Precoding for Full Duplex Multiuser MIMO Systems: Spectral and Energy Efficiency Maximization

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Precoding for Full Duplex Multiuser MIMO Systems: Spectral and Energy Efficiency Maximization Dan Nguyen, Student Member, IEEE, Le-Nam Tran, Member, IEEE, Pekka Pirinen, Senior Member, IEEE, and Matti Latva-aho, Senior Member, IEEE

Abstract—We consider data transmissions in a full duplex (FD) multiuser multiple-input multiple-output (MU-MIMO) system, where a base station (BS) bidirectionally communicates with multiple users in the downlink (DL) and uplink (UL) channels on the same system resources. The system model of consideration has been thought to be impractical due to the self-interference (SI) between transmit and receive antennas at the BS. Interestingly, recent advanced techniques in hardware design have demonstrated that the SI can be suppressed to a degree that possibly allows for FD transmission. This paper goes one step further in exploring the potential gains in terms of the spectral efficiency (SE) and energy efficiency (EE) that can be brought by the FD MU-MIMO model. Toward this end, we propose low-complexity designs for maximizing the SE and EE, and evaluate their performance numerically. For the SE maximization problem, we present an iterative design that obtains a locally optimal solution based on a sequential convex approximation method. In this way, the nonconvex precoder design problem is approximated by a convex program at each iteration. Then, we propose a numerical algorithm to solve the resulting convex program based on the alternating and dual decomposition approaches, where analytical expressions for precoders are derived. For the EE maximization problem, using the same method, we first transform it into a concave-convex fractional program, which then can be reformulated as a convex program using the parametric approach. We will show that the resulting problem can be solved similarly to the SE maximization problem. Numerical results demonstrate that, compared to a half duplex system, the FD system of interest with the proposed designs achieves a better SE and a slightly smaller EE when the SI is small. Index Terms—Energy efficiency, full duplex, linear precoding, MIMO, multiuser transmission, spectral efficiency.

I. INTRODUCTION

T

HE past few years have witnessed an ever increasing capacity that wireless communications networks can offer. In the perspective of physical layer design, this was accom-

Manuscript received August 06, 2012; revised January 07, 2013; accepted May 25, 2013. Date of publication June 11, 2013; date of current version July 19, 2013 . The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Rong-Rong Chen. This work was supported by the European Community’s Seventh Framework Programme FP7/2007-2013 under Grant 247733-Project EARTH and by the Academy of Finland under Grant 260755–Project Juliet. This paper was presented in part at the International Workshop on Small Cell Wireless Networks (SmallNets), IEEE ICC 2012, June 2012 [1]. The authors are with the Department of Communications Engineering and the Centre for Wireless Communications, University of Oulu, Oulu 90570, Finland (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2013.2267738

plished by the inventions of many powerful techniques such as strong channel coding (e.g., turbo code, low-density paritycheck code (LDPC) [2], [3]) and, particularly, multiple-input multiple output (MIMO) technologies [4], [5]. In current and emerging cellular networks, MIMO is realized in the form of multiuser MIMO (MU-MIMO) in the downlink (DL) and uplink (UL) channels. MU-MIMO techniques can simultaneously exploit the benefits brought by MIMO and multiuser diversity gains [6]. Although MU-MIMO has been shown to be a promising approach to increasing the system throughput, we seem to reach the boundary that MU-MIMO can provide in reality in not-so-far future. The fact is that we cannot integrate as many antennas as we want in both ends of a communications link due to practical limitations. Also, many other issues will arise when expanding the operating bandwidth or transmit power. Recall that the DL and UL channels of current cellular systems are designed to operate in half duplex (HD) mode, where the DL and UL channels run separately in time domain (time division duplex, TDD) or in frequency domain (frequency division duplex, FDD). Hence, such systems do not achieve the maximal spectral efficiency yet. So, we are still able to enhance the system capacity of cellular networks by allowing the DL and UL channels to work simultaneously over the same radio resources, referred to as full duplex (FD) transmission. That is to say, in a TDD system, DL and UL channels are designed to operate in the same time slot, and in a FDD system in the same bandwidth. Such a design, if possible, can greatly boost the system capacity of the HD systems and also resolve many problems of existing wireless communications networks such as reducing hidden terminals, congestion due to medium access control (MAC) scheduling, and large delays [7], [8]. The idea of designing the DL and UL channels that function over the same system resources basically means that base stations (BSs) should be able to transmit and receive data at the same time on the same frequency. This concept has been thought to be infeasible since the self-interference (SI) between transmit and receive antennas at BSs may severely affect the performance of the UL channel. The lack of advanced hardware techniques that can efficiently suppress the SI has remained until recently, when pioneer studies demonstrate the feasibility of a point-to-point FD wireless transmission in practice by eliminating the SI via advanced interference cancellation techniques [7]–[10]. This has resulted in many researchers studying the FD design for the future wireless communications. In particular, some FD approaches have been proposed in the context of MU-MIMO relay systems which have the potential to extend

