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Low-complexity QL-QR decomposition- s based beamforming design for two-way MIMO relay networks

Wei Duan1, Xueqin Jiang2, Ying Guo3, Yier Yan4, Kye-mun Cho1 and Moon Ho Lee1*

Abstract

In this paper, we investigate the optimization problem of joint source and relay beamforming matrices for a two-way amplify-and-forward (AF) multi-input multi-output (MIMO) relay system. The system, consisting of two source nodes and two relay nodes, is considered, and the linear minimum mean-square-error (MMSE) is employed at both receivers. We assume individual relay power constraints and study an important design problem, a so-called determinant maximization (DM) problem. Since this DM problem is nonconvex, we consider an efficient iterative algorithm by using an MSE balancing result to obtain at least a locally optimal solution. The proposed algorithm is developed based on QL, QR, and Choleskey decompositions which differ in complexity and performance. Analytical and simulation results show that the proposed algorithm can significantly reduce computational complexity compared with their existing two-way relay systems and have equivalent bit-error-rate (BER) performance to the singular value decomposition (SVD) based on a regular block diagonal (RBD) scheme.

Keywords: Two-way relay channel, MIMO, QL-QR decomposition, Choleskey decomposition, Determinant maximization, Amplify-and-forward

1 Introduction

Recently, wireless relay networks have been the focus of a lot of research because the relaying transmission is a promising technique which can be applied to extend the coverage or increase the system capacity. Various cooperative relaying schemes have been proposed, such as amplify-and-forward (AF) [1, 2], decode-and-forward (DF) [3], denoise-and-forward (DNF) [4], and compress-and-forward (CF) [5] cooperative relaying protocols. Among these approaches, an AF scheme is most widely used because it does not need to detect the transmitted signal. In addition, it requires less processing power at the relays compared to other schemes.

In a one-way relaying (OWR) approach, to completely exchange information between two base stations, four time slots are required in uplink (UL) and downlink (DL) communications, which leads to a loss of one-half spectral

Correspondence: moonho@jbnu.ac.kr

1 Division of Electronic and Information Engineering, Chonbuk National

University, Jeonju-si, South Korea

Full list of author information is available at the end of the article

resources [6]. In order to solve this problem, a two-way relaying approach has been considered in [7-9]. In a typical two-way relaying scheme, the communication is completed in two steps. First, the transmitters send their symbols to two relays, simultaneously. Upon receiving the signals, each relay processes them based on an efficient relaying scheme to produce new signals. After these processes, the processed signals are broadcasted to both receiver nodes.

Multi-input multi-output (MIMO) relay systems have been investigated in [10-13]. It is shown that, by employing multiple antennas at the transmitter and/or the receiver, one can significantly improve the transmission reliability by leveraging spatial diversity. Relay precoder design methods have been investigated in [14-16]. A problem in designing optimal beamforming vectors for multi-casting is challenging due to its nonconvex nature. In [14], the authors propose a transceiver precoding scheme at the relay node by using zero-forcing (ZF) and MMSE criteria with certain antenna configurations. The information theoretic capacity of the multi-antenna

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© 2015 Duan et al. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons

license, and indicate if changes were made.

multicasting is studied in [15], along with the achievable rates using lower complexity transmission schemes, as the number of antennas or users goes to infinity. In [16], the authors propose an alternative method to characterize the capacity region of a two-way relay channel (TWRC) by applying the idea of rate profile.

Joint optimizations of the relay and source nodes for the MIMO TWRC have been studied in [9, 17]. In [9], the authors develop a unified framework for optimizing two-way linear non-regenerative MIMO relay systems and show that the optimal relay and source matrices have a general beamforming structure. The joint source node and relay precoding design for minimizing the mean squared error in a MIMO two-way relay (TWR) system is studied in [17].

