Linear Precoding Designs for Amplify-and-Forward Multiuser Two-Way Relay Systems

# Linear Precoding Designs for Amplify-and-Forward Multiuser Two-Way Relay Systems

Rui Wang, Meixia Tao, and Yongwei Huang R. Wang and M. Tao are with the Department of Electronic Engineering at Shanghai Jiao Tong University, Shanghai, 200240, P. R. China. Emails:{liouxingrui, mxtao}@sjtu.edu.cn. Y. Huang is with the Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong. Email: huang@hkbu.edu.hk.This work is supported by the Joint Research Fund for Overseas Chinese, Hong Kong and Macao Young Scholars under grant 61028001, the NSF of China under grant 60902019, and the NCET Program under grant NCET-11-0331.Part of this work was presented at GLOBECOM 2011.
###### Abstract

Two-way relaying can improve spectral efficiency in two-user cooperative communications. It also has great potential in multiuser systems. A major problem of designing a multiuser two-way relay system (MU-TWRS) is transceiver or precoding design to suppress co-channel interference. This paper aims to study linear precoding designs for a cellular MU-TWRS where a multi-antenna base station (BS) conducts bi-directional communications with multiple mobile stations (MSs) via a multi-antenna relay station (RS) with amplify-and-forward relay strategy. The design goal is to optimize uplink performance, including total mean-square error (Total-MSE) and sum rate, while maintaining individual signal-to-interference-plus-noise ratio (SINR) requirement for downlink signals. We show that the BS precoding design with the RS precoder fixed can be converted to a standard second order cone programming (SOCP) and the optimal solution is obtained efficiently. The RS precoding design with the BS precoder fixed, on the other hand, is non-convex and we present an iterative algorithm to find a local optimal solution. Then, the joint BS-RS precoding is obtained by solving the BS precoding and the RS precoding alternately. Comprehensive simulation is conducted to demonstrate the effectiveness of the proposed precoding designs.

MIMO precoding, two-way relaying, non-regenerative relay, minimum mean-square-error (MMSE), convex optimization.

## I Introduction

Due to complex wireless propagation environments, such as multi-path fading, shadowing and interference, the signals received by a remote destination receiver are not always strong enough to be decoded correctly. This problem has been considered as a main obstacle in the development of modern wireless communication systems. Recently, relay assisted cooperative communication has been proposed as an efficient way to deal with this problem, which now has received great attention from both academia and industry. One example of the relay assisted cooperative communication is one-way relay system, which has been well studied in past decade [1, 2]. Although it has shown great potential in for example, transmission reliability, energy saving and coverage extension, one-way relaying on the other hand reduces spectral efficiency due to half-duplex constraint.

A promising technique to improve spectral efficiency of one-way relaying is to apply network coding [3], resulting in two-way relaying which has now attracted great attention [4, 5, 6, 7]. Two-way relaying applies the principle of network coding at the relay node so as to mix the signals received from the two source nodes who wish to exchange information with each other and then employs at each destination self-interference (SI) cancelation to extract the desired information. Compared with traditional one-way relaying, spectral efficiency of two-way relaying can be significantly improved since only two time slots instead of four time slots are needed to complete one round of information exchange.

In this work, we consider two-way relaying in multiuser systems. As in traditional multiuser systems, it is crucial to mitigate co-channel interference (CCI) for multiuser two-way relay system (MU-TWRS). An advanced method to suppress CCI is to apply multiple-input multiple-output (MIMO) technique. Therein, transceiver or precoding should be carefully designed at each multi-antenna station, especially at the relay station (RS) [8, 9, 10, 11, 12, 13, 14, 15]. In [8, 9], authors study linear relay precoding for MU-TWRS with decode-and-forward (DF) relay strategy. Since the received signals is fully decoded in the first time slot, the relay precoding only affects the transmission in the second time slot. Then, by using zero-forcing (ZF) precoding, the relay precoding studied in [8, 9] reduces to a power allocation problem. The amplify-and-forward (AF) relay precoding, however, differs considerably from DF case as the transmissions of the first and second time slots are tightly coupled and hence is more challenging. Using ZF and minimum mean-square-error (MMSE) criteria, authors in [10, 11, 12, 13] study precoding design for an AF based MU-TWRS with multiple pairs of users. In particular, the explicit and analytical results are derived in [13] for system performance evaluation. Relay precoding design for the AF based MU-TWRS with multiple pairs of users is also considered in our previous work [14]. Unlike [10, 11, 12, 13], we do not impose any structural constraint on the relay precoder and thus the obtained results can approach the optimal performance [14]. In [15], authors study an AF MU-TWRS model with one base station (BS) and multiple mobile stations (MSs). By using ZF precoding scheme, explicit analytical results are also provided as in [13]. It is worth noting that the aforementioned ZF based precoding designs all impose certain constraints on the number of relay antennas which may not be available for some scenarios.

