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, andMatti 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 withmultiple users in the downlink (DL) and uplink (UL) channels onthe same system resources. The system model of consideration hasbeen 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 demon-strated that the SI can be suppressed to a degree that possiblyallows for FD transmission. This paper goes one step further inexploring the potential gains in terms of the spectral efficiency(SE) and energy efficiency (EE) that can be brought by the FDMU-MIMO model. Toward this end, we propose low-complexitydesigns for maximizing the SE and EE, and evaluate their per-formance numerically. For the SE maximization problem, wepresent an iterative design that obtains a locally optimal solutionbased on a sequential convex approximation method. In this way,the nonconvex precoder design problem is approximated by aconvex program at each iteration. Then, we propose a numericalalgorithm to solve the resulting convex program based on thealternating and dual decomposition approaches, where analyticalexpressions for precoders are derived. For the EE maximizationproblem, using the same method, we first transform it into a con-cave-convex fractional program, which then can be reformulatedas a convex program using the parametric approach. We willshow that the resulting problem can be solved similarly to theSE maximization problem. Numerical results demonstrate that,compared to a half duplex system, the FD system of interest withthe proposed designs achieves a better SE and a slightly smallerEE 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 ca-pacity that wireless communications networks can offer.

In the perspective of physical layer design, this was accom-

plished by the inventions of many powerful techniques suchas strong channel coding (e.g., turbo code, low-density parity-check code (LDPC) [2], [3]) and, particularly, multiple-inputmultiple output (MIMO) technologies [4], [5]. In current andemerging cellular networks, MIMO is realized in the form ofmultiuser MIMO (MU-MIMO) in the downlink (DL) and up-link (UL) channels. MU-MIMO techniques can simultaneouslyexploit the benefits brought by MIMO and multiuser diversitygains [6].Although MU-MIMO has been shown to be a promising

approach to increasing the system throughput, we seem toreach the boundary that MU-MIMO can provide in reality innot-so-far future. The fact is that we cannot integrate as manyantennas as we want in both ends of a communications linkdue to practical limitations. Also, many other issues will arisewhen expanding the operating bandwidth or transmit power.Recall that the DL and UL channels of current cellular systemsare designed to operate in half duplex (HD) mode, where theDL and UL channels run separately in time domain (timedivision duplex, TDD) or in frequency domain (frequencydivision duplex, FDD). Hence, such systems do not achieve themaximal spectral efficiency yet. So, we are still able to enhancethe system capacity of cellular networks by allowing the DLand UL channels to work simultaneously over the same radioresources, referred to as full duplex (FD) transmission. Thatis to say, in a TDD system, DL and UL channels are designedto operate in the same time slot, and in a FDD system in thesame bandwidth. Such a design, if possible, can greatly boostthe system capacity of the HD systems and also resolve manyproblems of existing wireless communications networks suchas reducing hidden terminals, congestion due to medium accesscontrol (MAC) scheduling, and large delays [7], [8].The idea of designing the DL and UL channels that func-

tion over the same system resources basically means that basestations (BSs) should be able to transmit and receive data atthe same time on the same frequency. This concept has beenthought to be infeasible since the self-interference (SI) betweentransmit and receive antennas at BSs may severely affect theperformance of the UL channel. The lack of advanced hardwaretechniques that can efficiently suppress the SI has remained untilrecently, when pioneer studies demonstrate the feasibility of apoint-to-point FD wireless transmission in practice by elimi-nating the SI via advanced interference cancellation techniques[7]–[10]. This has resulted in many researchers studying theFD design for the future wireless communications. In partic-ular, some FD approaches have been proposed in the context ofMU-MIMO relay systems which have the potential to extend

cell coverage and enhance the cell-edge throughput [11]–[13].However, these schemes are only dedicated to one directiontransmission and mostly focus on the DL channel. Another mo-tivation to study FD schemes is due to a growing trend towardsdrastically smaller coverage cells which result in rather smallpower 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

FDMU-MIMO system have not been reported yet. By the singlecell FD MU-MIMO system, we mean a system where a BStransmits and receives signals from several HD terminals in theDL and UL channels over the same resources. In this paper, wemake an attempt to understand the benefits that can be achievedby the use of the FD-based transceivers. In traditional HD sys-tems, the optimal transmit strategy for the DL channel is with theuse of dirty paper coding, but it requires high complexity [14].Hence, we consider linear precoding for the DL channel. Wenote that the optimal linear precoder design for the DL channelis challenging due to the presence of multiuser interference, andthus many suboptimal schemes such as zero forcing (ZF) [15]and minimum mean square error (MMSE)[16] have been pro-posed. In our work, we adopt the ZFmethod proposed in [15] forthe DL channel for simplicity. For the UL channel, we assumea minimum mean square error with successive interference can-cellation (MMSE-SIC) receiver, which is known to achieve thecapacity of the UL channel [17]. Due to the existence of a certainamount of the SI, the DL and UL channels are coupled, makingthe design of transmission strategies for the FD-based BS diffi-cult.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 rateper bandwidth unit, has been a main focus for wireless commu-nications design. Very recently, EE, defined as the number ofbits transmitted per an energy unit, has received significant at-tention due to growing interest in green communications, whereenergy consumption is the ultimate goal to be optimized. In apractical communications system, besides actual transmit powerallocated for conveying data, circuit power radiated from elec-tronics devices for signal processing functionality also plays animportant role in EE transmission design. Traditionally, we as-sume the sum power constraint (SPC) in the DL channel andper user power constraints (PUPCs) in the UL channel. First,the SE and EE maximization problems are formulated as non-convex problems. Then, we propose effective algorithms basedon the concept of a sequential convex approximation [18]. Ourcontributions in this paper include, but are not limited to, thefollowing.• For the problem of the SE maximization, using the firstorder approximation, we derive a lower bound of the SEin each iteration, which turns out to be convex. Then,linear precoders are found to maximize the lower boundby solving the dual problem with the block coordinateascent (BCA) and dual decomposition methods [19]. Theproposed iterative design, referred to as the SE-optimaldesign, iteratively increases this lower bound until con-vergence. We demonstrate by numerical results that the

