Robust Designs of Beamforming and Power Splitting for Distributed Antenna Systems with Wireless Energy Harvesting

In this paper, we investigate a multiuser distributed antenna system with simultaneous wireless information and power transmission under the assumption of imperfect channel state information (CSI). In this system, a distributed antenna port with multiple antennas supports a set of mobile stations who can decode information and harvest energy simultaneously via a power splitter. To design robust transmit beamforming vectors and the power splitting (PS) factors in the presence of CSI errors, we maximize the average worst-case signal-to-interference-plus- noise ratio (SINR) while achieving individual energy harvesting constraint for each mobile station. First, we develop an efficient algorithm to convert the max-min SINR problem to a set of "dual" min-max power balancing problems. Then, motivated by the penalty function method, an iterative algorithm based on semi-definite programming (SDP) is proposed to achieve a local optimal rank-one solution. Also, to reduce the computational complexity, we present another iterative scheme based on the Lagrangian method and the successive convex approximation (SCA) technique to yield a suboptimal solution. Simulation results are shown to validate the robustness and effectiveness of the proposed algorithms.


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I Introduction

For the past decade, there has been a considerable evolution of wireless networks to satisfy demands on high speed data. Since resources shared among users are limited, a capacity increase is technically challenging in the wireless networks. Recently, a distributed antenna system (DAS) has received a lot of attentions as a new cellular communication structure to expand coverage and increase sum rates [1, 2, 3].

Unlike conventional cellular systems where all antennas are co-located at the cell center, distributed antenna (DA) ports of the DAS are separated geographically in a cell and are connected with each other by backhaul links [4]. Each DA port in the DAS is usually equipped with its own power amplifier at the analog front-end [4] [5]. Thus, individual power constraint at each antenna should be considered for the DAS unlike the conventional systems which normally impose sum power constraint [5].

In the meantime, one of the limits in current cellular communication systems is the short lifetime of batteries. To combat the battery problem of mobile users, simultaneous wireless information and power transmission (SWIPT) has been studied in [6, 7, 8, 9, 10, 11, 12, 13]. With the aid of the SWIPT, users can charge their devices based on the received signal [8] [9]. To realize the SWIPT, a co-located receiver has been proposed [10], which employs a power splitter to perform energy harvesting (EH) and information decoding (ID) at the same time [11]. By adopting the power splitting (PS) receiver, the SWIPT scheme for multiple-input single-output (MISO) downlink systems has been examined in [8] and [11]

where perfect channel state information at the transmitter (CSIT) was assumed. In practice, however, due to channel estimation errors and feedback delays, it is not possible to obtain perfect CSIT

[14, 15, 16, 17].

On the other hand, some recent works have investigated SWIPT in DAS [18, 19, 20, 21, 23, 22, 24, 25]. [18] has provided several intuitions and revealed the challenges and opportunities in DAS SWIPT systems. In order to improve energy efficiency of SWIPT, the application of advanced smart antenna technologies has been focused in [19]. In [20], a power management strategy has been studied to supply maximum wireless information transfer (WIT) with minimum wireless energy transfer (WET) constraint for adopting PS. Moreover, a tradeoff between the power transfer efficiency and the information transfer capacity has been introduced in [21]. The work in [22] examined a design of robust beamforming and PS for multiuser downlink DAS SWIPT. However, only one antenna was considered in each DA port. The authors in [23] investigated resource allocation for DAS SWIPT systems based on the worst-case model, where per-DA port power constraint was adopted. In [24], a few open issues and promising research trends in the wireless powered communications area with DAS were introduced. In addition, to achieve a balance between transmission power and circuit power, [25] studies a system utility minimization problem in a DAS SWIPT system via joint design of remote radio heads selection and beamforming. However, joint optimal design of transmit beamforming and the receive PS factor for SWIPT in DAS PS-based systems with multiple transmit antennas of each DA port, has not been considered in the literature yet.