In this paper, we consider linear precoding design for a cellular MU-TWRS where a multi-antenna BS intends to conduct bi-directional communications with multiple MSs via a multi-antenna RS. Our work differs from [9] in that we adopt AF relay strategy rather than DF for its simplicity in practical implementation. However, as mentioned previously, the precoding design with AF relay strategy is more challenging. Our work is also different from [15] since we do not impose any structures on precoders. Our design goal is to enhance uplink performance subject to individual signal-to-interference-plus-noise ratio (SINR) requirement for downlink signals. Specifically, total mean-square error (Total-MSE) and sum rate are chosen to measure the performance of uplink. Since linear precoding can be employed at the BS, RS or both, three associated optimization problems are considered. When precoding is only conducted at the BS with the RS precoder fixed, we show that this optimization problem can be converted to a standard second-order cone programming (SOCP), thus the optimal solution can be obtained efficiently. The RS precoding with the BS precoder fixed, on the other hand, is non-convex and we present an iterative algorithm to find a local optimal solution. Thirdly, we obtain the joint BS-RS precoding design by solving the BS precoding and the RS precoding alternately, the convergence of which is guaranteed. Simulation results show that the RS precoding scheme outperforms the BS precoding scheme in most cases and the joint precoding scheme outperforms the individual precoding scheme. Besides performance, practical implementation issues, including signaling overhead and design complexity, for the proposed precoding designs are also discussed and compared.

The rest of the paper is organized as follows. In Section II, we present the system model. Different precoding designs are presented in Section III. In Section IV, we discuss the overhead and design complexity. Extensive simulation results are illustrated in Section V. Finally, we conclude the paper in Section VI.

Notations: denotes the expectation over the random variables within the brackets. denotes the Kronecker operator. , , and stand for the trace, inverse, determinant and the rank of a matrix , respectively, and denotes a diagonal matrix with being its diagonal entries. Superscripts , and denote the transpose, conjugate and conjugate transpose, respectively. implies the zero matrix. denotes the identity matrix and if . denotes the squared Euclidean norm of a complex vector and denotes the Frobenius norm of a complex matrix . implies the norm of a complex number , and denote its real and imaginary part, respectively. denotes the space of matrices with complex entries. The distribution of a circular symmetric complex Gaussian vector with mean vector and covariance matrix is denoted by .

## Ii System Model

Consider a multiuser two-way relay system where an -antenna BS conducts bi-directional communication with single-antenna MSs under the assistance of an -antenna RS. For effective multiuser transmission, we let and . Moreover, we assume that all the MSs are cell-edge users. Thus, due to impairments such as multipath fading, shadowing and path loss of wireless channels, the direct-path link between the BS and each MS is ignored. It is also assumed that the RS operates in half-duplex mode. That is, it cannot transmit and receive simultaneously.

The bi-directional (i.e., uplink and downlink) communications take place in two time slots as shown in Fig. 1. In the first time slot, also referred to as multiple-access (MAC) phase, both the BS and MSs simultaneously transmit their signals to the RS. The received signal vector at the RS can be written as

 yR=H1xB+K∑k=1h2ksk+nR,

where represents the transmit signal vector from the BS, denotes the transmit signal from the MS . We assume that the transmission power at the MS is , i.e., . is the MIMO channel matrix from the BS to the RS, is the channel vector from the MS to the RS, and denotes the additive noise vector at the RS following . Here can be further expressed as

 xB=BsB,

where with is the modulated signal vector from the BS, denotes the transmit precoding matrix at the BS. Furthermore, the maximum transmission power at the BS is assumed to be , i.e.,

 Tr(BBH)≤PB. (1)

Upon receiving the superimposed signal , the RS performs linear processing by multiplying it with a precoding matrix and then forwards it in the second time slot, also referred to as broadcast (BC) phase. Therefore, the transmit signal vector from the RS is given by

 xR=FyR=FH1xB+K∑k=1Fh2ksk+FnR.