SE of the FD MU-MIMO system is superior to that of theconventional HD system.

• The EE maximization problem at hand belongs to the classof nonlinear fractional programs, and thus the optimal so-lution is generally hard to find. Herein, we propose a sub-optimal precoder design using the framework of concave-convex fractional programming, presented in [20]–[22].For this purpose, we approximate the EE maximizationproblem to be a concave-convex fractional program usingthe convex relaxation method as employed in the SE-op-timal design. Then the resulting problem is transformedinto an equivalent convex program by applying some ap-propriate transformations. As we show later, the resultingconvex programming for EE maximization has a similarstructure to that for SE maximization, and thus the mainsteps in the SE-optimal design can be applied. We referto the iterative precoder design for maximizing the EE asthe EE-optimal design. The simulation results reveal thatthe EE-optimal design outperforms the SE-optimal designin terms of EE. Moreover, the FD system yields slightlylower EE than the HD system when the SI is efficientlysuppressed.

As themain goal of this paper is to investigate the potential gainsof the FDMU-MIMO system along with the proposed precoderdesigns for the SE and EE maximization, we are not putting anemphasis on proposing a hardware technique that cancels theSI. Although other issues may still exist, we believe that the FDMU-MIMO system is a potential solution to improve the perfor-mance of current cellular networks. By the results presented inthis paper, we also call for in-depth studies of FD MU-MIMOsystems that, e.g., find the boundary SE of FD MU-MIMO sys-tems.The rest of this paper is organized as follows. Section II intro-

duces system model and linear precoding techniques. The pro-posed algorithms for the SE and EE maximization are derivedin Sections III and IV, respectively. In Section V, we presentour 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 trans-pose of , respectively; and are the trace anddeterminant of , respectively; means that is apositive semidefinite matrix.

II. SYSTEM MODEL AND LINEAR PRECODING TECHNIQUES

Consider a single cell FD MU-MIMO system with a BSsending data to users in the DL channel, and receivingdata from users in the UL channel at the same time onthe same frequency, as shown in Fig. 1. The sets of users inthe DL and UL channels are denoted byand , respectively. We assume that the BS isequipped with antennas, of which antennas are used totransmit data in the DL channel and antennas are used toreceive the data in the UL one, . The numberof antennas at user , , in the DL channel is

Fig. 1. FD MU-MIMO system model.

denoted by , and that at user , , in the ULchannel is denoted by .The design of transmission strategies for the FD MU-MIMO

system of interest is challenging. The reason is that there alwaysexists a certain amount of the SI between transmit antennasand receive ones at the BS after applying hardware cancella-tion techniques. A possibility is to design the DL and UL chan-nels separately without accounting for the SI. However, suchseparate designs of the DL and UL channels become ineffectivedue to an excessive amount of the induced SI [1]. In additionto the SI from the DL to the UL channel, the co-channel inter-ference caused by users in the UL channel to those in the DLchannel should also be considered. In this work, for simplicity,we assume that the users in the DL and UL channels are geo-graphically separated, meaning that co-channel interference isignore. A general design that takes both the SI and co-channelinterference into consideration is an interesting problem for fu-ture work.Assuming linear precoding at the BS, the received signal at

user is

(1)

where , , andare the vector of transmitted signal, the linear precoder andthe channel matrix of user , respectively. We can further as-sume that without loss of generality. The back-ground noise is assumed to be a zero-mean addi-tive white Gaussian noise (AWGN) vector with the single-sidednoise spectral density . Assuming perfect channel state in-formation (CSI) at both transmitter and receiver, the sum rate ofthe DL channel is given by

(2)

Throughout the paper, sum rate (spectral efficiency) is mea-sured in nats/s/Hz. The optimal linear precoder design is hard

to find even for the DL channel alone due to the multiuser infer-ence. In fact, the problem of precoder design for maximizing

in (2) has been proved to be NP-hard in [23]. Hence, tosimplify the design, we adopt the block diagonalization (BD)scheme, proposed in [15], which is widely used thanks to itssimplicity. In the BD scheme, ’s are designed such that

for all , i.e., the multiuser interference iscompletely canceled. For the ease of description, let us define

for user , and de-

note by , , an orthog-onal basis of the null space of . Then, the ZF constraintsin BD imply that , for all . Thus, we can write

