In the past ten years, the rapid development of various business systems such as the Internet of Things (IoT) and the Internet of Vehicles (IoV), as well as the advancement of wireless equipment manufacturing processes, have promoted the development and deployment of 5G mobile communication systems with high-speed, large connections and low latency. In general, the improvement of spectrum efficiency is achieved through network densification of micro-cell millimeter wave and massive MIMO technology [1]. The current low-frequency spectrum resources can no longer meet the people‚s needs for high-rate data transmission in people‚s lives. The development of wireless communication technology has forced researchers to include the millimeter wave frequency band into the scope of research. The millimeter wave frequency is distributed in the range of 30 to 300 GHz, and its ample bandwidth has become a hot spot in the academic and industrial circles. However, millimeter waves are severely absorbed by the atmosphere and rain in the process of space transmission, and the limited number of propagation paths, resulting in their short effective propagation distance, which is very suitable for micro-cell communication with small coverage and high data transmission rate. It provides sufficient array gain to form a needle beam to reduce interference, which is regarded as one of the key technologies of 5G [2]. When a large number of antennas are deployed at the millimeter wave transmitting end, all-digital precoding needs to be equipped with a dedicated radio frequency chain for each transmitting antenna, and a radio frequency chain with the number of antennas (composed of digital-to-analog converters, mixers, etc.) which has high cost and unacceptable energy consumption. Therefore, the research and design of low-dimensional baseband digital precoders and high-dimensional radio frequency analog precoders instead of all-digital precoders has aroused great interest. In a single-user communication system, there are ways to improve the spectrum efficiency by minimizing the Euclidean distance between hyrbid analog and digital precoding and fully-digital precoding [3‐6], and also through a joint transmitter analog precoder and receiver analog research on the maximized spectral efficiency of the combiner [7–9]. In multi-user communication systems, the hybrid precoding with a fully connected structure has also been studied [10–14]. For example, reference [15] directly uses the phase of the channel conjugate transpose as an analog precoder, and then uses the zero forcing (ZF) technique to design the baseband digital precoder. Reference [16] proposed a two-stage hybrid precoding and designed analog precoding in which the phases of all digital precoding obtained by maximum ratio transmission (MRT) and ZF precoding were extracted respectively, and then combined with channel moments. The equivalent matrix obtained by the matrix eliminates inter-user interference through ZF to obtain the baseband digital precoding. However, these hybrid precoding based on the fully connected structure require the use of more RF chains and high-precision phase devices, resulting in expensive hardware costs and power consumption proportional to accuracy, hindering the hybrid precoding structure in the base station and mobile end millimeter wave cellular network systems with strict size and power restrictions in the deployment.

In order to reduce the number and precision of the hardware used, further research on hybrid precoding of partial connection structures has been carried out [17,18]. Reference [19] first proposed the hybrid precoding of switch and inverter based on machine learning adaptive cross entropy. The authors in [20–22] applied machine learning adaptive cross entropy to the hybrid precoding of lens array switch structure, and further analyzed the impact of important parameters based on the sum rate and energy efficiency of the hybrid precoding of the switch and inverter structure. Reference [23] proposes a precoding with an adaptive connection structure, which can better achieve beam gain, but still requires a higher-precision phase shifter (at least 6-bit accuracy) to achieve close to the optimal fully-digital precoding and lower precision such as 1-bit quantized adaptive connection structure is achievable and rate performance is severely reduced. In order to solve the problem of poor accessibility a rate performance of the 1-bit quantized phase shifter of the adaptive connection structure, this paper proposes a 1-bit quantized phase shift based on machine learning adaptive cross-entropy hybrid precoding. The adaptive connection structure that obtains the matching relationship between the RF chain and the base station antenna by maximizing the user's received signal power is more flexible than the fixed sub-connection. According to the probability distribution, the analog precoder is randomly generated, and the classic ZF precoding is used to obtain the corresponding digital precoder. The analog precoder is adaptively weighted according to the reach and rate. Then, the probability distribution of the simulated precoding is updated by reducing the cross entropy and adding a constant smoothing parameter, and repeating this way, a hybrid precoder with almost optimal performance and rate is finally obtained. Numerical simulations are performed to evaluate the effectiveness of the proposed scheme.