Since singular value decomposition (SVD) and generalized SVD (GSVD) are widely used to find the orthogonal complement to solve an optimization problem [2, 9, 16, 33], but their computational complexity is extremely high. In order to reduce the complexity, the SVD can be replaced with a less complex QR decomposition [18] in this work. However, this approach leads to degrading the bit-error-rate (BER) performance. In addition, it is difficult to realize in the TWRC. In this paper, we investigate the joint source and relay precoding matrix optimization for a two-way relay amplify-and-forward relaying system where two source nodes and two relay nodes are equipped with multiple antennas. Also, in order to apply the QL-QR decomposition to the TWRC, we design a three-part relay filter. Compared with existing works such as [9-14], the contributions of this paper can be summarized as follows. Firstly, we investigate a two-way MIMO relay system using the criteria which minimize an MSE of the signal waveform estimation for both two source nodes. We prove an optimal sum-MSE solution can be obtained as the Wiener filter while the signal-to-noise ratios (SNR) at both source nodes are equivalent [20], which leads to an MSE balancing result. Secondly, we propose a new cooperative scenario, i.e., the QL-QR compared with the Choleskey decomposition which significantly reduces the computational complexity of the optimal design. In this proposed design, the channels of its left side are decomposed by the QL decomposition while those of its right side factorized by the QR decomposition. And the equivalent noise covariance is decomposed by the Choleskey decomposition. We also design the three-part relay filter, which is comprised of a left filter, a middle filter, and a right filter, to efficiently combine two source nodes and the relay nodes. By these approaches, the received signals at both two source nodes are able to be redeemed as either lower or upper triangular matrices. Stemming from one of the properties of triangular matrices such that their determinant is identical to the multiplication of their eigenvalues, we are able

to straightforwardly solve the optimization problem as a determinant maximization problem. Also, we can obtain the BER performance equivalent to that of the singular value decomposition-regular block diagonal (SVD-RBD) scheme.

The rest of this paper is organized as follows. Section 2 describes a system model of the TWRC and raises a sum-MSE problem. In Section 3, we propose an iterative QL-QR algorithm and a joint optimal beamforming design. In Section 4, we discuss the computational complexity of an efficient channel model. The simulation results are presented to show the excellent performance of our proposed algorithm for the TWRC in Section 5. Section 6 concludes this paper.

Notations: A*, AT, AH, E(A), tr(A), K(A), and det(A) denote the conjugate, transpose, Hermitian transpose, statistical expectation, trace, real part, and determinate of a matrix A, respectively. An N x N identity matrix is denoted by IN.

2 System model and sum-MSE

We consider a TWRC consisting of two source nodes, Si and S2, and two relay nodes, R1 and R2, as shown in Fig. 1. The source and relay nodes are equipped with M and N antennas, respectively. We adopt the relay protocol with two time slots introduced in [14]. In the first time slot, the information vector xi e CGx1, where G < M, is linearly processed by a precoding matrix, Vi e CMxM, and then transmitted to the relay nodes. In this paper, we assume that each transmit antenna satisfies the unity transmission power constraint, which is tr {x,xH} = IM. The received signals at Ri, i e {1,2} can be expressed as

y*1 = H11S1 + H1,2S2 +

YR2 = H2,1 S1 + H2,2S2 + nR2, (1)

where yR; e CNx1, i e {1,2}, indicates the received signal vector, Hy e CNxM, i, j e {1,2}, represents the channel matrix from source j to relay i, as shown in Fig. 1, Si = Vixi e CMx1 is the transmitted symbol vector from Si with a power constraint tr {E (s,sH^ < Pi, and

nR; ~ CN (0, aR,.In) represents the additive white Gaussian noise (AWGN) vector with zero mean and variance aR, at relay node i.

In the second time slot, the relay node R, linearly amplifies yR; with an N x N matrix F, and then broadcasts the amplified signal vector xRi to source nodes 1 and 2. The signal transmitted from relay node i can be expressed as

xRi = FiyRi. (2)

Fig. 1 Proposed QL-QR amplify-and-forward MIMO TWR system

Using (1) and (2), the received signal vectors at Si and S2 can be, respectively, written as

yi = Hj",i FiHi,iSi + HTiFiHi,2 S2 + HjiF2H2,iSi

+ HT,iF2H2,2S2 + HjiFinii, + HTiF2nR2 + ni y2 = HT2 FiHi,iSi + HT2FiHi,i S2 + H T,2 F2 H2,i Si

+ H T,2 F2 H2,i S2 + H^^FinRi + Hj,2F2nR2 + n2.

where HT;, i, j e {i, 2}, indicates the M x N channel matrix from the relay node i to the source node j and n,, i e {i, 2}, is an M x i noise vector at S,.