, where and reduces to

(3)

where is the effective channel matrixof user (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 channelallows us to arrive at a formulation that can be solved usingclosed-form expressions. This possibility eliminates the need ofinstalling a generic optimization package.Clearly, the BD scheme is feasible if

due to the condition for the exis-

tence of the null space of for all . This dimensionalityconstraint means that the BS can simultaneously transmit datato a limited number of users. In case that is larger than thesupportable number of users, a group of users must be chosenby a user scheduling algorithm. We note that existing userscheduling algorithms for BD in the literature (see [24]–[26]and references therein) can be employed in combinationwith our proposed designs. However, these user schedulingalgorithms, which were only devised for the DL channel, donot account for the SI. Obviously, a user selection algorithmthat considers the SI may result in better performance anddesigning such an algorithm constitutes a rich area for futureresearch.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)

where and is an AWGN vector with. In (4), specifies the SI channel from the

transmit antennas to the receive ones at the BS, and its entriesdepend on the effectiveness of the SI cancellation techniques. Inthis paper, we treat the SI as the background noise and assumeperfect CSI at the BS and users. As a result, the sum rate of theUL channel with a MMSE-SIC receiver is

(5)where is the transmit covariance matrix of user.Due the term in (5), the performance of

the DL and UL channels is coupled. If one only concentrates onmaximizing , this interference term can be so destructive thatit can make the UL channel useless. Thus, as one of our maincontributions in this paper, we consider a joint design of the DLand UL channels. Conventionally, we assume a SPC at the BS inthe DL channel and PUPCs in the UL channel, i.e., each user issubject to an individual power constraint. Letbe the performance measure of interest. Then, the problem ofthe joint design of and is generally expressed as

(6)

where is the maximum transmit power of the BS and isthe maximum transmit power of the user . In the followingsections, we present joint designs for maximizing the spectraland energy efficiency.

III. SPECTRAL EFFICIENCY MAXIMIZATION

In this section, we propose a low-complexity joint design forthe SE maximization problem, referred to as the SE-optimal de-sign. For this case, , and (6) be-comes

(7)

The optimization problem above is also known as the sum ratemaximization problem. It should be noted that all the constraintsare convex with respect to and . Moreover,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 solvein general. Solving (7) globally requires in-depth knowledgeof global optimization methods, which is beyond scope of thispaper. Instead, we resort to a local optimization method, i.e., we

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

A. Proposed Joint Design for SE Maximization

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

(8)

where and

. In (8), wehave used the inequality

for , where is an arbitrary oper-ating point. This inequality is due to the concavity offunction [28]. In fact, the idea of iteratively maximizing lowerbounds of a nonconvex function has been used in the literaturefor different contexts, e.g., in [29]–[31]. In the st iterationof the proposed algorithm, are found to maximize

the lower bound of the SE. Mathematically, are thesolutions of the following problem

(9)

where is the right side of (8) with the con-stant being ignored since it does not affect the optimizationproblem in (9). With the convex approximation of , problem(9) now becomes a convex program in each iteration, whichcan be solved using general standard convex optimization pack-ages, e.g., CVX [32] or YALMIP [33]. The values ofare updated until the iterative procedure converges. Regardingthe use of a generic method to solve (9), the rank constraints,

, for , must be guaranteed sothat we are able to extract from . This problem is fur-ther discussed in the Appendix.Generic methods do not exploit the specific structure of the

problem and thus require high computational complexity.More-over, such methods do not offer useful insights into the optimalsolutions. In this paper, exploiting the specific expression of

, we develop an efficient iterative algorithmto solve (9), in which analytical expressions of and are

found in each iteration. Moreover, the proposed precoder designguarantees that the rank constraints, for all

’s, are satisfied automatically. To start with, we rewrite (9)as

(10)

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

(11)

where is the dual variable associated with the constraint. Consequently, the dual objective of (10), de-

noted by , is the optimal value of the following problem

(12)

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 solvingits dual problem in (13). Due to the fact that a subgradient of

is given by

(14)

and that is a scalar, the minimization of over canbe carried out efficiently using one dimension search method,e.g., bisection method [34]. Thus, solving (9) boils down tofinding an efficient algorithm to evaluate , i.e., solving (12)efficiently. Before proceeding further, we outline the proposedjoint design of the DL and UL channels for the SE maximiza-tion problem in Algorithm 1.The proposed SE-optimal design is guaranteed to converge

since the lower bound is increased after every iteration, andthe total SE of system is bounded above due to the power con-straints. Moreover, due to the fact that the first order approxima-tion is employed, Algorithm 1 converges to a stationary point,i.e., a point that satisfies the Karush-Kuhn-Tucker (KKT) con-ditions of (7). The detailed proof is provided in [18].

Algorithm 1 The proposed SE-optimal design for theconsidered FD system.

1: Generate initial points for ; tolerance .

2: .

3: repeat

4: Generate initial points for , and .

5: while do

6:

7: Solve (12) to find optimal solutionsand using the alternating optimization algorithmpresented in Section III-B.

8: if then

9:

10: else

11:

12: end if

13: end while

14: .

15: Update covariance matrices: .

16: until convergence.

