Chengxiao Liu, Wei Feng*, Te Wei, Ning Ge

Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China

I. INTRODUCTION

With the fast development of maritime econo-my in recent years, there is a booming demand for broadband maritime communications so as to support video monitoring, salvage, and so on. Most existing maritime communication schemes are based on the satellite communication networks, which are not qualified for broadband service because of limited transmission rate. To solve this problem, some specialized communication network schemes have been designed to achieve broadband maritime coverage [1-4]. For the purpose of improving transmission rate in maritime communication systems, a massive MIMO base station (BS)with hybrid digital and analogue precoding is considered in this paper. On one hand, a terrestrial BS is much cheaper than a satellite. On the other hand, the coastwise users may share a higher quality of service in terms of data rate with the shore-based maritime BSs than traditional satellite communications, thanks to the greatly reduced transmission distance.

1.1 Related work

In [1-4], new schemes for broadband maritime communication network were studied.In [1], a scheme to extend the terrestrial wireless communication network to sea areas was proposed. In [2, 3], some novel ideas of maritime communication structure were described, which have inspired our ideas for system design. The authors of [4] proposed a maritime wireless communication network based on the trans-horizon maritime channel and virtual MIMO system. In [5-8], massive MIMO technologies were studied. Such technologies can significantly improve both the spectral efficiency and the energy efficiency.Inspired by the design of conventional cellular networks introduced in [5, 9, 10], we believe the massive MIMO technologies could be appropriately applied in maritime wireless communication networks.

In massive MIMO systems, serious interference caused by multiple antennas and multiple users will make sense. To eliminate such interference, precoding algorithms should be adequately designed. As conventional full digital baseband precoding (FD-BP) schemes have serious power consumption problem, hybrid digital and analogue precoding schemes are taken into consideration. In [11-16], various hybrid precoding schemes were proposed in conventional cellular networks.The authors of [12] proposed an effective hybrid precoding method that converts the optimization problem to a least projection problem, which has great potential for further research. Because the scheme proposed in [12]reduced the complexity of hybrid precoding,the authors of [13] proposed an alternating minimization algorithm for hybrid precoding utilizing manifold theory, which has lower complexity and better spectral efciency with stricter constraints. The scheme proposed in[14] is similar with those in [12, 13] besides minimum mean square error (MMSE) principle was applied. The authors of [16] proposed a two-stage hybrid precoding method utilized in multi-user massive MIMO systems, which splits digital precoding and analogue precoding into two stages rather than designing these two precoders simultaneously. Such method has inspired us an innovative way to design hybrid precoding algorithms with lower complexity. The method proposed in [11] designed a similar two-stage precoding algorithm and substituted the fully-connected structure of hybrid precoding system by adaptive-connected structure which has reduced the cost and power consumption. Different from conventional cellular networks, some outstanding characters of maritime communication network should be considered when designing hybrid precoding schemes.

One main character of maritime communication is the huge difference among the user sites as is shown in figure 1, which implies fairness dilemma. Fairness problem has already been discussed in conventional cellular networks on terrene before. In [17, 18], an optimal max-min fairness-oriented multiple user precoding method under the background of multi-cell cellular networks was proposed. The critical point to solve such max-min optimization problem is uplink-downlink (UL-DL) duality theory. The main idea of UL-DL duality comes out that the achievable capacity of DL system should be the same as its dual UL system when other conditions remain unchanged.Such theory is always applied in solving DL beamforming problems. After converting the DL beamforming problem to equivalent UL beamforming problem, the complexity in beamforming will be reduced. On the basis of UL-DL duality method, an iterative algorithm is proposed whose convergence is proved by Perron-Frobenius theory in [18].

Another character is related to the channel state information at the transmitter (CSIT).Users in maritime communication networks have much poorer transmission rate than users on terrene because of the remote distance between the users and the BSs. Basically the slowly-varying large-scale fading information in maritime channel which mainly depends on the locations of users is easily estimated [19-21]. However, the small-scale fading information can hardly be estimated when power attenuation is serious in maritime channels.Accordingly, the fading information of the CSIT in maritime communication systems can not be perfectly estimated. When the parameters describing small-scale fading effect are random variables at the transmitter, conventional precoding algorithms which are designed based on perfect CSIT will inevitably lose efcacy.