We assume that the relay nodes perfectly know the channel state information (CSI) of Hy. The relay node R, performs the optimizations of F, and V, and then transmits the information to the source nodes i and 2. Since source node i knows its own transmitted signal vector s, and full CSI, the self-interference components in (3) can be efficiently canceled. The effective received signal vectors are given by

yi = HT,iFiHi,2 S2 + HTiF2H2,2S2 + H^F^ + HT,iF2nR2 + ni = HiS2 +ni, (4)

y2 = Hj^Fi Hi,i Si + H^^HySi + H^F^

+ HT,2F2nR2 + n2

= H2Si + n2, (5)

where Hi = H^FiHi,2 + HT,iF2H2,2 and H2 = HT,2 FiHi,i + HT2F2H2,i are the equivalent MIMO channels seen at source nodes Si and S2, respectively The vectors ni = HTaFinRi + HTiF2nR2 + ni and y = H^FinRi + HT2F2nR2 + n2 are the equivalent noises at source nodes Si and S2, respectively.

Due to the lower computational complexity, linear receivers are applied at source node i to retrieve the transmitted signals sent from the other nodes. The estimated signal waveform vector is given as ' = wHyi, where Wi is an M x M weight matrix, with i = 2for i = i and i = i for i = 2. From (4), the MSE matrix of the signal waveform estimation is denoted by MSE- = E [' — s,) ' — s(-)h], which can be further written as

MSE, = (WHH, — im) (WHH, — im)H

+wHC„,.Wi (6)

where Cni = HTriFiFHHii + H-T.F;FHH*. + Im is the equiv-

i i,i i i,i i,i i i i,i

alent noise covariance. The sum-MSE of the two source nodes in the proposed system model can be written as

MSESUm = MSEi + MSE2. (7)

Note that the sum-MSE minimization criterion measures the overall transmission performance of both the DL and the UL. Since the two data streams are transmitted at different directions during the two time slots, MSEsum is considered in the TWR network.

3 Joint source and relay beamforming design

In this section, we develop an iterative QL-QR algorithm by using the MSE balancing result. The QL-QR algorithm involves two steps, i.e., the linear receiver matrix optimization and the joint source and relay beamformer design.

3.1 Proposed optimal detector and optimization problem

We would like to find the jointly optimal beamforming vectors W,, V,, and F, such that the following sum-MSE is minimized:

min MSESum. (8)

W;,F;,V;

According to (2), we consider the following individual transmission power constraint at relay node:

tr(F«D«FH) < PRi,

t<V«vH) < Pi, i = 1,2

where Pi is the available power at the ith source node. According to (8), (9), and (10), the joint optimization problem of the sum-MSE can be formulated as follows:

Wi,Fi,Vi

s.t. tr(FiDiFH < PRi, tr(ViVH) < Pi. (11)

It is shown in [20] that at the optimum, SNR1 = SNR2 holds true, thus leading to an SNR balancing result. Otherwise, if SNR1 > SNR2, then P2 can be reduced to retain SNR1 = SNR2, and this reduction of P2 will not violate the power constraint, i.e.,

Pi • SNRi = P2 • SNR2.

are functions of SNR , namely the MSE at the output of a linear-MMSE (LMMSE) filter of each user:

where Di = Hi iViVHHH + H7 7V7VHhH + In and the

i i,i i,i i i i,i

PRi denotes the individual power constraint at the relay node Ri. The transmission power constraint at two source nodes can be written as

In Fig. 2, we show two examples of the SNR regions with a1 = 0.5 and a2 = 0.3, where m7 e [0,1] is a Lagrange multiplier weight value and ai e [0,1] is an SNR weight value. We have assumed that the sum of SNR is a constant value. It is clear that the SNR region of a1 is larger than that of a2. For further details, see [20]. As discussed in [21], the optimization problems have the performance matrix that

1 + SNR

By these two approaches, the max-min optimization problem in (11) can be efficiently written as

min MSEi (14)

Wi,F;,V2

s.t. tr(F,D,FH) < PRi, MSE1 = MSE2, V, = 1,2.