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NGUYEN et al.: PRECODING FOR FULL DUPLEX MULTIUSER MIMO SYSTEMS: SPECTRAL AND ENERGY EFFICIENCY MAXIMIZATION

cell coverage and enhance the cell-edge throughput [11]–[13]. However, these schemes are only dedicated to one direction transmission and mostly focus on the DL channel. Another motivation to study FD schemes is due to a growing trend towards drastically smaller coverage cells which result in rather small power difference between transmit and receive antennas at a BS, making SI cancellation more efficient. To the best of our knowledge, potential gains of a single cell FD MU-MIMO system have not been reported yet. By the single cell FD MU-MIMO system, we mean a system where a BS transmits and receives signals from several HD terminals in the DL and UL channels over the same resources. In this paper, we make an attempt to understand the benefits that can be achieved by the use of the FD-based transceivers. In traditional HD systems, the optimal transmit strategy for the DL channel is with the use of dirty paper coding, but it requires high complexity [14]. Hence, we consider linear precoding for the DL channel. We note that the optimal linear precoder design for the DL channel is challenging due to the presence of multiuser interference, and thus many suboptimal schemes such as zero forcing (ZF) [15] and minimum mean square error (MMSE)[16] have been proposed. In our work, we adopt the ZF method proposed in [15] for the DL channel for simplicity. For the UL channel, we assume a minimum mean square error with successive interference cancellation (MMSE-SIC) receiver, which is known to achieve the capacity of the UL channel [17]. Due to the existence of a certain amount of the SI, the DL and UL channels are coupled, making the design of transmission strategies for the FD-based BS difficult. In this paper, we consider two main metrics that are widely used to evaluate the system performance: spectral efficiency (SE) and energy efficiency (EE). In fact, SE, defined as data rate per bandwidth unit, has been a main focus for wireless communications design. Very recently, EE, defined as the number of bits transmitted per an energy unit, has received significant attention due to growing interest in green communications, where energy consumption is the ultimate goal to be optimized. In a practical communications system, besides actual transmit power allocated for conveying data, circuit power radiated from electronics devices for signal processing functionality also plays an important role in EE transmission design. Traditionally, we assume the sum power constraint (SPC) in the DL channel and per user power constraints (PUPCs) in the UL channel. First, the SE and EE maximization problems are formulated as nonconvex problems. Then, we propose effective algorithms based on the concept of a sequential convex approximation [18]. Our contributions in this paper include, but are not limited to, the following. • For the problem of the SE maximization, using the first order approximation, we derive a lower bound of the SE in each iteration, which turns out to be convex. Then, linear precoders are found to maximize the lower bound by solving the dual problem with the block coordinate ascent (BCA) and dual decomposition methods [19]. The proposed iterative design, referred to as the SE-optimal design, iteratively increases this lower bound until convergence. We demonstrate by numerical results that the

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SE of the FD MU-MIMO system is superior to that of the conventional HD system. • The EE maximization problem at hand belongs to the class of nonlinear fractional programs, and thus the optimal solution is generally hard to find. Herein, we propose a suboptimal precoder design using the framework of concaveconvex fractional programming, presented in [20]–[22]. For this purpose, we approximate the EE maximization problem to be a concave-convex fractional program using the convex relaxation method as employed in the SE-optimal design. Then the resulting problem is transformed into an equivalent convex program by applying some appropriate transformations. As we show later, the resulting convex programming for EE maximization has a similar structure to that for SE maximization, and thus the main steps in the SE-optimal design can be applied. We refer to the iterative precoder design for maximizing the EE as the EE-optimal design. The simulation results reveal that the EE-optimal design outperforms the SE-optimal design in terms of EE. Moreover, the FD system yields slightly lower EE than the HD system when the SI is efficiently suppressed. As the main goal of this paper is to investigate the potential gains of the FD MU-MIMO system along with the proposed precoder designs for the SE and EE maximization, we are not putting an emphasis on proposing a hardware technique that cancels the SI. Although other issues may still exist, we believe that the FD MU-MIMO system is a potential solution to improve the performance of current cellular networks. By the results presented in this paper, we also call for in-depth studies of FD MU-MIMO systems that, e.g., find the boundary SE of FD MU-MIMO systems. The rest of this paper is organized as follows. Section II introduces system model and linear precoding techniques. The proposed algorithms for the SE and EE maximization are derived in Sections III and IV, respectively. In Section V, we present our numerical results under different simulation setups. Finally, conclusion is given in Section VI. Notation: Standard notations are used in this paper. Bold lower and upper case letters represent vectors and matrices, respectively; and are Hermitian and standard transpose of , respectively; and are the trace and determinant of , respectively; means that is a positive semidefinite matrix. II. SYSTEM MODEL AND LINEAR PRECODING TECHNIQUES Consider a single cell FD MU-MIMO system with a BS sending data to users in the DL channel, and receiving data from users in the UL channel at the same time on the same frequency, as shown in Fig. 1. The sets of users in the DL and UL channels are denoted by and , respectively. We assume that the BS is equipped with antennas, of which antennas are used to transmit data in the DL channel and antennas are used to receive the data in the UL one, . The number of antennas at user , , in the DL channel is

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to find even for the DL channel alone due to the multiuser inference. In fact, the problem of precoder design for maximizing in (2) has been proved to be NP-hard in [23]. Hence, to simplify the design, we adopt the block diagonalization (BD) scheme, proposed in [15], which is widely used thanks to its simplicity. In the BD scheme, ’s are designed such that for all , i.e., the multiuser interference is completely canceled. For the ease of description, let us define for user , and denote by , , an orthogonal basis of the null space of . Then, the ZF constraints in BD imply that , for all . Thus, we can write , where and reduces to

Fig. 1. FD MU-MIMO system model.

