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

: 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 mobilecommunication 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 exam-ines 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.


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][3][4][5][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][14][15][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. 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 N t antennas and N RF t radio frequency chains; the transmission data contains N s data streams and satisfies the following in equation:

Hybrid Precoding Model
The user receiver is configured with N r antennas and N RF r radio frequency chains to meet the following requirement: N s ≤ N RF r ≤ N r . The hybrid precoding, which is performed at the base station, consists of digital precoding, P D ∈ C N RF t ×N s , and analog precoding, P A ∈ C N t ×N RF t . The precoding of the corresponding base station is defined as follows: P = P A P D . Assuming that the base station sends a signal vector of s, then the hybrid precoding signal is expressed by (1).
where P is the base station transmit power, which satisfies tr PP H ≤ N s . Consequently, the signal vector received by the user is expressed by (2).
In the previous equations, (·) H is the complex conjugate transposition operation; ρ is the mean received power; H ∈ C N r ×N t is the transmission matrix of the mmWave channel; n ∈ C N r ×1 is the noise vector, which obeys the complex Gaussian distribution, i.e., CN 0, σ 2 I N r , where σ 2 is the noise variance, I N r is the unit dimensional matrix N r × N r . The analog precoding P A is implemented by a network of phase shifter, and respects the limit of the constant modulus, i.e.,

Channel Model
The mmWave channel follows the Saleh Valenzuela channel model, and the channel transfer matrix H can be expressed by (3) [26].
In (3), L is the number of (resolvable) channel paths, β i is the complex gain of the i-th path, and r i and t i are respectively the arrival angle (AoA) and departure angle (AoD) of the i-th path. α r r i and α t t i are response vectors of the antenna array of the user's receiver corresponding to r i and the base station transmitter corresponding to t i . For a simple uniform linear array (ULA) containing N elements, the array response vector is expressed as: 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].

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. 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 subchannels 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.

Optimization of Spectral Efficiency
From (5), we can see that for any arbitrary value having a rank of N s and a singular value of σ 1 ≥ σ 2 ≥ . . . ≥ σ N s . The complex channel matrix H can be expressed by (6).
In (6), values of matrix H. R 1 ∈ C N s ×N s is an upper triangular matrix with the same diagonal elements, and its diagonal elements contain the geometric mean of the first N s singular values of matrix is a triangular matrix, and its diagonal elements are the singular values of matrix H after N s , and the corresponding matrix elements are arbitrary values. The * denotes any random possible value within the matrix set. As there are only N s transmission data streams, only the geometric mean of the first N s 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).
Assuming that P = P A P D is very close to the optimal unit matrix Q res = Q 1 , the following two approximate conclusions can be obtained [6]: (1) The matrix I N s − Q H 1 P A P D P H D P H A Q 1 has a small eigenvalue. It can thus be expressed in a way equivalent to Q H 1 P A P D ≈ I N s in precoding mmWave. (2) Moreover, since the singular value of Q H 2 P A P D is low, it can be equivalent to Q H 1 P A P D ≈ 0, resulting in the transformation formula defined by (8).

Q H GMD P A P D P H D P H
Substituting Eq. (8) into Eq. (7), we obtain Eq. (9).
In (9), the first term can be obtained by setting Q res = R 1 , and the second term is the squared chord distance between two points Q res = R 1 and (P A P D ) on the distance of the Grassmann manifold coconut Q res − Q A Q D F . Therefore, the problem of maximizing the spectral efficiency can be converted into that of minimizing Q res − Q A Q D F .
Assuming that Q res is the optimal precoding based on GMD, and that Q A and Q D are respectively the corresponding analog precoding and digital precoding, the optimized objective function after transformation is defined by (10).

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 Lines of evidence: Calculate the geometric mean of the channel matrix r = σ 1 , σ 2 , . . . , σ N s 1 Ns , define the auxiliary matrix by (12).
In the previous equation, ":=" means is defined as, and M (i) is the corresponding permutation matrix. Let R 1 = 1 , Q 1 = V 1 , G 1 = U 1 , and define 2 replacements from I N s containing 4 sub-matrix generation. The four elements of the matrices N (i) R have the following two-dimensional matrix form (see (13)).
where r i,i = r i+1,i+1 = r, when c = 1, s = 0, then r i,i = r i+1,i+1 , therefore: Update R 1 , Q 1 and G 1 , then: Further: This shows that through N (i) R , we can make r i,i = r, and keep the other elements unchanged. Substitute R i for R 1 in the calculation in turn, where 1 ≤ i ≤ N s − 1.
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 Q D = Q D {S R } H .

Optimize Objective Function Solution
From (18), although Q A and Q D are coupled to each other, because the elements of Q A have constant modulus constraints, Q A and Q D can be designed separately by decoupling, i.e., when designing Q A , Q D remains unchanged [21]. The reverse is also true because the constant modulus elements of α t t i satisfy the constraint of , and the column vector of V 1 forms the orthogonal base of the channel row vector space, so that V 1 can be considered as a linear combination of α t t i . By approximating V 1 by Q A and Q D , the optimal solution of Q A and Q D can be obtained. For this reason, it is necessary to reasonably find the "optimal" N RF t array CMC, 2021, vol.69, no.1 687 response vector from the α t t i to form the column Q A , so that the design problem of analog precoding can be converted (see (19)).
In (19), is a matrix of order N t × L and sparse constrained indicates that T is impossible because there are non-zero rows with more than N RF t . When T has only N RF t non-zero rows, N RF t non-zero columns in A t 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 Q A is determined, the digital precoding design problem can be converted into a minimizing problem of the F-norm (see (20)).
The optimal solution of Eq. (20) is presented in the form of an LS method (see (21)) [26]. Furthermore: According to the derivation of the LS matrix: 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 Q opt D .

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 Q 1 obtained by the GMD transformation as the optimal precoding matrix, G H 1 , substituting the OMP method as a combined matrix to obtain Q A and Q D . The latter performs the transformation corresponding to Lemma 1 on the Q D 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 S R in the process, just apply the corresponding permutation and multiplication in each step of Q D . Therefore, the computational complexity of generating Q D is O ((N s + N t ) N s ) in the reference algorithm [31], and the proposed Algorithm 1 has a complexity of O N RF t 2 N t N s . The Golub-Kahan double-diagonalization scheme [32] (usually the first step of calculating SVD) has a computational complexity of O (N s N t K), which shows that GMDbased 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 Q A 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.

Algorithm 1: Proposed Algorithm for Fully Connected Structure
Input: The optimal precoding matrix Q res , N RF t , N t , N r , and the array response vector-matrix A t Initialization:

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 G 0 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 Q A and Q D 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 = N t /N RF t , the base station uses K RF chains to transmit K data streams, that is, K = N RF t = N s , 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 M 2 N RF t S + N r [28], and the computational complexity of generating Q D is O ((K + N t ) K), so the complexity of Algorithm 2 is O K 2 N RF t S + N r + K 2 + N t K , and the algorithm complexity proposed in [6] [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.

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 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/σ 2 n . 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. 1) 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]. 2) 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. 3) When the number of base station antennas increases to a certain value, the proposed scheme can approach the optimal precoding performance. (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. 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. 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.