Since the optimization problem (14)-(15) is nonconvex, it is difficult to obtain the globally optimal solution. In this paper, we present a locally optimal solution of the joint optimization problem over W,, V,, and F,, where i = 1,2, which can be solved by three stages, i.e., (1) the linear receiver weighted matrices are optimized with the fixed source precoding matrix V, and relay amplifying matrices F, (W, is not in constraint (15)). (2) With given W, and fixed F,, update V,. (3) With given W, and V,, obtain suboptimal F, to solve (14).

Lemma 1. For any fixed. V, and F,, the тШт,гаЫоп problems ,n (14) are convex quadratic problems and the optimal W, can be obtamedas the Wienerfilter [22] wh,ch ,s used to decode s, shown asfollows:

W = (Hi hH + C„i) 1 Hi,

Proof. For source node i, the MSE can be further expressed as

MSEi = WHHiHHWi - W^Hi - HHWi + Im + WHC„, Wi

(17) □

Based on (17), the derivation of an optimal MSE detection matrix Wopt is equivalent to solving the following equation [23]:

д MSE,-

= 2HiHHWi - 2h, + 2CniWi = 0. (18)

Then, we may obtain the closed-form solution of Wi, which is

WO = (Hi HH + C„J 1 Hi.

This completes the proof.

With the optimal Wo1 fixed, the outer minimization problem in (14) can be rewritten as

min F1,F2,V1,V2

s.t. tr(FiDiFH) < Pr,, MSE1 = MSE2, (20)

where MSE? is the MSE matrix using W°. By substituting (i6) into (6), we have

MSE1 = [IM + H" 1, i = 1,2.

min Vi

Im + vH^iV;

Lv = tr

+ X1 (tr {VHVi} - Pi) + X2 (tr {vHv^ - P^ + X3 {tr {VH^,Vi} - PRi} + {tr {vH^Vj} - PRT] ,

where T-TH = x2Im + We obtain the derivative of LV as

Note that the matrix inversion lemma is used to obtain (21).

3.2 Joint optimal source and relay beamforming matrices design and iterative algorithm

In this section, we focus on the source and relay beam-forming matrices design and develop an iterative algorithm which is suboptimal for the general case but has a much lower computational complexity. For the fixed F,-, the source precoding matrix V, is optimized by solving the following problem:

Im + vh^ivi

' Vh$; + VHTT-H = 0.

Since VH and Qi are nonsingular matrices, multiply both sides by (vl^ and i; we have

Im + vh$;v7

vH = t-tH î-1.

Since Q, is Hermitian and positive definite, we apply the Choleskey decomposition of Q, = Q1^,, where Q, is a lower triangular matrix. Consequently, we represent (28)

s.t. tr {vH^iVi} < PRi, tr {VHVi} < Pi, (22)

where Qi = HlC-1Hi, C-1 = (n,)1^, = hH-FH F,^,-,- + Hl FlFjH;,,

1 1.1 1 ¡J I 1 1,1

The Lagrangian function associated with the problem (22) is given by

Im + vh^h^iv;

QH ri (vH)

= (QH)—i tjtH (QHQi)—i QH. By the definition of the matrix identity as

[IM + XX1]-i X = X [IN + XHX]—i, for any M x N matrix X, we can rewrite (29) as

Im + ^¿vvh^h

H ^ 1 TT^a-1.

Solving (31) for V-, we have

V = V,-VH -

where / > 0 is the Lagrange multiplier.

Case 1: When / = 0, making the derivative of LV with respect to V, be zero, we obtain

Im + VH $;V7

'vH$i = 0.

Since V7 and are nonsingular matrices, (24) can be represented as

where V, = (tJhQ^ 2.

Figure 3 shows our proposed relay filter design, which forwards the received signal (input) from Si amplified by a left filter (LF) matrix Fl,(- and the signal from S2 amplified by a right filter (RF) matrix Fr- to a center filter (CF) FAi. FAi amplifies the outputs from the LF matrix Fl,(- and the RF matrix Fr- (i e {i,2} denotes relay node i) and forwards them to Si and S2 (output).i

Im + vh^iv, = 0.