As shown in Fig. 1, consider the massive MIMO system of multi-user downlink, the base station deploys N antennas, NRF radio frequency chains, and simultaneously serves K non-cooperative users with a single antenna. Generally, the massive MIMO system satisfies K≤NRF≤N, and NRF=K is set in this article. The analog precoder FRF is composed of a small number of radio frequency (RF) chains, an adaptive connection network and a large number of antennas [24]. It is assumed that the symbols sent to K users s=[s1,,s2…,sK]T are independently and identically distributed, and they are all symbols that obey the zero-mean complex Gaussian distribution, satisfying E(ssH)=P/KIK where P is the total transmitted signal power of the base station [25].

The signal received by K users can be expressed as (1)y=HTFRFFBBs+n Among them, HT=[h1,h2,…,hK]T∈CK×N represents the downlink channel from the base station to all users, hk∈CN×1 represents the channel gain from the base station to the k th user. FRF=[f1RF,f2RF,…,fNRFRF]∈CK×NRF represents a high-dimensional constant modulus analog precoder that can only adjust the signal phase, FBB=[f1BB,f2BB,…,fKBB]∈CNRF×K stands for a low-dimensional digital precoder that can adjust the signal amplitude and phase. n represents K×1 dimension vector additive white Gaussian noise AWGN, satisfying E(nnH)=σ2IK, σ2 represents the power of additive white Gaussian noise [26]. In order to ensure the total transmit power constraint, after the analog precoding matrix FRF is designed, the corresponding digital precoding matrix FBB should meet the power constraint ‖FRFFBB‖F2=K. For the channel gain vector, in order to better describe the characteristics of limited space selection and limited scattering caused by the extremely high propagation path loss of the millimeter wave channel, and the large number of antenna arrays of the millimeter wave transmitter leads to a large degree of antenna correlation, using the geometric Saleh-Valenzuela channel model, assuming that the transmitting end is a uniform linear array (ULA) antenna, the channel gain of the k-th user can be expressed as: (2)hk=NLk∑l=1Lk⁡αk(l)a(Φk(l)) Among them, N is the number of antennas of the base station ULA antenna. Lk represents the number of propagation paths of the k-th user, αk(l) and Φk(l) respectively represent the complex gain and departure azimuth angle of the l-th path of the k-th user, and a(Φ) represents the response vector of the transmission array with a dimension of N×1, which can be expressed for (3)a(Φ)=1N[1,ej2πdλsin⁡(Φ),ej4πdλsin⁡(Φ),…,ej2πd(N−1)λsin⁡(Φ)]T Here, λ is the wavelength of the millimeter wave, and d is the distance between the ULA antenna elements, usually set to half the wavelength d=λ/2 [27--28].

The analog precoding of the traditional low-precision phase shifter partial connection structure often fails to achieve the array gain of the millimeter wave large-dimensional antenna. Therefore, this article simulates the precoder adaptive connection and deploy adaptive connection network instead of fixed sub-connection switch and reverse vectorizer (equivalent to a phase shifter with 1-bit quantization). The same as the fixed sub-connection structure, the adaptive connection only requires N number of 1-bit quantized phase shifters, NRF number of radio frequency chains, and no adders. Compared with fully connected N×NRF high-precision phase shifters, NRF radio frequency chains, and N adders, it greatly reduces the hardware complexity, cost and energy consumption. The adaptive connection network can better match the downlink to improve the user's reach and rate. In order to make better use of adaptive cross-entropy optimization to apply to the adaptive connection structure, it is necessary to match the RF chain and the antenna under specific channel state information, that is, to find the position of the non-zero element in the corresponding analog precoding matrix FRF. FRF=[f1,1RFf1,2RF⋮f1,NRFf2,1RFf2,2RF⋮f2,NRF⋯⋯⋮⋯fK,1RFfK,2RF⋮fK,NRF]N×K→sumbyline∑k=1K⁡fk,jRF (4)fxRF→=[fx1,1RFfx2,2RF⋮fxN,NRF]N×1