17: Apply the Cholesky decomposition to to find ,and calculate the precoder for each user inthe DL channel.

B. Alternating Optimization

As the core of Algorithm 1, we now present an efficient al-gorithm to solve (12). We observe that the problem formula-tion in (12) lends itself to the BCA method since the constraintsfor individual and are separable. Moreover, as wewill show shortly, the optimization of one variable, when othersare fixed, can be expressed through analytical forms. Particu-larizing the BCA method to solve (12) leads to two differentcases. In the first case, for each , we find

that solves problem (12) by treating othersand as constants. Explicitly, we need to solvethe following problem

(15)

where ,

, and

. Itis easy to see that problem admits a solution based onthe water-filling procedure. To be specific, let and ,

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

(16)

where

(17)

consists of right singular vectors of , and the waterlevel 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 othersand are held fixed. This

amounts to solving the following problem

(18)

where ,,

, and

.The alternating procedure to solve (12) is outlined in Algorithm2.

Algorithm 2: The proposed alternating optimization algorithm.

1: Initialize: ; .

2: repeat

3: for to do

4: Solve using the water-filling algorithm to findwhile keeping all other variables fixed.

5: end for

6: for to do

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

8: end for

9: until desired accuracy is reached.

Since each iteration of the alternating optimization algorithmincreases the objective of (12), it converges to a locally optimalsolution of problem (12), which is also the optimal solution ofproblem (9) due to its convexity.

As mentioned in line 7 of Algorithm 2, we now present Al-gorithm 3 that can solve efficiently, i.e., via analyticalexpressions. First, under the framework of dual decompositionmethod, we rewrite (18) as

(19)

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 decom-position method, and are local versions of thecomplicating variable , along with a consistency constraint

that requires the two local versions to beequal. We also omit the constant in (18) since it does notaffect the optimization of . Next, let be the Lagrangemultiplier associated with the consistency constraint. Then, thepartial Lagrangian function of (19) is written as

(20)

and the dual function is the optimal value of the followingproblem

(21)

Since the objective function and all constraints in (21) are sep-arable, it can be decomposed into dual subproblems 1 and 2 asfollows

(22)

and

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

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

(24)

where is the dual variable associated with the constraint,, , and .

Let be an SVD of, where . Then, the optimal solution

to subproblem 1 is found as

(25)

where , and

. We note thatfor subproblem 1 to be solvable, must satisfy the constraint

.Similarly, let , and

, the objective of (23) is rewritten as

(26)

and thus the optimal of subproblem 2 is simply given by

(27)

where is theSVD of , , and

. We also note that

subproblem 2 is solvable if .We have shown that subproblems 1 and 2 can be solved ef-

ficiently using the water-filling algorithm with the fixed waterlevel. The problem now is to find the optimal which mini-mizes the dual problem of (19), which is formulated as

(28)

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

updated until convergence is reached. To be specific, at thest iteration, is expressed as

(29)

where is a positive step size, and is a proper subgra-dient. Explicitly, if is feasible, is a subgradient of ,i.e., . Otherwise, is set to be a subgra-dient of the violated constraint function at as follows [35].

(30)

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

(31)

where 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). Theproposed dual decomposition method to solve problem issummarized in Algorithm 3.

Algorithm 3: A dual decomposition method for solving .

1: Initialize such that .

2: .

3: repeat

4: Compute and using (25) and (27).

5: Update with and givenin (30) and (31).

6: .

7: until convergence

8: .

IV. ENERGY EFFICIENCY MAXIMIZATION

Clearly, the main use of the FD transmission in wireless com-munications systems is to improve their SE performance. Whilethe SE has been recognized for long time as one of the most im-portant design criteria, the EE is another metric that has drawnmuch attention recently due to increasing interest in green wire-less networks. Thus, it is interesting to investigate the EE of theFD system considered in this paper. We note that since the DLand UL channels in the FD system operate simultaneously in anactive mode, the FD system consumes more energy than a tra-ditional HD one. Hence, if the precoders are not designed prop-erly, the EE of the FD system can be much smaller than that ofthe HD system. For this purpose, we continue with the problemof precoder design for maximizing EE of the FD system withthe BD scheme being used in the DL channel and evaluate itsperformance numerically in the next section.

A. Power Model

The energy spent to send data comes from many hardwarecomponents involving in the data transmission. According to[36] and [37], apart from data-dependent transmit power, the cir-cuit power consumption, dissipated in all other electronics de-vices for signal processing (such as mixer, filter, analog-to-dig-ital converter (ADC), digital-to-analog converter (DAC), low-noise amplifier (LNA), etc.) also plays an important role in EEperformance. Noticeably, for short range communications suchas micro or femto cells, the circuit power consumption can becomparable to or even dominate the actual transmit power forthe data transmission. In this work, we adopt the linear powermodel as in [36] and [37], where the total power consumptionat the BS in the DL channel is modeled as

(32)

where is the power amplifier (PA) efficiency whichdepends on the design and implementation of the PA, isthe transmit power determined by linear precoders, i.e.,

. In (32), is called the cir-cuit power, where is the dynamic circuit power consump-tion, corresponding to the power radiation of all circuit blocksand proportional to the number of the transmit antennas, and

is the static circuit power spent by cooling system, powersupply, etc. Similarly, the total power consumed at the trans-mitter of the th user in the UL channel is denoted as

(33)

where is the transmit power allocated for datatransmission, and is the circuit power.