The maximum transmission power at the RS is given by , which yields

 Tr{F(H1BBHHH1+H2PPHHH2+σ2RIM)FH}≤PR, (2)

where we define and . Then the received signals at the BS and MS after the BC phase can be written as

 ~yB=K∑k=1G1Fh2ksk+G1FH1BsB+G1FnR+nB=G1FH2sM+G1FH1BsB+G1FnR+nB, (3)
 ~yk=K∑i=1gT2kFH1bisBi+K∑i=1gT2kFh2isi+gT2kFnR+nk. (4)

Here, , denotes the -th entry in , and are the channel matrix and vector from the RS to the BS and MS , respectively, and denotes the additive noise at the BS and MS , respectively, with and . Note that both the BS and MS know their transmit signals and , respectively. Therefore, the back propagated self-interference terms and can be subtracted from (3) and (4), respectively. The equivalent received signals at the BS and MS are yielded, respectively, as

 yB=G1FH2sM+G1FnR+nB, (5)
 yk=gT2kFH1bksBk+∑i≠kgT2kFH1bisBi+∑i≠kgT2kFh2isi+gT2kFnR+nk. (6)

From (6), we find that the received downlink signal at each MS not only consists of the CCI from the downlink transmission (i.e., the second term), but also the CCI from the uplink transmission (i.e., the third term). The downlink performance of each MS can be measured by SINR given by

 SINRk=|gT2kFH1bk|2∑l≠k(|gT2kFH1bl|2+Pl|gT2kFh2l|2)+σ2R||gT2kF||22+σ2k,k=1,2,⋯,K. (7)

As for the uplink transmission in (5), it can be viewed as a MIMO multiple-access channel. Depending on different performance requirements, various metrics can be used to evaluate its performance. Our first objective aims to minimize the Total-MSE of all the MSs by assuming linear minimum mean-square error (MMSE) receiver at the BS. Using Total-MSE for precoding design has been widely studied in multiuser systems [16, 10, 11, 17, 18]. By minimizing MSE

 e=EsM(||WyB−sM||22) (8)

with respect to the decoding matrix , the minimum Total-MSE is given by [19]

 e=Tr(E−1), (9)

where and the optimal in (8) is

 W=PHHH2FHGH1(G1FH2PPHHH2FHGH1+σ2RG1FFHGH1+σ2BIN)−1. (10)

Our second objective aims to maximize the sum rate of the uplink transmission. By applying successive interference cancelation (SIC) and linear MMSE filter at the BS, the sum rate at the BS is given by [20]

 r=0.5log2det(IK+PHHH2FHGH1(σ2RG1FFHGH1+σ2BIN)−1G1FH2P), (11)

where the factor is due to the fact that the MSs use two time slots to complete the uplink transmission. Note that (11) can be re-expressed as with defined in (9). We will see that the precoding designs proposed for Total-MSE minimization can be extended for sum rate maximization.

## Iii Linear Precoding Designs

From Section II, it is seen that the downlink performance of each MS depends on both the BS precoder and the RS precoder . While for the uplink transmission, it is only related to the RS precoder , thus less design freedom can be exploited compared with the downlink. In theory, the BS precoder and the relay precoder should be jointly designed such that the downlink and uplink performance can be optimized simultaneously. However, there is no single figure of merit to measure the overall performance of the multiuser bidirectional transmission. In this paper, we choose to ensure the downlink quality-of-service (QoS) for each individual MS while at the uplink minimizing the Total-MSE or maximizing the sum rate of all the users. This is because in practice the downlink data traffic usually is more dominant than the uplink traffic. As such, the optimization problem is formulated as

 minB,F  e or −r (12) s.t. SINRk≥λk, k=1,2,⋯,K, Tr(BBH)≤PB Tr{F(H1BBHHH1+H2PPHHH2 +σ2RIM)FH}≤PR

where is a preset threshold for the MS .