Simplifying (25), Im > 0 and VQiV, > 0. Consequently, in Case 1, the optimal solution is not existent.

Case 2: When / > 0, we rewrite the Lagrangian function as

Im + vh^iv,

- X1Pi - X2P7 - X3PR - X4Pr7

+ vH T7THV7 + X3VH^iV; + X1VHVi,

Fig. 3 The relay filter design of the proposed QL-QR technique

Lemma 2. The optimal relay filter constructive of FL, Fr, and FD matrices, i.e., for R1 and R2, can be designed as

FL,1 = Q£i, FL,2 = QL,2,

FR,1 = qH,1, FR,2 = qH,2,

Fd,1 and Fd,2 are diagonal matrices,

where Ql,« and Qr,« for i = 1,2 are the unitary matrices which relate to QL and QR decompositions for the dependent channel coefficients.

Proof. For Fl,< and Fr,«, the proof is similar to Theorem 3.1 in [24]. For Fd,«, using Theorem 2 in [9], the structure of FDi is optimal for the two cases of

(a) : RCH1V1HRCH2V2); (H^)±R(H*) and R(H3Vi)±R(H4V2); (H*)±R(H*)

(b) : R(HiVi) IRCH2V2); (H^)±R(H*) and R(HiVi) ||R(H2V2); (H*1)±R(H2).

(33) □

Fd,, —

fd,u 0 0 fd,i,2

If N = 2a, a = 2,3,..., the optimal Fd,, is a 2 x 2 block diagonal matrix given as

Fd,, —

FD,i,1 0

0 Fd,,,2

where FD,i,1 and FD,,,2 are N x N matrices.

Case b: The optimal FD,, is defined as

с* A FD,i —

fd,i,1 fd,i,2

IF1I = |Fi,1FD,1FR,1| = | Fd,1 | I F2 I = |FI,2FD,2FR,2| = |FD,2|.

Now, let us introduce the following QL decomposition: [HUV1, H2,1V1] = [Ql,1L1, Ql,2L2] , (38)

where Ql., for i = 1,2 is a unitary matrix with a dimension CNxN and {L1, L2} e CNxM are lower triangular matrices.

Similarly, let us introduce another decomposition, namely QR decomposition, as

[H1,2 V2, H2.2V2] = [QR.1R1, Qr,2R2] , (39)

where Qr,7 e CNxM for i = 1,2 is a unitary matrix and {R1, R2} e CMxM are upper triangular matrices. Substituting (38) and (39) back into (4), we can get equivalent received signals shown as

y = (LT Fd,,R, + lTFd.Rï) x,

+ LTFD,i nR,. + lT FD,i nRr + nj = Hi,7xî + yi,

Case a: If N = 2, the optimal FD,, is a diagonal matrix given as

where H,- = L^F^R« + LTFDiR| and n« = LTFD,inR; + LjFDlnR7 + ni are efficient channel and noise coefficients; obtained from the covariance of Q, we have

c, = HiHH

= LTFD,iFH,iL* + L^FD-FH^LÎ + IN. (41)

For fixed Vi, using (40) and Property 1, the optimal problem (20) becomes

FD,1,FD,2

(In + H^H,

s.t. tr(FH,iDiFB,0 < PRi, MSE1 = MSE2, (43)

where we have employed the principle min (a) = max (a-1), for a = 0. By using the lemma tr(A + B) = tr(A) + tr(B), (42) can be represented as

(hHC^h,

Discussion 1: In Case b, since F*Di is optimal, but the computational complexity will be considerably increased compared with Case a, so we exclude it.