B. Problem Formulation

In this subsection, our key objective is to optimize the EEby jointly designing linear precoders under SPC in the DLchannel and PUPCs in the UL channel for the considered FDMU-MIMO system, referred to as the EE-optimal design. Forsimplicity, we only consider EE at the transmitter side in theactive mode. Since the DL and UL channels operate simultane-ously, we propose a definition of the overall energy efficiencyof the FD MU-MIMO system as

(34)

where is the overall circuit powerof the system, assumed to be constant for simplicity. In thispaper, energy efficiency is measured in nats/J. The performancemeasure now is and the problem ofinterest (6) is reformulated as

(35)

We note that, like (7), problem in (35) is also nonconvex. For-tunately, we will show shortly that the convex approximationmethod presented in the previous section for maximizing SE isuseful 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 func-tion 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 . Motivatedby this observation, we propose to iteratively replaceby the lower bound given in (8). In this way, after iterations,the energy efficiency of the considered FD MU-MIMO systemis lower bounded by

(36)

where is the right side of (8). Consequently, the values ofand in the st iteration are the solution to the

following problem

(37)

For mathematical convenience, let us define,

and rewrite(37) as

(38)

We recall that is a concave function ofand , and is an affine function

of and . Accordingly, problem (38) belongs to theclass of concave-convex fractional programs. There are severalalgorithms to globally solve concave-convex fractional pro-grams such as the parametric convex program, parameter-freeconvex program or duality program, etc. [20]–[22]. In thispaper, for simplicity, we adopt the parametric convex approachto solve problem (38). In this way, for a fixed , we definethe following concave function as

(39)

and consider the following problem

(40)

The optimal value of (40) for a given is denoted by . Aresult of [21] states that and are optimal solutionsto (38) if and only if . Moreover, it can be shownthat 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 formulationof the SE maximization in (9). Hence, our proposed algorithmfor SE maximization presented in Section III can be slightlymodified to solve (40) for a fixed . The proposed algorithmfor EE maximization is outlined in Algorithm 4.

Algorithm 4: The proposed EE-optimal design.

1: Randomly initialize ; tolerance .

2: .

3: repeat

4: Randomly initialize and such that.

5: while

6:

7: Solve (40) using the proposed algorithm presented inSection III to find optimal solutions , and

.

8: if then

9:

10: else

11:

12: end if

13: end while

14: .

15: Update covariance matrices: .

16: until convergence.

17: Calculate the linear precoder for the user in the DLchannel: where is obtained fromCholesky decomposition to .

V. PERFORMANCE EVALUATION

A. System Parameters

In this section, we carry out numerical simulations to inves-tigate the potential gains of the FD MU-MIMO system with

the proposed designs over the conventional HD MU-MIMOscheme in terms of SE and EE. The channels are assumed to bestandard Rayleigh fading of unit variance, i.e., the channel coef-ficients of and are modeled as i.i.d. complex Gaussianrandom variables with zero mean and unit variance. To the bestof our knowledge, no standard reference channel model for theSI has been reported. Hence, in this paper, the entries of aresimply generated as ), where represents the capa-bility of the SI cancellation technique. Unless otherwise men-tioned, the number of users in the DL and UL channels is setto with 2 antennas for each user in the DL andUL channels, and the BS is equipped with antennas, 4of which are used in the DL and UL channels, respectively, i.e.,

and . The initial values for in the pro-posed iterative method are generated randomly for each channelrealization. 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 allthe antennas in both DL and UL channels to communicate withthe users.

B. Convergence Results

Fig. 2 shows the convergence rate of our proposed SE- andEE-optimal algorithms for a random channel realization. At theBS, the maximum transmit power is set to , dy-namic circuit power to . The is fixed at 0dB. Each point in Figs. 2(a) and 2(b) is computed using Algo-rithms 2 and 3, in which the error tolerance for convergenceis set to . For Algorithm 3, we use a constant step size

. It is observed that our proposed algorithms formaximizing SE (Algorithm 1) and EE (Algorithm 4) exhibitmonotonic convergence to a locally optimal point within a fewiterations. This convergence rate is typical for other channel re-alizations.

C. Spectral EfficiencyIn Fig. 3, we plot the overall SE of the FD and conventional

HD systems with the SE-optimal designs versus self interfer-ence channel gain for a fixed maximum transmit power atthe BS for two scenarios.1 In the first setup, thenumber of antennas at the BS is set to , and theDL and UL channels serve 2 users each. In the second one, thenumber of the BS is reduced to , and the DL andUL channels serve 1 user each. The purpose of considering thetwo cases is to study the spatial multiplexing gain provided bythe FD MU-MIMO, i.e., how the SE varies with and . Ascan be seen in Fig. 3, the SE of the FD system with Algorithm 1is greatly improved, about 25% when is relatively small. Itis worth mentioning that even when the SI is negligible, the FDsystem cannot attain a double gain of SE as compared to the HD

1For the SE-optimal design of the HD system, the conventional water-fillingprocedure 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 theUL channel [38].