Since linear precoding can be conducted at the BS, RS or both, three associated precoding designs are considered respectively in the following three subsections. Note that for each design, the system needs different computational complexity and signaling overhead, such that they are suitable to different scenarios.

### Iii-a BS precoding

In this subsection, we assume that precoding is only employed at the BS, while the RS precoder is given as where is an arbitrary fixed precoder applied at the RS, and is a non-negative scalar used to scale the received signals at the RS to satisfy relay power constraint. Note that besides maintaining the downlink SINR, a properly designed can reduce the RS power consumption by the signal from the BS. Then the uplink transmission can share more power at the RS, which is helpful for improving its performance.

The optimization problem can be formulated as:

 minB,α  f1(α)  or −f2(α) (13) s.t. ρk≥λk,∀k Tr(BBH)≤PB Tr{α2~F(H1BBHHH1+H2PPHHH2+ σ2RIM)~FH}≤PR

where and with and

 ρk=α2|gT2k~FH1bk|2ξ+α2σ2R||gT2k~F||2+σ2k,

where . To proceed to solve (13), we first give the following lemma, the proof of which is given in Appendix A.

Lemma 1: and are monotonically decreasing functions with respect of .

Based on Lemma 1, it is easy to see that minimizing or in (13) is equivalent to maximizing the scalar . By defining , problem (13) can be re-expressed as:

 max~B,α  α (14) s.t. Tr(~B~BH)≤α2PB Tr(~FH1~B~BHHH1~FH) +α2Tr(~F(H2PPHHH2+σ2RIM)~FH)≤PR K∑i=1|gT2k~FH1~bi|2+α2⎛⎝∑i≠kPi|gT2k~Fh2i|2+

Although (14) is still a non-convex problem, we can use the observation made in [21] that any phase shift of , i.e., , does not affect the optimality of the primal problem. Therefore, for any optimal solutions, there always exists a phase shift version of to make the term real and positive while not affecting the value of the objective function and keeping the constraints satisfied. Thus, we can convert problem (14) into the following equivalent form

 maxxB  α (15) s.t. ||~B||2F≤α2PB, ||~FH1~B||2F +α2Tr(~F(H2PPHHH2+σ2RIM)~FH)≤PR K∑i=1|gT2k~FH1~bi|2+α2⎛⎝∑i≠kPi|gT2k~Fh2i|2+ σ2R||gT2k~F||22)+σ2k≤(1+1λk)(gT2k~FH1~bkreal and >0)2,∀k

where . It is not hard to verify that (15) is a standard second-order cone programming [22] and the optimal solution can be obtained by using available software package [23]. Then, dividing by , we finally get the optimal .

### Iii-B RS precoding

In this subsection, we consider the precoding design at the RS with the BS precoder fixed. In the following, we first consider the precoding design for Total-MSE minimization, then extend it to sum rate maximization.

#### Iii-B1 Total-MSE minimization

The RS precoding to minimize Total-MSE can be formulated as:

 minF Tr(E−1) (16) s.t. τ≤PR ζk≥λk, ∀k

where is defined in (9), and

 ζk=|gT2kFH1bk|2∑i≠k(|gT2kFH1bi|2+Pi|gT2kFh2i|2)+σ2R||gT2kF||22+σ2k.

Note that the power constraint at the BS is irrelevant here since is fixed. It is not hard to verify that the objective function and SINR constraints in (16) are both non-convex. To make (16) more tractable, we substitute the linear MMSE decoding matrix back into (16) and rewrite it as:

 minF,W f(F,W) (17) s.t. τ≤PR ζk≥λk, ∀k

where

 f(F,W)=Tr{WG1FH2PPHHH2FHGH1WH+σ2RWG1FFHGH1WH+σ2BWWH+IK−WG1FH2P−PHHH2FHGH1WH}. (18)

Note that (18) can also be computed from (8). Although the two design matrices and are coupled together in (17), the advantage of introducing is that we can apply alternating optimization to solve two decoupled subproblems iteratively in what follows.