For Case a, Before we develop a numerical method to solve vector FD,i, let us have some insights into the structure of this suboptimal relay beamforming matrix. To simplify relay beamforming matrix FD,,-, we introduce the following properties:

Property 1: The statistical behavior of a unitary matrix U remains unchanged when multiplied by any unitary matrix T independent of U. In other words, TU has the same distribution as U, i.e., in (33),

Since the matrix C7 is Hermitian and positive definite, we can decompose this matrix using the Cholesky factorization as

where Ei denotes a lower triangular matrix. By substituting (45) back into (44), we can simply rewrite the optimal problem as

max (MSE;)-1 = max tr (hH (3H3i)-1 H;) + n

= max tr ^(hHs-1) (i^-1)

tr (BiBH),

where = denotes n has nothing to do with the maximum solution and B. = Hj^S—1. Therefore, the optimal problem can be represented as the determinant maximization of B2 [29].

In Case a, since Fd,i is the block diagonal matrix, its determinant can be written as

detF^,,- = detF^y • detF^y

detFD,(,(, for i e 1,2, as

detFo,i,i =

A D B C

I A-1D 0 C - BA-1D

= |A| |C - BA-1D|,

A 0 B I

min tr (UB£BAH (UaSaA ^ min tr (Sb (Sa)-1 Sd)

_1 UdSd AH

A • V^ bidi

= min L —,

Consequently, we have

detH(- = |~| (li,m,mfD,i,mri,m,m + l~i,m,m^D~i,mr~i,m,m)

= 11 ta +

Let A, B, C, and D be an ^ x N matrix. We can define

where I stands for an N x N identity matrix. In (48), to obtain maximum detFD,i,i, we should minimize BA-1D. Let us introduce the SVD of B, A, and D as

B = UbsbaH, A = UaSaAH, D = Udsdad, (49)

where U,, Ai, i e {A, B, D}, are the unitary matrices and S. is an N x N diagonal matrix. Substituting (49) back into BA-1D, we have

where gi = li,m,mfD,i,mri,m,m, = h,m,mfD,i, ,mri,m,m, li,m,m, fD ,i,m, ri,m,m, limm, Dim and rI,m,m are diagonal elements

of L,, Fd,,, R,, Li, Fd-, and R-, respectively. Since S. and S—1 are also lower triangular matrices, we have

ÏHJ„+ c-1

detBi = detHfdet: i

= n (Si + Ç-tëm,

where fm is the diagonal element of S—1.

After introducing slack the variable t-, the objective problem can be equivalently converted into the optimization one with respect to an individual relay power constraint, shown as follows:

Hi (sr1)1

(B'-1H

-1ÎÏH)\

^ 0 (55)

where b-, d-, and a, are the diagonal elements of SB, SD, and SA, respectively. To simplify our discussion, we assume that FD,i,i is a semi-positive matrix; thus, we have the minimum solution as bidi = 0. Interestingly, if both bi and d, are 0, FD,i,i is a diagonal matrix. Otherwise, it is a lower/upper triangular matrix. In addition, for S1, the equivalent channel H1, since the terms LT, LT, R1, and R2 are upper triangular matrices, the optimal FD,i should be an upper triangular matrix. Since the equivalent channel H2, L1, L2, Rj1, and RT are lower triangular matrices for S2, the optimal FD,i is a lower triangular matrix. Therefore, if and only if Fd,i is a diagonal matrix, the sum-MSE is optimal in our proposed method.

This completes the proof for Lemma 2.

Property 2: For any M x N rectangular matrices G and J, matrices A and B are lower/upper triangular matrices based on QR or QL decomposition of G and J. If ai,i+bi,i = 0, where ai,i and bi,i are diagonal elements of matrices A and B, respectively, we can easily obtain

det (A + B) = Y\ (an + b,-,,-) > detA + detB. (51) i=1

tr (FD,iD.-Fg,,.) < PRt, detB2 > T, t, > 0, V, = 1,2.

From (54) to (56), it is easy to see that t, is only dependent on the beamformer FD,t and with respect to (55) and (56). Thus, by using (53), we have

Ti < O ^ (ji,m,mfD,i,mri,m,m + lI,m,mfD~i,mrI,m,m) X ^ (fm) .

From (57), it is easy to follow that with fixed FD,i/FD~i, t- is convex with regard to FD~i/FD,i. In summary, we outline the iterative beamforming design algorithm as follows (QL - QR Algorithm):

Clearly, Algorithm 1 will converge to a sub-optimal solution as t& — T^nax1"1 < e. Therefore, e is initialized to be a small value and Nmax is set to limit the number of iterations.