(3) denoted by , and that at user , , in the UL channel is denoted by . The design of transmission strategies for the FD MU-MIMO system of interest is challenging. The reason is that there always exists a certain amount of the SI between transmit antennas and receive ones at the BS after applying hardware cancellation techniques. A possibility is to design the DL and UL channels separately without accounting for the SI. However, such separate designs of the DL and UL channels become ineffective due to an excessive amount of the induced SI [1]. In addition to the SI from the DL to the UL channel, the co-channel interference caused by users in the UL channel to those in the DL channel should also be considered. In this work, for simplicity, we assume that the users in the DL and UL channels are geographically separated, meaning that co-channel interference is ignore. A general design that takes both the SI and co-channel interference into consideration is an interesting problem for future work. Assuming linear precoding at the BS, the received signal at user is (1) where , , and are the vector of transmitted signal, the linear precoder and the channel matrix of user , respectively. We can further assume that without loss of generality. The background noise is assumed to be a zero-mean additive white Gaussian noise (AWGN) vector with the single-sided noise spectral density . Assuming perfect channel state information (CSI) at both transmitter and receiver, the sum rate of the DL channel is given by

where of user

is the effective channel matrix (cf. [15] for more details). In (3), we have denoted . In order to recover from , we impose a rank constraint on that . As we will show later, the adoption of BD in the DL channel allows us to arrive at a formulation that can be solved using closed-form expressions. This possibility eliminates the need of installing a generic optimization package. Clearly, the BD scheme is feasible if due to the condition for the exis-

for all . This dimensionality tence of the null space of constraint means that the BS can simultaneously transmit data to a limited number of users. In case that is larger than the supportable number of users, a group of users must be chosen by a user scheduling algorithm. We note that existing user scheduling algorithms for BD in the literature (see [24]–[26] and references therein) can be employed in combination with our proposed designs. However, these user scheduling algorithms, which were only devised for the DL channel, do not account for the SI. Obviously, a user selection algorithm that considers the SI may result in better performance and designing such an algorithm constitutes a rich area for future research. For user in the UL channel, let be the channel matrix and be the transmitted symbols. Then, the received vector at the BS is given by

(4) (2) Throughout the paper, sum rate (spectral efficiency) is measured in nats/s/Hz. The optimal linear precoder design is hard

where

and . In (4),

is an AWGN vector with specifies the SI channel from the

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transmit antennas to the receive ones at the BS, and its entries depend on the effectiveness of the SI cancellation techniques. In this paper, we treat the SI as the background noise and assume perfect CSI at the BS and users. As a result, the sum rate of the UL channel with a MMSE-SIC receiver is

(5) is the transmit covariance matrix of user

where . Due the term in (5), the performance of the DL and UL channels is coupled. If one only concentrates on maximizing , this interference term can be so destructive that it can make the UL channel useless. Thus, as one of our main contributions in this paper, we consider a joint design of the DL and UL channels. Conventionally, we assume a SPC at the BS in the DL channel and PUPCs in the UL channel, i.e., each user is subject to an individual power constraint. Let be the performance measure of interest. Then, the problem of the joint design of and is generally expressed as

(6) is the maximum transmit power of the BS and is where the maximum transmit power of the user . In the following sections, we present joint designs for maximizing the spectral and energy efficiency.

propose a joint design that solves (7) locally. The proposed joint design is based on a successive convex approximation (SCA) of (7). More specifically, we employ the first order approximation of to find a lower bound of the SE and update this lower bound iteratively until the algorithm converges. We note that the SCA method has been shown to be efficient for the problem of beamformer design for MISO downlink channels (see [27] and references therein). A. Proposed Joint Design for SE Maximization Let SE after

denote the value of after iterations. Then the iterations is bounded below as

(8) where

and

. In (8), we have used the inequality for , where is an arbitrary operating point. This inequality is due to the concavity of function [28]. In fact, the idea of iteratively maximizing lower bounds of a nonconvex function has been used in the literature for different contexts, e.g., in [29]–[31]. In the st iteration are found to maximize of the proposed algorithm, the lower bound of the SE. Mathematically, are the solutions of the following problem

III. SPECTRAL EFFICIENCY MAXIMIZATION In this section, we propose a low-complexity joint design for the SE maximization problem, referred to as the SE-optimal design. For this case, , and (6) becomes

(7) The optimization problem above is also known as the sum rate maximization problem. It should be noted that all the constraints and . Moreover, are convex with respect to and are also concave with and , respectively. However, due to the interference from the DL channel, in (5) is neither convex nor concave with . Consequently, problem (7) is a nonconvex program, which is difficult to solve in general. Solving (7) globally requires in-depth knowledge of global optimization methods, which is beyond scope of this paper. Instead, we resort to a local optimization method, i.e., we

(9) where is the right side of (8) with the constant being ignored since it does not affect the optimization problem in (9). With the convex approximation of , problem (9) now becomes a convex program in each iteration, which can be solved using general standard convex optimization packages, e.g., CVX [32] or YALMIP [33]. The values of are updated until the iterative procedure converges. Regarding the use of a generic method to solve (9), the rank constraints, , for , must be guaranteed so that we are able to extract from . This problem is further discussed in the Appendix. Generic methods do not exploit the specific structure of the problem and thus require high computational complexity. Moreover, such methods do not offer useful insights into the optimal solutions. In this paper, exploiting the specific expression of , we develop an efficient iterative algorithm to solve (9), in which analytical expressions of and are

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found in each iteration. Moreover, the proposed precoder design guarantees that the rank constraints, for all ’s, are satisfied automatically. To start with, we rewrite (9) as

Algorithm 1 The proposed SE-optimal design for the considered FD system. 1: Generate initial points for

; tolerance

.