Due to the special structure of the adaptive connection and the normal mode constraint of the elements in the analog precoder, the corresponding analog precoding FRF is a sparse matrix, as shown in Eq. (4), fk,jRF represents the element in the j th row and k th column, and the k th radio frequency chain is connected to the j th base station antenna, and its element fk,jRF∈1N{−1,0,1},1≤k≤K;1≤j≤N. Sum each row of FRF to get vector fxRF→=[fx1,1RF,fx2,2RF,…,fxN,NRF]T, and its j th element fxj,jRF∈1N{−1,1}. Therefore, the adaptive connection structure simulation precoding has two constraints: (5){∑j=1N⁡|fk,jRF|=MN∑k=1K⁡|fk,jRF|=1N

Indicates the connection relationship between all radio frequency chains and all antennas of the base station. Assuming that N/K=M is an integer, divide the N base station antennas into K independent sets, use Sk to represent the set of k-th radio frequency chain connected to base station antennas, and Sk(j) to denote the k-th radio frequency chain connected to j th base station antenna, and |Sk|=M. So Eq. (5) is also equivalent to (6)S={{Sk}k=1K|∪k=1K⁡Sk={1,2,…,N}},Sk∩Sq=ψ,∀k≠q The designed analog precoding FRF and digital precoding FBB should maximize the downlink reachability and rate of the K users served (7){FRF,FBB=arg⁡FRF,FBBmax∑k=1K⁡log⁡(1+SINRk)s.t.FRF∈FRF‖FRFFBB‖F2=K

where SINRk represents the signal-to-interference and noise ratio of the k-th user, which is expressed as (8)SINRk=P/K|hkTFRFfkBB|2P/K∑k≠i⁡|hkTFRFfkBB|2+σ2

It can be seen that Eq. (7) is a non-convex optimization problem under the constraints of power constraints and adaptive connection structure. To maximize the total reachability and rate, that is, to maximize the SINR of the symbols received by each user, the k-th column fkRF of the FRF must satisfy (9)fkRF=arg⁡fkRFmax{|hkTfkRF|:costraints(5)}

There are only M non-zero elements in fkRF, and the first M maximum values of the channel hk element modulus need to be matched, one element is matched each time and the antenna position is returned, and stored in the set Sk (10)Sk=arg⁡1≤j≤Nmax{|hk,j|:∑k′=1K⁡|fk′,jRF|=0}

The constraint ∑k′=1K⁡|fk′,jRF|=0 means that the j th line of the FRF has not been evaluated or the j th antenna has not been matched.

Performing Eq. (10) once can match the k-th radio frequency chain with a base station antenna, that is, get a non-zero element position in the FRF. To ensure the fairness of the radio frequency chain, K radio frequency chains are matched in turn, and M turns are performed. The position of the N non-zero elements of the FRF that meets the constraint (5) can be obtained, or the set of matching relationships between the RF chain and the antenna that meets the constraint (6) {Sk}k=1K.

In the ACN-MLACE algorithm, since the phase shifter is quantized by 1 bit, the non-convex optimization problem of Eq. (7) has been transformed into a combinatorial optimization problem. How to obtain the FRF and FBB with the largest reach and rate? It is necessary to search exhaustively to find N determinations. The 2N combinations of non-zero element values in the position involve extremely high computational complexity. For millimeter-wave massive MIMO systems, N is often large, such as N = 56, and an exhaustive search is required for 256≈7.2×1016. Take another path, deploy ML adaptive algorithm based on cross-entropy optimization to intelligently look for the best adaptive connection precoding. In each iteration, Z⋅N×1 dimensional vectors are generated according to a probability distribution. These vectors can be set according to the matching relationship between each radio frequency chain and the base station antenna {Sk}k=1K to obtain Z candidates analog precoding matrices FRF, each candidate analog precoding matrix FRF has a corresponding reach and rate. Select the Zelite⋅FRF with the best performance and rate, and update the probability distribution by minimizing cross entropy and adding smoothing parameters. In this way, it will be generated with probability 1 and close to the optimal candidate simulates the probability distribution of non-zero elements in the precoding matrix. The vector fxRF→=[fx1,1RF,fx2,2RF,…,fxN,NRF]T composed of non-zero elements in each row of the analog precoder is reconstructed according to the found matching relationship, and p=[p1,p2,…,pN]T is used to represent the probability of the corresponding element in fxRF→, and the j-th element fxj,jRF=1/N is Bernoulli random variables, that is, the probability of fxj,jRF=1/N is pj, and the probability of fxj,jRF=−1/N is 1−pj.