Motivated by the existing literature [18, 19, 20, 21, 23, 22, 24, 25], in this paper, we study a joint design of robust transmit beamforming at the DA port and the receive PS factors at mobile stations (MSs) in multiuser DAS SWIPT systems with imperfect CSI. Channel uncertainties are modeled by the worst-case model as in [22]. Our aim is to maximize the worst-case signal-to-interference-and-noise ratio (SINR) subject to EH constraint and per-DA port power constraint. The contributions of this work are summarized as follows:

  • For a given SINR target, the original problem is decomposed into a sequence of min-max per-DA port power balancing problems. In order to convert the non-convex constraint into linear matrix inequality (LMI), Schur complement is used to derive the equivalent forms of the SINR constraint and the EH constraint. Furthermore, we prove that a solution of the relaxed semi-definite program (SDP) is always rank-two. Also, to recover a near-optimal rank-one solution, we employ a penalty function method instead of the conventional Gaussian randomization (GR) technique.

  • To reduce the computational complexity, another formulation is expressed for the minimum SINR maximization problem. By employing the Lagrangian multiplier method and the first order Taylor expansion, the SINR constraint can be approximately reformulated into two convex forms with linear constraints. Then, we propose an iterative algorithm based on the successive convex approximation (SCA) to find a suboptimal solution.

Simulations evaluation have been conducted to provide the robustness and effectiveness of the proposed algorithms. The performance is also compared with other recent conventional schemes in this area. We show that the proposed algorithms has the superior performances in terms of average worst-case rate by extensive simulation results.

The remainder of this paper is organized as follows: in Section II, we describe a system model for the multiuser DAS SWIPT and formulate the worst-case SINR maximization problem subject to per-DA port power and EH constraint. Section III derives the proposed robust joint designs. In Section IV, we present the computational complexity of the proposed algorithms. Simulation results are presented in Section V. Finally, Section VI concludes this paper.

Notation: Lower-case letters are denoted by scalars, bold-face lower-case letters are used for vectors, and boldface upper-case letters means matrices. represents the Euclidean norm of a complex vector x and denotes the diagonal matrix whose diagonal element vector is x. stands for the norm of a complex number . For a matrix M, , , , and are defined as trace, transpose, conjugate transpose, rank, and the -th element, respectively.

denotes the maximum eigenvalue of

M, and stacks the elements of M in a column vector. I

defines an identity matrix.

, and are the set of complex matrices, Hermitian matrices and real matrices of size , respectively. equals the set of positive semi-definite (PSD) Hermitian matrices. is a null matrix with size .

Ii System Model and Problem Formulation

In Fig. 1, we describe a single cell system model for the multiuser downlink DAS scenario with SWIPT. The DAS consists of DA ports and single-antenna MSs. It is assumed that each DA port is equipped with antennas, which have individual power constraint. All DA ports are physically connected to the main processing unit (MPU) through fiber optics or an exclusive radio frequency (RF) link. Furthermore, all DA ports share the information of user distance and user data, but do not require CSI of all MSs as in [4]. The MS distance information can be simply obtained by measuring the received signal strength indicator [5]. Note that one MS can be supported by several DA ports.

We consider the channel model for DAS which contains both small scale and large scale fading [5]. We denote the channel between the -th DA port and the -th MS as , where stands for the distance between the -th DA port and the -th MS, indicates the path loss exponent, and equals the channel vector for small scale fading. For the -th MS, the channel vector is given as .

Due to channel estimation and quantization errors, CSI is imperfect at each DA port and we assume that the uncertainty of the channel vectors is determined by as an Euclidean ball [10] [14] as


where the ball is centered around the actual value of the estimated CSI vector from DA ports to the -th MS, is the norm-bounded uncertainty vector, defines the orientation of the region, and represents the radius of the ball.