In the alternating optimization, the first step is to update the BS decoding matrix for a given . From (17), it is seen that the constraints are independent of . Thus, the optimal can be readily obtained as in (10) by equating the gradient of the objective function in (17) to zero.

Secondly, we need to optimize with fixed. This problem is equivalently rewritten as:

 minF Tr{GH1WHWG1F(H2PPHHH2 +σ2RIM)FH−FH2PWG1 −GH1WHPHHH2FH+σ2BWWH+IK} s.t. τ≤PR ζk≥λk, ∀k

where we have used the fact that for (18). Although we can verify that the objective function in (III-B1) is convex based on [24], while due to the non-convex SINR constraints, the optimal is still not easy to obtain. To proceed, we need to recast (III-B1) into a suitable form such that efficient optimization tools can be applied. After certain transformation as detailed in Appendix B, problem (III-B1) can be rewritten into the following inhomogeneous quadratically constrained quadratic program (QCQP) form [22]:

 minf fHQ0f−fHq0−qH0f+q0 (20a) s.t. fHQxf≤PR (20b) fHQkf≥λkσ2k, ∀k (20c)

where , , and are defined in (34), (36) and (38) in Appendix B, respectively. By checking the positive semidefiniteness of and the positive definiteness of , we can verify that both the objective function (20a) and the RS power constraint (20b) are convex. However, the constraint (20c) is not concave due to that defined in (38) is not necessarily negative semidefinite. Hence, optimization problem (20) is non-convex. To solve (20), we rewrite (20) into a standard QCQP form as follows:

 minxF xHF~Q0xF (21) s.t. |t|2=1 xHF~QxxF≤0 xHF~QkxF≤0,∀k

where , , and . Note that (III-B1) and (21) are equivalent to each other. If we get an optimal solution of (21), we can always obtain an optimal solution of (20) by selecting appropriate entries from no matter is real or complex. By a close inspection of (21), we find that (21) can be transformed into the following semidefinite programming (SDP) form [22]:

 minXF⪰0 Tr(~Q0XF) (22) s.t. Rank(XF)=1, Tr(QXF)=1 Tr(~QxXF)≤0 Tr(~QkXF)≤0,∀k

where . Due to the rank-one constraint, it is not easy to obtian an optimal solution of (22). We therefore resort to relaxing it by deleting the rank-one constraint, namely,

 minXF⪰0 Tr(~Q0XF) (23) s.t. Tr(QXF)=1 Tr(~QxXF)≤0 Tr(~QkXF)≤0,∀k

Note that (23) is a standard SDP problem, thus its optimal solution can be easily obtained by using the available software package [23]. If the optimal solution of (23) is rank-one, the optimal RS precoder can be obtained by using eigenvalue decomposition. Otherwise, certain techniques are required to find the optimal RS precoder.

In what follows, we first consider a system with no more than two MSs (i.e., ) for which an optimal solution of (20) can be obtained in most cases. Then, we extend the results to a more general system with where the randomization technique is applied to find a quasi-optimal solution.

##### K≤2

We first give the following theorem.

Theorem 1: Suppose that the considered cellular MU-TWRS has at most two MSs, i.e., , an optimal rank-one solution of the non-convex optimization problem (22) can be derived in polynomial time from the relaxed SDP problem (23) in the following cases: 1) problem (23) has an optimal rank-one solution; 2) problem (23) has at least one inactive constraint at the optimal solution; 3) problem (23) has an optimal solution of rank higher than two if all the constraints are active. {proof} Please refer to Appendix C. From Theorem 1, we find that we cannot obtain an optimal rank-one solution if the SDP relaxation problem (23) happens to have an optimal solution of rank two with all the constraints being active. However, our simulations show that this case has rarely occurred. Nonetheless, we can propose a procedure of producing a suboptimal rank-one solution in Appendix D for that special case.

Now, the iterative RS precoding algorithm to minimize Total-MSE for can be outlined as follows.