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 al-lowed to use all antennas to send data in the DL channel (whilethat of the FD system only uses half of them). Moreover, the SEof the FD system is degraded and becomes lower than that ofHD system as the SI increases. The main reason is that if theSI suppression is not efficient enough, the high power from thetransmitted signals in the DL channel overwhelms the receivedsignals in the UL channel, which makes it erroneous to recoverthe signals from users in the UL channel. This results in a lossof the SE of the FD system. We also observe that the SE of theFD system for the case nearly doubles that for thecase , implying that the proposed precoder designfor the FD system can exploit the spatial multiplexing gain.Fig. 4 compares the SE of the FD and conventional HD sys-

tems for a fixed , but in this case we vary themaximum transmit power at the BS, (the maximum transmitpower at each user in the UL channel, i.e., , is changed ac-cordingly due to their relationship mentioned earlier). We cansee that increasing also leads to a remarkable gain of the SEof the FD over that of the HD system.

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

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 theconsidered FD system and that of the HD one, to which the pro-posed EE-optimal scheme is applied, as a function of self in-terference channel gain . At the BS, we set the maximumtransmit power to be and the dynamic circuitpower to . An interesting, but unsurprising, ob-servation is that HD system achieves better EE than the FD one,which can be explained as follows. Comparing the both systemsfor the specific setup of Fig. 5, we have numerically seen thatthe FD system increases the SE approximately by 16.7%, but itconsumes 28% more energy. This then leads to an EE gain of9.7% for the HD system as shown in Fig. 5. We recall that theDL and UL channels of the FD system are active continuouslywhile those of the HD one operate in an on-off fashion. As a re-sult, the HD system consumes less energy than the FD one forthe 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 circuitpower at the BS. The maximum transmit power of the BS

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

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

is set to and is fixed at 20 dB. As mentionedabove, the circuit power model of the whole system is definedas where ,

, and . By this relation-ship, Fig. 6 implicitly shows the performance of EE over thetotal circuit power consumption , which is a key param-eter in designing energy efficient schemes. Clearly, the SE-op-timal precoder design is inferior to the EE-optimal one. Thisis because the SE-optimal precoder design does not take intoaccount the effect of other sources of power consumption. Weobserve that the EE of the two designs is degraded as in-creases. When is small, the EE-optimal precoder designyields the remarkable gain over the SE-optimal design. When

is sufficiently large, the SE- and EE-optimal designs ob-tain nearly the same EE performance. We note that if islarge, the totally consumed power is mostly constituted by thecircuit power. Thus, from (34), we can see that maximizing theEE is equivalent to maximizing the SE.To obtain further insights into the performance of the trans-

mission designs of the different systems in comparison, weevaluate the EE of two approaches with the maximum transmitpower 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 channelgain is chosen to be . We observe that, in the lowtransmit power regime, the EE of the two designs increases asincreases. This is due to the fact that the power consumption

is largely determined by the circuit power consumption in thisregion. An increase in transmit power leads to an improvementon the SE, and thus also on the EE. Particularly, the EE of theFD SE-optimal design achieves maximum EE for a certaintransmit power, and decreases after that. This is because theSE-optimal design always transmits with full power to maxi-mize the SE. However, the SE increases only logarithmicallywith the transmit power while the total power consumptiongrows up linearly with the transmit power (in high transmitpower regime). As a result, the EE of the SE-optimal designis reduced. On the other hand, the EE of the FD EE-optimaldesign remains constant after reaching its peak value. Bythe optimization mechanism of the EE-optimal design, theEE-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 MIMOsystem in which the DL and UL channels are designed to op-erate in the FD mode. Taking the natural coupling of the DL andUL channels into consideration, we have presented joint designsof linear precoders to optimize SE and EE subject to a SPC inthe DL channel and PUPCs in the UL channel, referred to asthe SE-optimal and EE-optimal designs, respectively. For theSE-optimal design, since the problem is formulated as a non-convex program, we have proposed an iterative algorithm to findthe suboptimal solutions based on a convex relaxation method.Particularly, in each iteration, the relaxed convex program issolved using the alternating and dual decomposition method,by which we obtain the analytical solutions for precoders withgiven dual variables. For the EE-optimal design, we first ap-proximate the nonlinear fractional program of EE as a con-cave-convex fractional program, which is then transformed intoa parametric convex program by applying the parametric ap-proach. We have shown that the resulting optimization problem

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

APPENDIX

II. ON THE RANK OF OPTIMAL SOLUTIONS TO (7)

Asmentioned earlier, the proposed precoder design offers op-timal solutions that meet the rank constraint automatically. Inthis appendix, we show that the iterative algorithm in which ageneric method is used to compute the optimal precoders in eachiteration can also yield the same result for a specific condition.Explicitly, let be the optimal dual variable associated withthe sum power constraint and ’s bethe optimal solutions to (9) after the iterative procedure con-verges. 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) forstep of the iterative procedure, which is given by

(41)

where is the dual variables associated with the power con-straint in the UL channel, andare the dual variables for the positive semidefinite constraints

and , respectively. We can then rewrite (41)as

(42)

Taking the first derivative ofwith respect to

and set it to zero, we have

(43)

where. In fact, (43) is the stationary property of the op-

timal solutions for (9) and we have used the fact that. Next,

using the complementary slackness property , weobtain

(44)

From the definition of , the following inequality is obvious

(45)

Note that the equality in (45) holds true due to the fact that theiterative process has converged. Since , we can conclude

that .Then, it follows from (44) that

, which completes the proof. Interestingly, weobserve that always holds true in the simulation setupsconsidered 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 limiterror-correcting coding and decoding: Turbo-codes (1),” in Proc. IEEEICC’93, Geneva, Switzerland, May 1993, pp. 1064–1070.