Fig. 1: Structure of a multi-user DAS downlink system

During one time slot, independent signal streams are conveyed simultaneously to MSs. Specifically, the transmit beamforming vector is allocated for the -th MS at the -th DA port. Thus, we denote the joint transmit beamformer vector used by DS ports for the -th MS as . Then, the transmitted signal to the -th MS is obtained by

where indicates the corresponding transmitted data symbol for the

-th MS, which is independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian (CSCG) random variable with zero mean and unit variance. We assume that each DA port has its own power constraint

. Let us define an square matrix . Then, per-DA per power constraint is given as .

The received signal at the -th MS is expressed as

where represents the additive white gaussian noise (AWGN) with variance at the -th MS. It is also assumed that each MS splits the received signal power into two parts using a power splitter, one for the EH and the other for the ID [8] [11]. The PS divides the portion and the portion of the received signal power to the ID and the EH, respectively.

Therefore, the split signal for the ID of the -th MS is written as

where stands for the AWGN with variance during the ID process at the -th MS. Then, the received SINR for the -th MS is defined as


Also, due to the broadcast nature of wireless channels, the energy carried by all signals, i.e., the portion of , can be harvested at the -th MS, and the split signal for the EH of the -th MS is thus given as

Then, the harvested energy by the EH of the -th MS is obtained as

where is the constant that accounts for the energy conversion efficiency for the EH of the -th MS.

In this paper, we assume that the harvested power at each MS should be larger than a given threshold, and each DA port also needs to satisfy per-DA port power constraint. Hence, our aim is to jointly optimize the transmit beamforming vector and the PS factor by maximizing the minimum SINR subject to EH constraint and per-DA power constraint. Then, by incorporating the norm-bounded channel uncertainty model in (1), the robust optimization problem is expressed as


where represents the required harvested power of the -th MS. Problem (3) is non-convex due to coupled variables and in both the objective function and the EH constraint, and thus, is difficult to solve efficiently.

Iii Proposed Robust Joint Designs

In this section, we propose two robust joint design algorithms for problem (3). First, we present a bisection search method which generates a local optimal rank-one solution. To reduce the computational complexity, we then introduce an SCA based algorithm to achieve a suboptimal solution.

Iii-a Proposed Method Based on Bisection Search

To make problem (3) tractable, we decompose the problem into a set of the min-max per-DA port power balancing problems, one for each given SINR target [15]. Using bisection search over , the optimal solution to problem (3) can be obtained by solving the corresponding min-max per-DA port power balancing problem with different . Then, for a given , we focus on the following min-max per-DA port power balancing problem as


We represent as the optimal objective value of problem (4). Note that based on the equation [22, Lemma 2], we can obtain the optimal beamforming solution for problem (3). Problem (4) is still non-convex in terms of the non-convex objective function (4a). First, we tackle the objective function (4a) by introducing an auxiliary variable . Then, the min-max per-DA port power balancing problem (4) can be rewritten as


We can see that problem (5) has semi-infinite constraints (4b) and (4c), which are non-convex. To make the constraint (4b) tractable, the following lemma is introduced to convert (4b) into a quadratic matrix inequality (QMI).

Lemma 1: (Schur complement [26]) Let N be a complex Hermitian matrix as

Then, we have if and only if with , or with .

Let us define an square matrix as . By utilizing Lemma 1, the constraint (4b) can be converted into


where . Note that (6) is still non-convex. In order to remove the channel uncertainty in (6), the following lemma is required to convert the constraint (6) into linear matrix inequality (LMI).

Lemma 2: [30, Theorem 3.5] Let us denote , for . If for , then the following QMI

are equivalent to the LMI

where .

To proceed, we set , , , , , , , , . Then, by exploiting Lemma 2, the constraint (6) can be equivalently modified as the following convex LMI


where is a slack variable.

Next, we transform the constraint (4c) to the convex one. Due to the definition of and , the constraint (4c) can be recast as

and thus, it follows


where .

Also, we utilize a similar methodology for (8) as follows. By applying Lemma 1, the constraint (8) can be changed into


In order to get rid of the channel uncertainty in (9), Lemma 2 is adopted, and the constraint (9) is equivalently modified as


where is a slack variable.