Algorithm 1 (RS precoding with )

• Initialize

• Repeat

• Update the BS decoding matrix using (10) for a fixed ;

• Update the RS precoder with fixed as follows: If the obtained in (23) is rank-one, using eigenvalue decomposition to get . Otherwise, using the procedures presented in Appendix C or D to get ;

• Until termination criterion is satisfied.

Lemma 2: Algorithm 1 is convergent and the limit point of iteration is a stationary point of (17). {proof} Since for , the optimal solution in (III-B1) can be obtained in most cases as claimed in Theorem 1, the solution in each iteration in Algorithm 1 can be viewed as being optimal. Thus the Total-MSE at the BS is strictly reduced after each iteration before convergence. On the other hand, the objective function is lower-bounded (at least zero). Therefore, we conclude that Algorithm 1 is convergent. We assume that the limit point of Algorithm 1 is . At the limit point, the solution will not change if we continue the iteration. Otherwise, the Total-MSE can be further decreased and it contradicts the assumption of convergence. The optimal solution in each iteration further means that and are local minimizers of each subproblem. Hence, we have

 Tr{▽Wf(¯W;¯F)T(W−¯W)}≥0,Tr{▽Ff(¯F;¯W)T(F−¯F)}≥0,

Summing up the two inequalities, we get

 Tr{▽Xf(¯X)T(X−¯X)}≥0, (24)

where . Condition (24) implies the stationarity of in (17) (e.g., see Theorem 3 of [25]).

##### K>2

Now we consider a more general case with . Since at least five constraints are contained in (23), it is difficult to find an optimal rank-one solution if the optimal solution in (23) has higher rank than one. Next we propose to apply the randomization technique in [26] to find a quasi-optimal rank-one solution of (20). We first transform (20) into the following equivalent form:

 minf Tr(Q0~F)−fHq0−qH0f+q0 (25) s.t. Tr(Qx~F)≤PR Tr(Qk~F)≥λkσ2k,∀k ~F=f×fH

Relaxing the constraint to and applying the Schur complement theorem, we get the following optimization problem:

 min~F,f Tr(Q0~F)−fHq0−qH0f+q0 (26) s.t. Tr(Qx~F)≤PR Tr(Qk~F)≥λkσ2k,∀k [~FffH1]≥0

Note that (26) is convex, thus the obtained solution is optimal. If we generate enough samples of Gaussian variable following with and being an optimal solution of (26), and choose the best candidate from the samples as a solution of (20), will optimally solve (20) on average, i.e.,

 minf E(fHQ0f−fHq0−qH0f+q0) (27) s.t. E(fHQxf)≤PR E(fHQkf)≥λkσ2k,∀k

Finally, the proposed iterative algorithm for is outlined as:

Algorithm 2 (RS precoding with )

• Initialize

• Repeat

• Update the BS decoding matrix using (10) for a fixed ;

• Update the RS precoding matrix with fixed using the following steps: First, form an optimization problem as (23), if the obtained is rank-one, the optimal RS precoder is obtained by applying eigenvalue decomposition. Otherwise, apply the randomization procedures (25)-(26) to get a quasi-optimal solution;

• Until termination criterion is satisfied.

Note that although the obtained from the second step in Algorithm 2 may not be optimal, our simulation results show that the obtained by using randomization is always good enough to make the iteration convergent.

#### Iii-B2 Sum-rate maximization

Motivated by the relationship between sum rate and weighted MMSE in MIMO-BC system recently found in [27], we next try to extend the proposed RS precoding design for Total-MSE minimization to sum rate maximization. The sum-rate maximization problem is re-stated as:

 maxF log2det(IK+PHHH2FHGH1 (σ2RG1FFHGH1+σ2BIN)−1G1FH2P) s.t. τ≤PR ζk≥λk, ∀k

where the constraints are the same with (16). It is not hard to verify that (III-B2) is non-convex. To solve (III-B2), we introduce the following lemma.

