to obtain the optimal digital precoding matrix QDopt.

4  Hybrid Precoding Optimization Algorithm Based on GMD

4.1 Optimization Algorithm Under Fully Connected Structure

In the fully connected structure, hybrid precoding only takes into account the base station coding, and the user receiver can perform the corresponding decoding processing based on the received signal. The optimal solution algorithm of Eq. (18) consists of two links. The first one uses the for loop to use the residual matrix Q1 obtained by the GMD transformation as the optimal precoding matrix, G1H, substituting the OMP method as a combined matrix to obtain QA and QD. The latter performs the transformation corresponding to Lemma 1 on the QD and normalizes the effective precoding matrix to meet the transmission power constraints. Algorithm 1 provides the pseudocode for the fully connected structure.

It should be pointed out that due to the calculation using the conversion relation in Lemma 1, it is not necessary to calculate SR in the process, just apply the corresponding permutation and multiplication in each step of QD. Therefore, the computational complexity of generating QD is O((Ns+Nt)Ns) in the reference algorithm [31], and the proposed Algorithm 1 has a complexity of O((NtRF)2NtNs). The Golub-Kahan double-diagonalization scheme [32] (usually the first step of calculating SVD) has a computational complexity of O(NsNtK), which shows that GMD-based hybrid precoding can optimize the algorithm complexity of spectral efficiency. Compared with the traditional SVD-based hybrid precoding, only a small additional algorithm complexity is added, because the pseudo-inverse of QA needs to be calculated. In terms of overall system complexity, the GMD precoding scheme using this Algorithm 1 can effectively avoid complex bit allocation problems, reduce the difficulty of encoding/decoding, and the complexity is much lower than SVD precoding schemes.

4.2 Optimization Algorithm Under Partially Connected Structure

Under the partially connected structure, the previous step of the hybrid precoding optimal solution algorithm is to transform the channel H by selecting the matrix G0 to obtain the required auxiliary matrix G [26]. In the latter step, according to the relationship between analog and digital precoding under part of the connection structure in [26], the unitary matrix obtained by the GMD transformation is used to further calculate the qualified QA and QD under the transformation, and then execute Lemma 1 corresponding transformation to normalize the effective precoding matrix to meet the constraints of transmit power. Assuming that the number of antennas connected to each chain is M=Nt/NtRF, the base station uses K RF chains to transmit K data streams, that is, K=NtRF=Ns, and the base station has complete channel state information H. The specific steps are shown in Algorithm 2.

In Algorithm 2, the computational complexity of generating the auxiliary matrix G is O(M2(NtRFS+Nr)) [28], and the computational complexity of generating QD is O((K+Nt)K), so the complexity of Algorithm 2 is O(K2(NtRFS+Nr)+K2+NtK), and the algorithm complexity proposed in [6] is O((NtRF)4M+(NtRF)2L2+(NtRF)2M2L)considering the typical millimeter-wave communication system where NtRF = 8, M = 8, Nt = 64, Nr = 16, L = 3, S = 5 [26]. By analyzing the above-mentioned complexity analytical expressions, it can be seen that the complexity of the proposed algorithm is lower than that of the algorithm proposed by [21], which shows that GMD also has a better performance in the complexity of processing partial connection structure applications.

5  Simulation Results

This section analyses the performance of the GMD-based hybrid precoding scheme and spatial sparse precoding through simulation. The simulation environment is set as follows: the base station adopts ULA transmitting antenna array, the number of antennas is 64 and 256, respectively, the number of user end antennas is 16 and 64, and the antenna interval d=λ/2d. The number of RF chains at the base station and the user end are both NtRF=NrRF=8, and the carrier frequency is 28 GHz. Using the Saleh Valenzuela channel model, the number of effective paths is L=3, the complex gain of each path obeys the distribution CN(0,1), and the azimuth angles AoA and AoD of the antennas at both ends obey [−π/2,π/2]. The signal to noise ratio is ρ/σn2. For the optimal unconstrained precoding scheme, the water injection power allocation scheme is adopted. For all sub-channels based on SVD and GMD precoding schemes, the 16QAM modulation method is adopted.