Defining as , problem (5) is thus reformulated as


The above optimization problem is difficult to solve in general due to the rank-one constraint. Therefore, we employ the semi-definite relaxation (SDR) technique [27] which simply drops the constraints for all ’s. Then, problem (11) becomes a convex problem which can be solved efficiently by a convex programming solver such as CVX [28]. In the following theorem, we show that a solution to problem (11) satisfies .

Theorem 1: If problem (11) is feasible, the rank of a solution to problem (11) via rank relaxation is less than or equal to 2.

Proof: See Appendix A.

After is obtained, if rank, we can compute an optimal transmit beamforming solution by eigenvalue decomposition (EVD) of . If rank, we use the conventional Gaussian randomization (GR) technique [27] to find for . In particular, the GR technique generates a suboptimal solution. Hence, when rank, we will propose an iterative algorithm to recover the optimal rank-one solution by following the approach in [34].

First, since is always semi-positive definite, we have . Thus, we can prove that rank if . Then, we can transform the constraint rank into the single reverse convex constraint as

Note that the function on the set of Hermitian matrices is convex. When is small enough, will approach , where

represents the eigenvector corresponding to the maximum eigenvalue

with . Then the optimal transmit beamformer vector can be expressed by


which satisfies the rank-one constraint.

Thus, in order to make as small as possible, we adopt the exact penalty method [26]. First, introducing a sufficiently large penalty ratio , the alternative formulation is considered as


We can find from (13c) that the difference will be minimized when is large enough. Clearly, (13c) is set to minimize . Note that (13c) is non-convex due to the coupled and . To eliminate the coupling between and , we apply the following lemma to provide an effective approximation of (13c).

Lemma 3: Let us define and . Then, it always follows , where denotes the eigenvector corresponding to the maximum eigenvalue of E.

According to Lemma 3, we propose an iterative algorithm to recover a local optimal solution. For given some feasible to problem (13), we get


where the superscript represents the -th iteration.

Hence, the following SDP problem generates an optimal solution that is better than to problem (13) as


Now, problem (15) can be further simplified to


To summarize, we can solve problem (3) with a given , and a bisection search algorithm is applied to update for the objective value . Then, this process is repeated until convergence. For the bisection method, we need to determine an upper bound as . Then, we can see that

From this, we can set as . Due to monotonicity of , the bisection search algorithm needs iterations, where is a small positive constant which controls the accuracy of the bisection search algorithm. It is noted that this bisection search algorithm converges to the optimal solution for problem (3). The proposed algorithm based on bisection search is summarized in Algorithm 1.111The proposed optimization algorithm is performed by MPU. Then, the MPU can send the beamforming solutions to individual transmitters through fiber optics or an exclusive radio frequency (RF) link. Also, it can transmit the PS factor solution to individual receivers through the estimated instantaneous channel.

  Set , , , , a prescribed accuracy tolerance and . Randomly generate an initial value in (16). Repeat Set .
   Solve problem (16) with to obtain a solution and .
   If , set .
   Update n n+1.
   Set , , and .
   Solve problem (16) with to obtain a solution , , and .
   Update n n+1.
   If , set .
   else set .
  Until Calculate according to (12).
Algorithm 1 Proposed algorithm based on bisection search

Iii-B Robust Iterative Algorithm Based on Successive Convex Approximation

To reduce the computational complexity of Algorithm 1, we consider another formulation for the minimum SINR maximization problem. Based on the SCA method, the optimization can also be reformulated into a convex form with linear constraints. Thus, the robust SINR maximization problem can be rewritten as


In this problem, we minimize the numerator of SINR while maximizing the denominator of SINR [9]. Based on a tight approximation, the minimum and the maximum for each term can be determined by employing the Lagrangian multiplier method. In addition, to equivalently convert the objective function (17a), we introduce the exponential variables and as


Thus, in order to circumvent the non-convex objective function (17a), problem (17) is expressed by introducing a slack variable as


Note that (18b) is in concave form. Defining as the variables at the -th iteration for an SCA iterative algorithm, a Taylor series expansion