1.2 Contribution

In this paper, we focus on designing a hybrid digital and analogue precoding algorithm to maximize minimum transmission rate for the BS in maritime communication networks. An optimization problem is formulated with unit modulus constraints, which makes the problem impossible to be solved directly [13]. Moreover, random variables in channel vectors will affect transmission rate when only large-scale CSIT is known at the BS.

To tackle these challenges, we first reformulate the expression of signal-to-interference-noise-ratio (SINR) in a way of taking expectation inspired by [19] to eliminate random variables in channel vectors. Then the max-min problem is solved in an iterative way based on the theories proposed in [17,18] after the analogue precoding matrix is fixed. The proposed algorithm in this paper utilizes the characters of maritime channel and also takes fairness among users into account. Although the solution is not globally optimal, simulation results depict the significant performance of the proposed algorithm regarding system’s minimum transmission rate.

1.3 Organization

The rest of this paper is organized as follows.Section II introduces the massive MIMO system model and a typical 3-ray scattering channel model for maritime communication system. In Section III, a max-min fairness-oriented hybrid precoding problem is formulated as an optimization problem with unit modulus constraints, and solved in an iterative way.Simulation results are presented in Section IV demonstrating the extraordinary performance of the proposed algorithm compared with conventional precoding schemes. Finally, Section V gives the conclusion of this article.

Fig. 1. Diagram of a maritime communication system.

II. SYSTEM MODEL

2.1 System model

We consider a typical fully-connected multi-user (MU) DL massive MIMO system,where a BS hasMtransmit antennas andKRF chains withKsingle antenna users. The whole DL channel under hybrid precoding scheme can be expressed as

We concentrate on the DL rate in order to evaluate the performance of the MU-MIMO system. Actually the performance of a MU-MIMO system is closely related to the SINR of every user. For thek-th user,we have the relationship between DL rateRkand the SINR of thek-th userSINRkas the BS station can only use large-scale CSIT to design precoding matrices. So we assume that the parameters representing small-scale fading information as random variables, then eliminating randomness by taking expectation.HenceSINRkin this paper can be dened as

based on the original definition of SINR in conventional MU-MIMO systems.

Furthermore, we have some practical constraints over analogue precoding matrix FRF. Analogue precoding in massive MIMO systems is realized through analogue phase shifters (APS). Hence every elementin matrix FRFis located at unit circle on the complex plane, i.e.Such constraints are named as unit modulus constraints and not convex.

2.2 Channel model

The maritime DL system regarded in this paper contains a BS withMtransmit antennas andKsingle-antenna users in typical massive MIMO channel environment. As a result, a clustered channel model named the Saleh-Valenzuela model [22] is taken into account. In this paper, the channel vector of thek-th user can be modelled as

wheredkstands for the distance between thek-th user and the BS,αl,kdenotes the smallscale fading parameter,Ldenotes the number of paths,βdenotes the fading parameter and a()θis an array response vector which can be written as

As was demonstrated above, small-scale fading parametersαl,kin maritime channel are hardly estimated, but the distribution of these parameters is much easier to acknowledge. Hence the assumption that the smallscale parametersαl,k～CN(0,1) are i.i.d.random variables is reasonable.

Besides, the parameterLhere denotes the number of paths in maritime channel. In conventional maritime communication systems, a two-ray channel model is commonly applied as is mentioned in [23], which meansL=2 under conventional cases. The authors of [23]proved that a three-ray model which has taken refraction path into consideration as is shown ingure 2 is more consistent with actual maritime channel environment than conventional two-ray model. Thus we haveL=3 when the three-ray channel model is applied in maritime communication networks.