.

2: 3: repeat

(10) where are newly introduced optimization variables. The partial Lagrangian of (10) is given by

4:

Generate initial points for

5:

while

noted by

is the dual variable associated with the constraint . Consequently, the dual objective of (10), de, is the optimal value of the following problem

if

then

9: else

11: 12:

end if

13:

end while

14: 15: (12)

do

7: Solve (12) to find optimal solutions and using the alternating optimization algorithm presented in Section III-B.

10: where

. .

Update covariance matrices:

16: until convergence. 17: Apply the Cholesky decomposition to and calculate the precoder the DL channel.

and the dual problem of (10) is

(13) It can be easily seen that strong duality holds for the problem (9), and thus the optimal solution of (9) can be found by solving its dual problem in (13). Due to the fact that a subgradient of is given by (14) over can and that is a scalar, the minimization of be carried out efficiently using one dimension search method, e.g., bisection method [34]. Thus, solving (9) boils down to finding an efficient algorithm to evaluate , i.e., solving (12) efficiently. Before proceeding further, we outline the proposed joint design of the DL and UL channels for the SE maximization problem in Algorithm 1. The proposed SE-optimal design is guaranteed to converge since the lower bound is increased after every iteration, and the total SE of system is bounded above due to the power constraints. Moreover, due to the fact that the first order approximation is employed, Algorithm 1 converges to a stationary point, i.e., a point that satisfies the Karush-Kuhn-Tucker (KKT) conditions of (7). The detailed proof is provided in [18].

.

6:

8: (11)

, and

to find , for each user in

B. Alternating Optimization As the core of Algorithm 1, we now present an efficient algorithm to solve (12). We observe that the problem formulation in (12) lends itself to the BCA method since the constraints for individual and are separable. Moreover, as we will show shortly, the optimization of one variable, when others are fixed, can be expressed through analytical forms. Particularizing the BCA method to solve (12) leads to two different cases. In the first case, for each , we find that solves problem (12) by treating others and as constants. Explicitly, we need to solve the following problem (15) where

, , and

. It is easy to see that problem admits a solution based on the water-filling procedure. To be specific, let and ,

NGUYEN et al.: PRECODING FOR FULL DUPLEX MULTIUSER MIMO SYSTEMS: SPECTRAL AND ENERGY EFFICIENCY MAXIMIZATION

be the rank and singular values of respectively. Then the optimal solution of is given by

,

(16)

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As mentioned in line 7 of Algorithm 2, we now present Algorithm 3 that can solve efficiently, i.e., via analytical expressions. First, under the framework of dual decomposition method, we rewrite (18) as

where (17) (19) consists of right singular vectors of , and the water level is chosen to meet the power constraint, i.e., . We have also used the notation . In the remaining case, for each , the problem is to update while others and are held fixed. This amounts to solving the following problem

where

and

. In (19), we have introduced two variables and , and imposed the equality constraint to make (19) equivalent to (18). In the context of the dual decomposition method, and are local versions of the complicating variable , along with a consistency constraint that requires the two local versions to be equal. We also omit the constant in (18) since it does not affect the optimization of . Next, let be the Lagrange multiplier associated with the consistency constraint. Then, the partial Lagrangian function of (19) is written as

(18) ,

where , ,

(20)

and

. The alternating procedure to solve (12) is outlined in Algorithm 2.

and the dual function problem

is the optimal value of the following

Algorithm 2: The proposed alternating optimization algorithm. 1: Initialize:

;

.

(21)

2: repeat to

do

3:

for

4:

Solve using the water-filling algorithm to find while keeping all other variables fixed.

5:

end for

6:

for

to

(22)

do

using Algorithm 3 to find 7: Solve keeping all other variables fixed 8:

Since the objective function and all constraints in (21) are separable, it can be decomposed into dual subproblems 1 and 2 as follows

while and

end for

9: until desired accuracy is reached. Since each iteration of the alternating optimization algorithm increases the objective of (12), it converges to a locally optimal solution of problem (12), which is also the optimal solution of problem (9) due to its convexity.