By initializing the probability distribution parameter p(0) to p(0)=1/2×IN×1, (I is a vector of all 1 s), according to the probability distribution ξ(fxRF→,p(i)), generate Z candidate vectors {fxRF,z→}z=1Z samples, the matching relationship is reconstructed to generate Z candidate analog precoding {fRFz}z=1Z samples, and then according to the equivalent channel Heqz=HHFRFz, 1≤z≤Z, the corresponding Z digital precoding {FBBz}z=1Z. In this article, the classic digital ZF precoding is used to eliminate the inter-user interference to obtain the corresponding z-th digital precoding matrix (11)FBBtemp,z=(Heqz)H((Heqz)(Heqz))−1 Put power constraints on it: FBBz=K/‖FRFzFBBtemp,z‖FFBBtemp,z.

After that, the achievable sum rate R(FBBz) is obtained by substituting FRFz and FBBz into Eqs. (7) and (8), and sort Z and {R(fRFz)}z=1Z in descending order. In order to adaptively update the next probability p(i+1), it is necessary to obtain the first Zelite and {R(fRFz)}z=1Zelite with the best achievable rate, and define the best analog precoding achievable rate average value is T which is expressed as (12){T=1Zelite∑z=1Zelite⁡R(FRFz)T=1Zelite∑z=1Zelite⁡R(fxRF,z→)

Here, the z-th analog precoding matrix FRFz is the first vector fxRF,z→ obtained according to the matching relationship between the radio frequency chain and the base station antenna, so R(FRFz)=R(fxRF,z→). The weight of the achievable rate corresponding to the z-th analog precoding matrix is wz=R(FRFz)/T. Update the probability of the next time adaptively according to the current probability distribution and weight (13)p(i+1)=p(i)max1Z∑z=1Zelite⁡wzln⁡ξ(fxRF,z→,p(i))

There is also ξ(fxRF,z→,p(i))=ξ(FRFz,p(i)), where (14)fxRF,z→=∏j=1N⁡(pj(i))12(1+Nfxj,jRF,z)(1−pj(i))12(1−Nfxj,jRF,z)

Substituting Eq. (14) into Eq. (13), and then Eq. (13) finds the first derivative of its j-th probability element pj(i) to obtain (15)1Z∑z=1Zelite⁡wz(1+Nfxj,jRF,z2pj(i)−121−Nfxj,jRF,z2(1−pj(i)))

Setting Eq. (15) equal to zero, the j-th element of the next probability distribution can be obtained (16)pj(i+1)=∑z=1Zelite⁡wz(1+Nfxj,jRF,z)2∑z=1Zelite⁡wz

In order to ensure that the adaptive cross entropy optimization converges to the optimal solution to avoid local convergence, a constant smoothing parameter Ӫ can be further added between the current probability distribution and the next probability distribution. (17)p(i+!)=θ¨p(i+!)+(1−θ¨)p(i) Here 0<θ¨≤1, until the end of the l iteration is reached, the probability distribution p(l) for generating the optimal analog precoding will be obtained, and the optimal analog precoding FRFl and the optimal number in the l generation sample will be selected precoding FBBl, which is the almost optimal adaptive connection hybrid precoding under the corresponding channel state information. The specific algorithm is as follows in Algorithm 1.

This section provides the simulation results and analysis. The proposed machine learning based precoding algorithm is compared with fully digital precoding, hybrid precoding of adaptive connection structure, and the conventional OMP precoding of structure. The combined precoding has the same lower hardware complexity and eliminates the NNRF phase shifters and N adders required by the fully connected hybrid structure. Therefore, the sum rate and complexity are used here as a comparison of the performance of different precoding schemes. The simulation parameters are set as follows in Tab. 1.

This paper proposes an adaptive connection network hybrid precoding with 1-bit quantization, and applies the adaptive algorithm based on machine learning to the adaptive connection structure hybrid precoding, which improves the 1-bit quantization phase shift of the adaptive connection structure. Under the same low hardware complexity, the proposed solution has a higher computational complexity than the switch and inverter hybrid precoding based on the fixed sub-connection of machine learning and the hybrid precoding based on the adaptive connection structure and achievable rate performance. Recently, highly efficient deep learning methods have been applied to hybrid precoding, and precoding with lower computational complexity and better spectral efficiency is worthy of further research.