III. MAX-MIN FAIRNESS-ORIENTED HYBRID PRECODING WITH LARGESCALE CSIT

3.1 Problem formulation

An optimization problem PHPis formulated in this section. The digital and analogue precoding matrices will be designed after solving PHP. Since the system model here is (1), the SINR of thek-th user is expressed as (2).A common idea to design hybrid precoding scheme is looking for the optimal solution to a formulated optimization problem based on the system model. In this paper, a max-min fairness-oriented optimization problem under hybrid precoding scheme with large-scale CSIT can be formulated as follows.

Fig. 2. Diagram of a typical three-ray maritime channel model.

Note that the unit modulus constraintof the elements in FRFwill make PHPunsolvable explained in [13].

The obstacle caused by unit modulus constraints should be eliminated in order to solve PHP. In this paper, a feasible algorithm which fixes analogue precoding matrix FRFfirst is proposed. Such method will effectively reduce the complexity at the expense of performance.When FRFis fixed or designed in advance,PHPcan be solved on the basis of a FD-BP algorithms proposed in [17, 18].

3.2 Analogue beamforming design

Considering conventional hybrid precoding methods, analogue beamforming vectors in FRFis designed based on channel matrix in[11], according to pre-designed codebook in[12] or jointly with baseband precoding matrix in [13] when CSIT is perfectly acknowledged.Limited by imperfect channel state information, pre-designing a codebook is more feasible with large-scale CSIT . Similar with the method in [12, 16], the codebook Fthere consists of all array response vectorsin (3) which can be expressed as

The principle of selecting beamforming vector is maximizing the equalized channel gain,which was analyzed in [11, 16] that such principle wouldn’t cause serious performance loss compared with other analogue beamforming methods. Thek-th column fkof matrix FRFis selected to maximize the equalized gain of thek-th user, i.e.Specic algorithm of analogue beamforming is concluded in Alg.1.

3.3 Digital precoding design

Before designing the digital precoding matrix in hybrid precoding algorithm, werstly focus on the optimal solution to a FD-BP optimization problem oriented by max-min fairness with large-scale CSIT. In fact, a similar problem with perfect CSIT was proposed and solved in [17], but their algorithm is not applicable considering large-scale CSIT with the adjusted denition of SINR. The received symbol of thek-th user under FD-BP scheme in massive MIMO system can be written as

wherexkis transmit symbol with normalized constraintis digital precoding vector of thek-th user. So the SINR of thek-th user can be expressed as

where hkis channel vector with partial channel knowledge as is described in Section 2.2.Accordingly an optimization problem for FDBP scheme can be formulated as

We’ll give an optimal solution to P as follows. Firstly, the expectationable to be expanded and simplied as follows after substituting (3) into (7):

because of i.i.d. normally Gaussian distributed random variablesαl,k

In fact, wkcan be split into power mul-tiplying normalized beamformer vector,expressed asThen we defineand also define the elements in G and Ω as

Let

then (7) will be rewritten in a simplied form as

So our FD-BP max-min optimization problem P can be reformulated in a vector form as

Actually, the vector p here denotes the DL power ofKusers while the vectorhere denotes the dual UL power ofKusers which is virtual and utilized as a tool vector. The reason why (13)-(17) holds results from UL-DL duality theory,which was proved in [18].

It is clear that (13) and (15) can be written in stable point form as

indicating that applying stable point iteration method can solve optimal power allocation vectors p and q. Such algorithm converges well proved in [17].

Finally we need to solve the optimal beamforming matrix U. We reformulate (19) as follows

Note thatτshould be maximized in P, the optimal beamformming vectors are solved by

utilizing generalized eigenvector theory. In fact if we defineandindicates that the maximum value ofτis the maximum generalized eigenvalue of matrix pair [R,Q] whose generalized eigenvector is uk. Eventually we will have an optimal solution to P if we calculate (18) (19) and (21) iteratively.

Now let us turn back to designing the digital precoding matrix in hybrid precoding design. After analogue precoding matrix isxed in the way Section 3.2 mentioned, PHPcan be reformulated substituting Aiin (10) byThe same changes will also take place in (18)-(21). As a result, a new optimization problem PBPwhich solves optimal digital baseband precoding matrix for hybrid precoding algorithm is formulated as

It is not difficult to rewrite FBBin vector form as

After substituting the vector-form expres-sion of FBBinto (22), PBPwill be reformulated equivalently as

Hence PBPis equivalent to P after a series of transformation. Consequently similar methods can be applied to solve PBPand P. The proposed digital and analogue hybrid precoding algorithm is concluded in Alg.1.