(23) with optimal values and , respectively. We will show shortly that subproblems 1 and 2 can be solved efficiently

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by analytical forms. Firstly, we consider the Lagrangian function of subproblem 1 that is given by

updated until convergence is reached. To be specific, at the st iteration, is expressed as (29) is a positive step size, and is a proper subgrawhere is feasible, is a subgradient of , dient. Explicitly, if i.e., . Otherwise, is set to be a subgradient of the violated constraint function at as follows [35]. (30)

(24) where is the dual variable associated with the constraint, , , and . be an SVD of Let , where . Then, the optimal solution to subproblem 1 is found as

The step size rule is chosen in many ways. In this work, the value of our step size is chosen as [35]

(31)

where

(25) ,

where

and

. We note that must satisfy the constraint

for subproblem 1 to be solvable, . Similarly, let , and , the objective of (23) is rewritten as

and thus the optimal

Algorithm 3: A dual decomposition method for solving 1: Initialize

such that

Compute

of subproblem 2 is simply given by

5:

Update in (30) and (31).

is the , and . We also note that subproblem 2 is solvable if . We have shown that subproblems 1 and 2 can be solved efficiently using the water-filling algorithm with the fixed water level. The problem now is to find the optimal which minimizes the dual problem of (19), which is formulated as

(28) where and . The two constraints in (28) are actually the ones to make subproblems 1 and 2 solvable as mentioned earlier. There are many approaches to solve problem (28), e.g., the subgradient and cutting-plane methods [19], [35]. In this paper, we use the subgradient method for constrained optimization [35]. In this way, is iteratively

.

3: repeat 4:

,

.

.

2:

(26)

(27) where SVD of

is the constant that can be chosen sufficiently small, is the largest eigenvalue of at , and is a small positive margin (see [35] for more details). The proposed dual decomposition method to solve problem is summarized in Algorithm 3.

6:

and

using (25) and (27). with

and

given

.

7: until convergence 8:

. IV. ENERGY EFFICIENCY MAXIMIZATION

Clearly, the main use of the FD transmission in wireless communications systems is to improve their SE performance. While the SE has been recognized for long time as one of the most important design criteria, the EE is another metric that has drawn much attention recently due to increasing interest in green wireless networks. Thus, it is interesting to investigate the EE of the FD system considered in this paper. We note that since the DL and UL channels in the FD system operate simultaneously in an active mode, the FD system consumes more energy than a traditional HD one. Hence, if the precoders are not designed properly, the EE of the FD system can be much smaller than that of the HD system. For this purpose, we continue with the problem of precoder design for maximizing EE of the FD system with the BD scheme being used in the DL channel and evaluate its performance numerically in the next section.

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A. Power Model The energy spent to send data comes from many hardware components involving in the data transmission. According to [36] and [37], apart from data-dependent transmit power, the circuit power consumption, dissipated in all other electronics devices for signal processing (such as mixer, filter, analog-to-digital converter (ADC), digital-to-analog converter (DAC), lownoise amplifier (LNA), etc.) also plays an important role in EE performance. Noticeably, for short range communications such as micro or femto cells, the circuit power consumption can be comparable to or even dominate the actual transmit power for the data transmission. In this work, we adopt the linear power model as in [36] and [37], where the total power consumption at the BS in the DL channel is modeled as (32) is the power amplifier (PA) efficiency which where depends on the design and implementation of the PA, is the transmit power determined by linear precoders, i.e., . In (32), is called the circuit power, where is the dynamic circuit power consumption, corresponding to the power radiation of all circuit blocks and proportional to the number of the transmit antennas, and is the static circuit power spent by cooling system, power supply, etc. Similarly, the total power consumed at the transmitter of the th user in the UL channel is denoted as

We note that, like (7), problem in (35) is also nonconvex. Fortunately, we will show shortly that the convex approximation method presented in the previous section for maximizing SE is useful to obtain a locally optimal solution to (35). C. Proposed Joint Design for EE Maximization We first observe that the denominator in (34) is a linear function of and . From results reported in [20]–[22], problem (35) becomes tractable if the nominator, i.e., in (34) is a concave function of and . Motivated by this observation, we propose to iteratively replace by the lower bound given in (8). In this way, after iterations, the energy efficiency of the considered FD MU-MIMO system is lower bounded by (36) where

is the right side of (8). Consequently, the values of and in the st iteration are the solution to the following problem

(33) where transmission, and

is the transmit power allocated for data is the circuit power.

(37) For

B. Problem Formulation

mathematical

convenience, ,

let

us

define

and rewrite

In this subsection, our key objective is to optimize the EE by jointly designing linear precoders under SPC in the DL channel and PUPCs in the UL channel for the considered FD MU-MIMO system, referred to as the EE-optimal design. For simplicity, we only consider EE at the transmitter side in the active mode. Since the DL and UL channels operate simultaneously, we propose a definition of the overall energy efficiency of the FD MU-MIMO system as (34)

(37) as

(38)

where is the overall circuit power of the system, assumed to be constant for simplicity. In this paper, energy efficiency is measured in nats/J. The performance measure now is and the problem of interest (6) is reformulated as

We recall that is a concave function of and , and is an affine function of and . Accordingly, problem (38) belongs to the class of concave-convex fractional programs. There are several algorithms to globally solve concave-convex fractional programs such as the parametric convex program, parameter-free convex program or duality program, etc. [20]–[22]. In this paper, for simplicity, we adopt the parametric convex approach to solve problem (38). In this way, for a fixed , we define the following concave function as

(35)

(39)

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and consider the following problem

(40) The optimal value of (40) for a given is denoted by .A result of [21] states that and are optimal solutions to (38) if and only if . Moreover, it can be shown that is strictly decreasing in , meaning that the equation can be solved efficiently using bisection method over . From the expression of in (39), it is clear that the convex problem in (40) is similar to the formulation of the SE maximization in (9). Hence, our proposed algorithm for SE maximization presented in Section III can be slightly modified to solve (40) for a fixed . The proposed algorithm for EE maximization is outlined in Algorithm 4. Algorithm 4: The proposed EE-optimal design. 1: Randomly initialize

; tolerance

.