[3] R. G. Gallager, “Low density parity check codes,” Trans. IRE Profes-sional 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 communicationsin 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, “Achievingsingle channel, full duplex wireless communication,” in Proc. MO-BICOM’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 wire-less,” in Proc. MOBICOM’11, Las Vegas, NV, USA, Sep. 2011, pp.301–312.

[9] M. Duarte and A. Sabharwal, “Full-duplex wireless communicationsusing off-the-shelf radios: Feasibility and first results,” in Proc.Asilomar Conf. Signals Syst. Comput. (ASILOMAR), CA, USA, Nov.2010, pp. 1558–1562.

[10] A. Sahai, G. Patel, and A. Sabharwal, “Pushing the limits of full-du-plex: design and real-time implementation” 2011 [Online]. Available:http://arxiv.org/abs/1107.0607

[11] J. Sangiamwong, T. Asai, J. Hagiwara, Y. Okumura, and T. Ohya,“Joint multi-filter design for full-duplexMU-MIMO relaying,” inProc.IEEE VTC 2009-Spring, Apr. 2009, pp. 1–5.

[12] J.-H. Lee and O.-S. Shin, “Distributed beamforming approach to full-duplex relay in multiuser MIMO transmission,” in Proc. WCNC 2012Workshop 4G Mobile Radio Access Netw., Apr. 2012, pp. 278–282.

[13] C.-H. Lee, J.-H. Lee, Y.-W. Kwak, Y.-H. Kim, and S.-C. Kim, “The re-alization of full duplex relay and sum rate analysis in multiuser MIMOrelay channel,” in Proc. IEEE VTC 2010-Fall, Sep. 2010, pp. 1–5.

[14] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region ofthe Gaussian multiple-input multiple-output broadcast channel,” IEEETrans. Inf. Theory, vol. 52, no. 9, pp. 3936–3964, Sep. 2006.

[15] Q. Spencer, A. Swindlehurst, and M. Haardt, “Zero-forcing methodsfor downlink spatial multiplexing in multiuser MIMO channels,” IEEETrans. Signal Process., vol. 52, no. 2, pp. 461–471, Feb. 2004.

[16] C. Peel, B. Hochwald, and A. Swindlehurst, “A vector-perturbationtechnique for near-capacity multiantenna multiuser communication-Part I: Channel inversion and regularization,” IEEE Trans. Commun.,vol. 53, no. 1, pp. 195–202, Jan. 2005.

[17] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.Cambridge, U.K.: Cambridge Univ. Press, 2005.

[18] A. Beck, A. Ben-Tal, and L. Tetruashvili, “A sequential parametricconvex approximation method with applications to nonconvex trusstopology design problems,” J. Global Optimiz., vol. 47, no. 1, pp.29–51, 2010.

[19] D. P. Bertsekas, Nonlinear Programming. Nashua, NH, USA:Athena Scientific, 1999.

[20] C. Isheden, Z. Chong, E. A. Jorswieck, and G. Fettweis, “Frame-work for link-level energy efficiency optimization with informedtransmitter,” IEEE Trans. Wireless Commun., vol. 11, no. 8, pp.2946–2957, Aug. 2012.

[21] J. B. G. Frenk and S. Schaible, “Fractional programming,” HandbookGeneral. Convexity General. Monotonicity, pp. 335–386, 2006.

[22] S. Schaible, “Parameter-free convex equivalent and dual programs offractional programming problems,” Z. Operations Res. Ser. A-B, vol.18, pp. A187–A196, 1974.

[23] Z.-Q. Luo and S. Zhang, “Dynamic spectrum management: Com-plexity and duality,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 1,pp. 57–73, Feb. 2008.

[24] L.-N. Tran, M. Bengtsson, and B. Ottersten, “Iterative precoder de-sign and user scheduling for block-diagonalized systems,” IEEE Trans.Signal Process., vol. 60, no. 7, pp. 3726–3739, Jul. 2012.

[25] Z. Shen, R. Chen, J. Andrews, R. W. Heath, Jr., R. W. , and B. Evans,“Low complexity user selection algorithms for multiuser MIMO sys-tems with block diagonalization,” IEEE Trans. Signal Process., vol.54, no. 9, pp. 3658–3663, Sep. 2006.

[26] X. Zhang and J. Lee, “Low complexityMIMO scheduling with channeldecomposition using capacity upperbound,” IEEE Commun. Lett., vol.56, no. 6, pp. 871–876, Jun. 2008.

[27] L.-N. Tran, M. F. Hanif, A. Tölli, and M. Juntti, “Fast converging al-gorithm for weighted sum rate maximization in multicell MISO down-link,” IEEE Signal Process. Lett., vol. 19, no. 12, pp. 872–875, Dec.2012.