Lemma 3: If a satisfies the Karush-Kuhn-Tucker (KKT) conditions of (III-B2), it will also satisfy the KKT conditions of the following problem:

 minF Tr(AE−1) (29) s.t. τ≤PR ζk≥λk, ∀k

where is defined in (9), if the weight matrix is set to

 A=1log2(IK+PHHH2¯FHGH1(σ2RG1¯F¯FHGH1+σ2BIN)−1G1¯FH2P). (30)
{proof}

The proof is similar to the MIMO BC precoding design problem in [27], thus we omit for brevity. Lemma 3 implies that using the weight matrix in (30), (III-B2) shares the same stationary point with (29). Then alternating optimization can be used to get the final solution of (III-B2) as in [27], which is presented as follows:

Algorithm 3 (RS precoding for maximizing sum rate)

• Initialize

• Repeat

• Update the BS decoder matrix using (10) for fixed and ;

• Update the weight matrix using (30) for fixed and ;

• Update the RS precoder matrix as in Algorithm 1 or 2;

• Until termination criterion is satisfied.

According to the convergence analysis provided in [27], the convergence of Algorithm 3 can be ensured.

### Iii-C Joint precoding

Obviously, the previously presented two precoding designs can be combined to realize the joint BS-RS precoding design to obtain better performance. In this case, if the RS has enough capability to enable the joint design, it can collect all the required CSI and optimize and jointly. Then besides , the RS should also broadcast to the BS and MSs. On the other hand, the joint optimization can also be conducted at the BS and the RS helps to collect CSI and transmits them to the BS. Then, the BS needs to transmit and to the RS, and the RS further broadcasts them to the MSs. Nevertheless, such joint precoding design requires more feedback overheads although it leads to better performance.

According to the algorithms proposed in Subsections A and B, the joint precoding design is outlined as:

Algorithm 4 (Joint precoding scheme)

• Initialize

• Repeat

• Update the RS precoder for a fixed BS precoder by using Algorithm 1 or 2 for Total-MSE minimization and Algorithm 3 for sum rate maxmization;

• Update the BS precoder for a fixed relay precoder by using the SOCP optimization as in Subsection A;

• Until termination criterion is satisfied.

Lemma 4: The proposed joint precoding design algorithm is convergent. {proof} For convenience of presentation, we take Total-MSE minimization as example. The proof can be easily extended to the case of sum rate maximization. Firstly, for a fixed , updating must decrease the Total-MSE at the BS by increasing in (13), otherwise, the BS precoder should not be changed. Thus, we have

 e(B(n+1),F(n))≤e(B(n),F(n)),

where denotes the iteration index. Then, we apply the proposed RS precoding design to update by initializing . Since the proposed iterative RS precoding design algorithm decreases Total-MSE after each iteration, we have111On the case of solving (20) through randomization at , if we cannot find a solution decreasing the objective value in (20), we can just set .

 e(B(n+1),F(n+1))≤e(B(n+1),F(n)).

Therefore, we conclude that the joint precoding design algorithm is convergent.

## Iv Discussion on Signaling Overhead and design complexity

As mentioned previously, each precoding design has its own merit. Choosing which precoding scheme is not only dependent on the processing capability of the BS and the RS, but also the design complexity and signaling overhead. In this section, we provide a comprehensive comparison between these designs. It is assumed that the channel characteristics of each link change slowly enough so that they can be perfectly estimated by using pilot symbols or training sequences. Besides, the information of channel state and precoders can be exchanged accurately between the BS and the RS, the RS and the MSs through some lower rate auxiliary channels. For completeness, two transmission modes, i.e., time-division duplex (TDD) mode and frequency-division duplex (FDD) mode, are considered, respectively. The overall comparisons are presented in Table I, where “Overhead-I” denotes the overhead used to feed back the CSI and “Overhead-II” denotes the overhead used to feed back the precoding information. Moreover, we suppose that the BS and MSs can estimate their local CSI and , , respectively.

Since the BS precoding design is a SOCP problem, according to [28], the design complexity can be approximated as

 nBS=(NK+1)2(K+2)0.5(2NK+K2+2K+4)log(1/ϵ), (31)

where denotes the solution accuracy. For the RS precoding design, the design complexity mainly comes from solving the SDP problem and using the randomization technique. Thus, according to [29], it can be approximated as

 nRS=lRS(max(M2,K+2)4Mlog(1/ϵ)+nrd), (32)

where denotes the complexity of randomization and denotes the iteration number required in Algorithm 1, 2 or 3. Note that when ,