.

2:

B. Convergence Results

3: repeat 4:

Randomly initialize .

5:

while

and

such that

6: 7: Solve (40) using the proposed algorithm presented in Section III to find optimal solutions , and . 8:

then

if

9: 10:

else

12:

end if

13:

end while

15:

Fig. 2 shows the convergence rate of our proposed SE- and EE-optimal algorithms for a random channel realization. At the BS, the maximum transmit power is set to , dynamic circuit power to . The is fixed at 0 dB. Each point in Figs. 2(a) and 2(b) is computed using Algorithms 2 and 3, in which the error tolerance for convergence is set to . For Algorithm 3, we use a constant step size . It is observed that our proposed algorithms for maximizing SE (Algorithm 1) and EE (Algorithm 4) exhibit monotonic convergence to a locally optimal point within a few iterations. This convergence rate is typical for other channel realizations. C. Spectral Efficiency

11:

14:

the proposed designs over the conventional HD MU-MIMO scheme in terms of SE and EE. The channels are assumed to be standard Rayleigh fading of unit variance, i.e., the channel coefficients of and are modeled as i.i.d. complex Gaussian random variables with zero mean and unit variance. To the best of our knowledge, no standard reference channel model for the SI has been reported. Hence, in this paper, the entries of are simply generated as ), where represents the capability of the SI cancellation technique. Unless otherwise mentioned, the number of users in the DL and UL channels is set to with 2 antennas for each user in the DL and UL channels, and the BS is equipped with antennas, 4 of which are used in the DL and UL channels, respectively, i.e., and . The initial values for in the proposed iterative method are generated randomly for each channel realization. For the sake of simplicity, we assume , , and for all and , and , where . In addition, the noise power of each user in the DL channel and BS is taken as unity, i.e., . The parameters in the linear power model are set to , , and . For a fair comparison, the BS in the HD system is allowed to use all the antennas in both DL and UL channels to communicate with the users.

. Update covariance matrices:

.

16: until convergence. 17: Calculate the linear precoder for the user in the DL channel: where is obtained from . Cholesky decomposition to V. PERFORMANCE EVALUATION A. System Parameters In this section, we carry out numerical simulations to investigate the potential gains of the FD MU-MIMO system with

In Fig. 3, we plot the overall SE of the FD and conventional HD systems with the SE-optimal designs versus self interference channel gain for a fixed maximum transmit power at the BS for two scenarios.1 In the first setup, the number of antennas at the BS is set to , and the DL and UL channels serve 2 users each. In the second one, the number of the BS is reduced to , and the DL and UL channels serve 1 user each. The purpose of considering the two cases is to study the spatial multiplexing gain provided by the FD MU-MIMO, i.e., how the SE varies with and . As can be seen in Fig. 3, the SE of the FD system with Algorithm 1 is greatly improved, about 25% when is relatively small. It is worth mentioning that even when the SI is negligible, the FD system cannot attain a double gain of SE as compared to the HD 1For the SE-optimal design of the HD system, the conventional water-filling procedure is employed to maximize the SE of the BD scheme in the DL channel [15], and the iterative water filling algorithm is applied to obtain the SE of the UL channel [38].

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Fig. 3. Impact of self interference on spectral efficiency of the FD MU-MIMO system.

Fig. 2. Convergence rate of Algorithm 1 (for spectral efficiency maximization) and Algorithm 4 (for energy efficiency maximization). (a) Spectral efficiency; (b) Energy efficiency.

system. This is due to the fact that the BS of the HD system is allowed to use all antennas to send data in the DL channel (while that of the FD system only uses half of them). Moreover, the SE of the FD system is degraded and becomes lower than that of HD system as the SI increases. The main reason is that if the SI suppression is not efficient enough, the high power from the transmitted signals in the DL channel overwhelms the received signals in the UL channel, which makes it erroneous to recover the signals from users in the UL channel. This results in a loss of the SE of the FD system. We also observe that the SE of the FD system for the case nearly doubles that for the case , implying that the proposed precoder design for the FD system can exploit the spatial multiplexing gain. Fig. 4 compares the SE of the FD and conventional HD systems for a fixed , but in this case we vary the maximum transmit power at the BS, (the maximum transmit power at each user in the UL channel, i.e., , is changed accordingly due to their relationship mentioned earlier). We can see that increasing also leads to a remarkable gain of the SE of the FD over that of the HD system.

Fig. 4. Spectral efficiency versus maximum transmit power of the BS,

.

D. Energy Efficiency In Fig. 5, we show the comparison between the EE of the considered FD system and that of the HD one, to which the proposed EE-optimal scheme is applied, as a function of self interference channel gain . At the BS, we set the maximum transmit power to be and the dynamic circuit power to . An interesting, but unsurprising, observation is that HD system achieves better EE than the FD one, which can be explained as follows. Comparing the both systems for the specific setup of Fig. 5, we have numerically seen that the FD system increases the SE approximately by 16.7%, but it consumes 28% more energy. This then leads to an EE gain of 9.7% for the HD system as shown in Fig. 5. We recall that the DL and UL channels of the FD system are active continuously while those of the HD one operate in an on-off fashion. As a result, the HD system consumes less energy than the FD one for the same operation period. In Fig. 6, we investigate the EE of the FD system with the SE- and EE-optimal designs as a function of the dynamic circuit power at the BS. The maximum transmit power of the BS

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Fig. 5. Impact of self interference on energy efficiency of the FD MU-MIMO system.