Algorithm 1Max-min Fairness-oriented Hybrid Precoding Algorithm With Largescale CSIT}

IV. SIMULATION RESULTS

In this section, we will present the simulation results to demonstrate the advantages of max-min fairness-oriented hybrid precoding algorithm proposed in Section III. We assume that the locations of the users in maritime communication networks are uniformly distributed between 1km and 100km away from the BS. The number of RF chains here isKwhich is the same as the number of users. Forsimplicity, the noise power parameter users are assumed to be identical, which is set-106dBm.

The simulation of conventional precoding schemes in this paper differs from common simulations in [11]–[15] because the CSIT cannot be perfectly acknowledged by BSs in maritime communication networks. Without the full estimation of CSIT, conventional precoding matrices cannot be calculated because the small-scale parameterαl,iin channel vector h is unknown. Hence we disregard the small-scale parameter in h and reformulate the expression of thek-th user’s channel vector as

for the simulation of conventional precoding schemes in this paper. Such assumption is reasonable because it has efficiently utilized large-scale CSIT without the knowledge of small-scale parameters.

When the CSIT is imperfect, the performance of the conventional precoding algorithms will be significantly affected, which will be discussed as follows.

Figure 3 demonstrates the minimum downlink rate curve of maritime communication systems when the transmit power changes from 0dBm to 50dBm utilizing different precoding algorithms. Every point on the curve is the average value of 1000 types of users’distribution making the result convinced.From figure 3 we can see that the proposed algorithm shows enormous performance gain over conventional FD-BP and hybrid precoding algorithms due to the inuence of imperfect channel state knowledge and max-min fairness. Apparently such algorithm performs better when transmit power is over 30dBm,resulting from the serious power attenuation when most users in maritime communication networks are far away from the BS.

In figure 4, the relationship between the number of usersKand the minimum DL rate of maritime communication system is considered with the transmit powerPset at 40dBm typically. For most hybrid precoding schemes,the increasing number in users represents worse multi-user interference, which will directly affect the DL rate in massive MIMO system. From the curves we can see that the proposed method maintains a stable gap over conventional methods when the number of users grows, which implies the ability to endure the interference among multiple users of the proposed algorithm.

Figure 5 shows the relationship between the number of transmit antennasMand the minimum DL rate utilizing different precoding algorithms. The number of antennas here changes from 8 to 128 with constant 40dBm transmit power. The curve indicates that the proposed algorithm performs better when more antennas are utilized, which means such algorithm is able to be appropriately applied in massive MIMO systems.

Fig. 3. Comparison of different precoding algorithms considering the relationship between the transmit power and minimum rate.

Fig. 4. Comparison of different precoding algorithms considering the relationship between the number of users and minimum rate.

Fig. 5. Comparison of different precoding algorithms considering the relationship between the number of antennas and minimum rate.

V. CONCLUSION

In this paper, a novel max-min fairness-oriented hybrid precoding algorithm with large-scale CSIT is proposed. When only the large-scale CSIT is known at BS, we reformulate the de-nition of SINR. Then an optimization problem under max-min fairness is formulated to derive optimal hybrid digital and analogue precoding matrices. However, such problem is limited by unit modulus constraints and difcult to be solved directly. Finally an efficient iterative algorithm is proposed which can give optimal digital and analogue precoding matrices. Simulation results indicate that the proposed algorithm provides a dramatic performance gain compared with conventional hybrid precoding methods in terms of minimum achievable rate,which inspires a novel way to realize broadband coverage in maritime communication networks. Furthermore, the fixed user set of the proposed algorithm inspired us a joint design method applying user scheduling and clustering, which provides a critical point for further research.

ACKNOWLEDGMENT

This work was supported in part by the National Science Foundation of China under grant No. 91638205, and grant No. 61771286,and grant No. 61701457, and grant No.61621091.

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