Fig. 3 shows the spectral efficiency performance of the proposed Algorithm 1 under different iterations when the number of transmitting antennas is 256 and the number of receiving end antennas is 64. It can be seen from Fig. 3 that as the number of iterations increases, spectral efficiency gradually increases. When the number of iterations reaches 100, Algorithm 1 converges. It shows that the proposed algorithm is feasible for spectral efficiency under the GMD channel processing method.

Figure 3: Comparison of spectral efficiency under different number of iterations

Fig. 4 shows the comparison of the spectral efficiency with SNR when the number of RF chains at both ends is 8 and the transmission data streams Ns=4, the number of base station antennas is different. From Fig. 4, the following conclusions are:

The spectral efficiency obtained by the GMD-based hybrid precoding method when the number of base station antennas is better than the reference scheme [21].

As the number of transmitting antennas increases, the spectral efficiency of the system gradually increases. With the increase of the SNR, the increase of the spectral efficiency becomes progressively more important, which means that the larger the scale of the antenna, the better the spectral efficiency performance, but due to the limitation of the maximum transmit power, the increase of the effect has extreme values.

When the number of base station antennas increases to a certain value, the proposed scheme can approach the optimal precoding performance.

Fig. 5 shows the comparison of the spectral efficiency performance with SNR in the two antenna connection modes when the number of RF chains at both ends is 8, the transmission data stream Ns=4, and the number of transmitting antennas is 64. This can be seen from the results:

1.    Under the partial connection structure, the performance of the GMD-based hybrid precoding scheme is better than the traditional analog precoding;

2.    The spectral efficiency performance of the proposed precoding scheme is equivalent to the optimal precoding scheme under the partial connection structure, indicating that the proposed scheme has achieved near-optimal performance;

3.    When the SNR reaches 10 dB, the spectral efficiency of the proposed scheme under the partially connected transmission structure is approximately 80% of the scheme in [21] in the fully connected structure.

Figure 4: Spectral efficiency performance of fully connected structure system under different SNR

Figure 5: Spectral efficiency performance under different connection structures

Overall, the proposed system has good scalability and spectral efficiency, which guarantees the complexity of the system.

Fig. 6 compares the computational complexity of the algorithms under an increasing number of RF chains. As shown in Fig. 6, the complexity of all algorithms increases with the increasing number of RF chains. In addition, the complexity of the proposed hybrid precoding system has reduced performance compared to the optimal precoding system, which means that the proposed system is computationally efficient and requires a lower number of iterations and information signal processing time. On the other hand, the reference analog precoding scheme [21] and the traditional analog precoding scheme are very complex.

Figure 6: Complexity comparison of the algorithms vs. the number of RF chains

6  Conclusions

Millimeter wave (30 GHz–300 GHz) is used for high-speed (5G) wireless communications by allocating more bandwidth to deliver faster and higher quality video and multimedia content and services.

Due to the significant changes in the signal-to-noise ratio of the different sub-channels in the singular value decomposition, which leads to increased system complexity and encoding/decoding difficulties, a hybrid precoding scheme, which is based on geometric mean decomposition (GMD), is proposed.

Compared to the system proposed by [28], the hybrid precoding system proposed here can effectively equalize the signal-to-noise ratio of the sub-channels and reduce the overall complexity of the system. At the same time, in order to be better applied to real communication scenarios, the code has been extended to some connection transmission structures, and better frequency performance has also been obtained. The simulation results show that the frequency efficiency of the hybrid precoding scheme proposed under different base station antenna numbers is better than that of the orthogonal matching pursuit scheme [28], and that it can be applied to both existing transmission structures with a high applicability.

In perspective, the algorithms and conclusions proposed in this paper are established under ideal channel conditions. Further research is more than necessary for more complex and universal communication scenarios.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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