Fig. 6. Energy efficiency versus dynamic circuit power of the BS,

.

is set to and is fixed at 20 dB. As mentioned above, the circuit power model of the whole system is defined as where , , and . By this relationship, Fig. 6 implicitly shows the performance of EE over the total circuit power consumption , which is a key parameter in designing energy efficient schemes. Clearly, the SE-optimal precoder design is inferior to the EE-optimal one. This is because the SE-optimal precoder design does not take into account the effect of other sources of power consumption. We observe that the EE of the two designs is degraded as increases. When is small, the EE-optimal precoder design yields the remarkable gain over the SE-optimal design. When is sufficiently large, the SE- and EE-optimal designs obtain nearly the same EE performance. We note that if is large, the totally consumed power is mostly constituted by the circuit power. Thus, from (34), we can see that maximizing the EE is equivalent to maximizing the SE. To obtain further insights into the performance of the transmission designs of the different systems in comparison, we evaluate the EE of two approaches with the maximum transmit power at the BS , as shown in Fig. 7. The dynamic circuit

Fig. 7. Energy efficiency versus maximum transmit power of the BS,

.

power of the BS is fixed at 38 dBm and the SI channel gain is chosen to be . We observe that, in the low transmit power regime, the EE of the two designs increases as increases. This is due to the fact that the power consumption is largely determined by the circuit power consumption in this region. An increase in transmit power leads to an improvement on the SE, and thus also on the EE. Particularly, the EE of the FD SE-optimal design achieves maximum EE for a certain transmit power, and decreases after that. This is because the SE-optimal design always transmits with full power to maximize the SE. However, the SE increases only logarithmically with the transmit power while the total power consumption grows up linearly with the transmit power (in high transmit power regime). As a result, the EE of the SE-optimal design is reduced. On the other hand, the EE of the FD EE-optimal design remains constant after reaching its peak value. By the optimization mechanism of the EE-optimal design, the EE-optimal design can find an optimal transmit power . If , the EE-optimal design will not transmit at full power, and thus its EE remains unchanged. VI. CONCLUSION We have investigated potential gains of the FD MU MIMO system in which the DL and UL channels are designed to operate in the FD mode. Taking the natural coupling of the DL and UL channels into consideration, we have presented joint designs of linear precoders to optimize SE and EE subject to a SPC in the DL channel and PUPCs in the UL channel, referred to as the SE-optimal and EE-optimal designs, respectively. For the SE-optimal design, since the problem is formulated as a nonconvex program, we have proposed an iterative algorithm to find the suboptimal solutions based on a convex relaxation method. Particularly, in each iteration, the relaxed convex program is solved using the alternating and dual decomposition method, by which we obtain the analytical solutions for precoders with given dual variables. For the EE-optimal design, we first approximate the nonlinear fractional program of EE as a concave-convex fractional program, which is then transformed into a parametric convex program by applying the parametric approach. We have shown that the resulting optimization problem

NGUYEN et al.: PRECODING FOR FULL DUPLEX MULTIUSER MIMO SYSTEMS: SPECTRAL AND ENERGY EFFICIENCY MAXIMIZATION

for EE maximization can be efficiently solved by applying the main techniques in the SE-optimal design. The numerical results indicate that the proposed joint linear precoder designs for the considered FD MU-MIMO system achieve better SE and slightly lower EE than the HD system.

Taking

the

first

of to

and set it to zero, we have

APPENDIX II.

derivative with respect

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(43)

ON THE RANK OF OPTIMAL SOLUTIONS TO (7)

As mentioned earlier, the proposed precoder design offers optimal solutions that meet the rank constraint automatically. In this appendix, we show that the iterative algorithm in which a generic method is used to compute the optimal precoders in each iteration can also yield the same result for a specific condition. Explicitly, let be the optimal dual variable associated with the sum power constraint and ’s be the optimal solutions to (9) after the iterative procedure converges. Then, we have the following claim. Claim 1: If , then it follows that , for all . The proof of Claim 1 follows similar arguments as in [39], [40]. We begin by forming the Lagrangian function of (9) for step of the iterative procedure, which is given by

where . In fact, (43) is the stationary property of the optimal solutions for (9) and we have used the fact that . Next, , we using the complementary slackness property obtain

(44) From the definition of

, the following inequality is obvious

(45)

(41) where is the dual variables associated with the power constraint in the UL channel, and are the dual variables for the positive semidefinite constraints and , respectively. We can then rewrite (41) as

(42)