[28] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge,U.K.: Cambridge Univ. Press, 2004.

[29] C. T. K. Ng and H. Huang, “Linear precoding in cooperative MIMOcellular networks with limited coordination clusters,” IEEE J. Sel.Areas Commun., vol. 28, no. 9, pp. 1446–1454, Dec. 2010.

[30] L.-N. Tran, “An iterative precoder design for successive zero-forcingprecoded systems,” IEEE Commun. Lett., vol. 16, no. 1, pp. 16–18, Jan.2012.

[31] A. Khabbazibasmenj, F. Roemer, S. A. Vorobyov, and M. Haardt,“Sum-rate maximization in two-way AF MIMO relaying: Polynomialtime solutions to a class of DC programming problems,” IEEE Trans.Signal Process., vol. 60, no. 10, pp. 5478–5493, Oct. 2012.

[32] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convexprogramming, version 2.0” Dec. 2012 [Online]. Available: http://cvxr.com/cvx

[33] J. Löfberg, “YALMIP : A toolbox for modeling and optimization inMATLAB,” in Proc. CACSD Conf., Sep. 2004, pp. 284–289 [Online].Available: http://users.isy.liu.se/johanl/yalmip

[34] W. Yu, “Sum-capacity computation for the Gaussian vector broadcastchannel via dual decomposition,” IEEE Trans. Inf. Theory, vol. 52, no.2, pp. 754 –759, Feb. 2006.

[35] S. Boyd and A. Mutapcic, “Subgradient methods” 2007 [On-line]. Available: http://www.stanford.edu/class/ee364b/notes/sub-grad_method_notes.pdf

[36] O. Arnold, F. Richter, G. Fettweis, and O. Blume, “Power consumptionmodeling of different base station types in heterogeneous cellular net-works,” in Proc. 19th Future Netw. MobileSummit (ICT Summit’10),Florence, Italy, Jun. 2010, pp. 1–8.

[37] J. Xu, L. Qiu, and C. Yu, “Improving energy efficiency through mul-timode 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 forGaussian 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,” IEEETrans. Commun., vol. 59, no. 5, pp. 1268–1274, May 2011.

[40] L.-N. Tran, M. Juntti, M. Bengtsson, and B. Ottersten, “Successivezero-forcing DPCwith per-antenna power constraint: Optimal and sub-optimal designs,” in Proc. IEEE ICC 2012, Jun. 2012, pp. 3746–3751.

Dan Nguyen (S’12) received the B.S. degree inElectrical Engineering from Ho Chi Minh NationalUniversity of Technology, Vietnam in 2003, and M.Sin Radio Engineering from Kyung Hee University,Republic of Korea, in 2008. Since August 2011,she has been working as a researcher toward thePhD degree in Department of Communicationsengineering at University of Oulu, Finland. Hercurrent research interests focus on energy efficiencycommunications, and full duplex wireless systems.

Le-Nam Tran (M’10) received the B.S. degree inElectrical Engineering from Ho Chi Minh NationalUniversity of Technology, Vietnam in 2003, and M.Sand PhD in Radio Engineering from Kyung Hee Uni-versity, Republic of Korea, in 2006 and 2009, respec-tively.In 2009, he joined the Department of Electrical En-

gineering, Kyung Hee University, Republic of Korea,as a lecturer. From September 2010 to July 2011, hewas a postdoc fellow at the Signal Processing Labo-ratory, ACCESS Linnaeus Centre, KTH Royal Insti-

tute of Technology, Sweden. Since August 2011, he has been with Centre forWireless Communications and Department of Communications Engineering,University of Oulu, Finland. His current research interests include multiuserMIMO 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) receivedMaster of Science, Licentiate of Science, and Doctorof Science in Technology degrees in electricalengineering from the University of Oulu, Finland,in 1995, 1998, and 2006, respectively. Since 1994he has been with the Telecommunication Laboratoryand since 1995 with the Centre for Wireless Com-munications, University of Oulu, working in variousEuropean and national wireless communicationsresearch projects. Currently he is a Senior ResearchFellow 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 bornin Kuivaniemi, Finland in 1968. He received theM.Sc., Lic.Tech. and Dr. Tech (Hons.) degreesin Electrical Engineering from the University ofOulu, Finland in 1992, 1996 and 1998, respectively.From 1992 to 1993, he was a Research Engineerat Nokia Mobile Phones, Oulu, Finland. Duringthe years 1994–1998 he was a Research Scientistat Telecommunication Laboratory and Centre forWireless Communications at the University of Oulu.Currently he is the Department Chair Professor

of Digital Transmission Techniques at the University of Oulu, Departmentfor Communications Engineering. Prof. Latva-aho was Director of Centrefor Wireless Communications at the University of Oulu during the years1998–2006. His research interests are related to mobile broadband wirelesscommunication systems. Prof. Latva-aho has published over 200 conferenceor journal papers in the field of wireless communications. He has been TPCChairman for PIMRC’06, TPC Co-Chairman for ChinaCom’07 and GeneralChairman for WPMC’08. He acted as the Chairman and vice-chairman ofIEEE Communications Finland Chapter in 2000–2003.