Note that the equality in (45) holds true due to the fact that the iterative process has converged. Since , we can conclude that . Then, it follows from (44) that , which completes the proof. Interestingly, we observe that always holds true in the simulation setups considered in this paper and in various setups as well. REFERENCES [1] D. Nguyen, L.-N. Tran, P. Pirinen, and M. Latva-Aho, “Transmission strategies for full duplex multiuser MIMO systems,” in Proc. Int. Workshop Small Cell Wireless Netw. (IEEE ICC 2012), Jun. 2012, pp. 6825–6829. [2] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes (1),” in Proc. IEEE ICC’93, Geneva, Switzerland, May 1993, pp. 1064–1070. [3] R. G. Gallager, “Low density parity check codes,” Trans. IRE Professional Group Inf. Theory, vol. IT-8, pp. 21–28, Jan. 1962. [4] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, pp. 585–595, Nov. 1999. [5] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, pp. 311–335, Mar. 1998. [6] D. Gesbert, M. Kountouris, R. W. Heath, Jr., C. B. Chae, and T. Sälzer, “Shifting the MIMO paradigm,” IEEE Signal Process. Mag., vol. 24, no. 5, pp. 36–46, Sep. 2007. [7] J. I. Choi, M. Jain, K. Srinivasan, P. Levis, and S. Katti, “Achieving single channel, full duplex wireless communication,” in Proc. MOBICOM’10, Chicago, IL, USA, Sep. 2010, pp. 1–12. [8] M. Jain, J. I. Choi, T. Kim, D. Bharadia, S. Seth, K. Srinivasan, P. Levis, S. Katti, and P. Sinha, “Practical, real-time, full duplex wireless,” in Proc. MOBICOM’11, Las Vegas, NV, USA, Sep. 2011, pp. 301–312. [9] M. Duarte and A. Sabharwal, “Full-duplex wireless communications using off-the-shelf radios: Feasibility and first results,” in Proc. Asilomar Conf. Signals Syst. Comput. (ASILOMAR), CA, USA, Nov. 2010, pp. 1558–1562.

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[37] J. Xu, L. Qiu, and C. Yu, “Improving energy efficiency through multimode transmission in the downlink MIMO systems,” EURASIP J. Wireless Commun. Netw., 2011. [38] W. Yu, W. Rhee, S. Boyd, and J. Cioffi, “Iterative water-filling for Gaussian vector multiple-access channels,” IEEE Trans. Inf. Theory, vol. 50, no. 1, pp. 145–152, Jan. 2004. [39] M. Vu, “MISO capacity with per-antenna power constraint,” IEEE Trans. Commun., vol. 59, no. 5, pp. 1268–1274, May 2011. [40] L.-N. Tran, M. Juntti, M. Bengtsson, and B. Ottersten, “Successive zero-forcing DPC with per-antenna power constraint: Optimal and suboptimal designs,” in Proc. IEEE ICC 2012, Jun. 2012, pp. 3746–3751. Dan Nguyen (S’12) received the B.S. degree in Electrical Engineering from Ho Chi Minh National University of Technology, Vietnam in 2003, and M.S in Radio Engineering from Kyung Hee University, Republic of Korea, in 2008. Since August 2011, she has been working as a researcher toward the PhD degree in Department of Communications engineering at University of Oulu, Finland. Her current research interests focus on energy efficiency communications, and full duplex wireless systems.

Le-Nam Tran (M’10) received the B.S. degree in Electrical Engineering from Ho Chi Minh National University of Technology, Vietnam in 2003, and M.S and PhD in Radio Engineering from Kyung Hee University, Republic of Korea, in 2006 and 2009, respectively. In 2009, he joined the Department of Electrical Engineering, Kyung Hee University, Republic of Korea, as a lecturer. From September 2010 to July 2011, he was a postdoc fellow at the Signal Processing Laboratory, ACCESS Linnaeus Centre, KTH Royal Institute of Technology, Sweden. Since August 2011, he has been with Centre for Wireless Communications and Department of Communications Engineering, University of Oulu, Finland. His current research interests include multiuser MIMO systems, energy efficient communications, and full duplex transmission. He received the Best Paper Award from IITA in August 2005. Pekka Pirinen (S’96–M’05–SM’09) received Master of Science, Licentiate of Science, and Doctor of Science in Technology degrees in electrical engineering from the University of Oulu, Finland, in 1995, 1998, and 2006, respectively. Since 1994 he has been with the Telecommunication Laboratory and since 1995 with the Centre for Wireless Communications, University of Oulu, working in various European and national wireless communications research projects. Currently he is a Senior Research Fellow at the University of Oulu. His research interests cover multi-access protocols, capacity evaluation, resource sharing, heterogeneous networks, full duplex systems and small cells. Matti Latva-aho (S’96–M’98–SM’06) was born in Kuivaniemi, Finland in 1968. He received the M.Sc., Lic.Tech. and Dr. Tech (Hons.) degrees in Electrical Engineering from the University of Oulu, Finland in 1992, 1996 and 1998, respectively. From 1992 to 1993, he was a Research Engineer at Nokia Mobile Phones, Oulu, Finland. During the years 1994–1998 he was a Research Scientist at Telecommunication Laboratory and Centre for Wireless Communications at the University of Oulu. Currently he is the Department Chair Professor of Digital Transmission Techniques at the University of Oulu, Department for Communications Engineering. Prof. Latva-aho was Director of Centre for Wireless Communications at the University of Oulu during the years 1998–2006. His research interests are related to mobile broadband wireless communication systems. Prof. Latva-aho has published over 200 conference or journal papers in the field of wireless communications. He has been TPC Chairman for PIMRC’06, TPC Co-Chairman for ChinaCom’07 and General Chairman for WPMC’08. He acted as the Chairman and vice-chairman of IEEE Communications Finland Chapter in 2000–2003.