Discussion 2: Figure 4 displays two extended system models. One is the multi-pair scenario with two relay nodes and K pairs of source nodes. The other is a Z relay node scenario with two source nodes. In Fig. 4a, each pair of sources and two relay nodes can be seen as a group. Since each pair of source nodes are independent of one

MSE(a)

s.t. tr(FiDtF") < Pr,, tr(V«vH) < Pi, (58)

W, ,F, Vt

where MSESum = Eli det (B,-)2 + jc+i det (By)2 is

the sum-MSE of the multi-pair scenario. It is clear that (58) is a bi-convex problem which is similar to (54)-(56) with different effective channel coefficients and can be solved by our proposed algorithm.

Algorithm 1 QL-QR Algorithm

1. Initialize: F(n), W(n), V(n), e = 10-3, NmaX fori,j = 1,2, set n = 0;

2. Repeat:

1: for n = 0 to Nmax do

2: for fixed F(n-1), V(n-1) update W(n) via solving (19);

3: for given W(n) and F(n-1) update V(n) via solving (32); i i i

4: for given w(n) and V(n) do

5: fix F(n-1) and update F(n) and t(n) via solving

(54)-(56); i i i

end for

7: for given W(n), V(n) and F(n) do 8: update F(n) and r-n) via solving (54)-(56);

if r(n) _ r_(n)

Otherwise t^L break;

< 0, then t,

(n) max

end if

end for

if T(n) - T(n-i) < e then if Tmax Tmax < fc, then

break;

13: end if 14: end for

In Fig. 1b, a TWRC consisting of two source nodes and L relay nodes is considered. Obviously, by employing the MSE balancing result, the objective of the extended system model (b) is to minimize the sum-MSE, which is subject to the individual relay transmit power constraint, shown as

W, ,Fi,Vt

s.t. tr (FiD,-FH) < Pr,,, tr (VtVH) < Pt, MSElb) = MSE,

(b) e2 .

another, we can design each relay node comprised of K RFs, LFs, and one CF. Therefore, the extended system model (a) can be classified as another version of our proposed system model with K parallel nodes. The objective problem can be expressed as

where MSEt = (wHfit - Im) (wHfit - Im) H + WHC Wi, for t = 1,2, Ht = EL=1 HJTiFiHt,i, and C„t = Y^L=1 H^'j-F,FhH*7+Im, is the equivalent noise covariance, for i = 1,..., L. Due to the semi-infinite constraints at the relay node, the objective problem (59) is non-convex. In this scenario, we consider a two-stage solution where in the first step, the semi-infinite constraints are converted to linear matrix inequalities (LMI) and in the second step, we use our proposed iterative algorithm to solve it. By using the S-Lemma [28], the relay power constraint can be converted into the LMI version, we have

Ei=i PRi -Ei=i -vec (Fl)1 -vec (Fi)

-vec (Fi) 0

-vec(Fi)H ' 0

where vec(-) denotes to stack the columns of a matrix into a single vector. Now, the objective problem (59) becomes

W, ,Fi ,Vt

s.t. (60), tr(VtvH) < Pt, > 0.

In step 2, we use an iterative algorithm based on alternating convex search (ACS) to solve the resulting convex problem. The algorithm is almost the same as our proposed one which only converts [Algorithm 1, Steps 4-11] update F(n) via solving (61)".

into "for fixed W(n), V(n)

4 Computational complexity analysis

In this section, we measure the performance of the proposed QL-QR scheme in terms of the computational complexity compared with existing algorithms by using the total number of floating point operations (FLOPs). A flop is defined as a real floating operation, i.e., a real addition, multiplication, and division. In [30], the authors show the computational complexity of the real Choleskey decomposition. For complex numbers, a multiplication followed by an addition needs eight FLOPs, which leads to four

Table 1 Computational complexity of the proposed QL-QR algorithm

Step Operations FLOPS Case: (2,2,2) x 6

1 Vl, V2 2 x K (40N3 - 24N2 + 17N,) 1560

2 Ql,1 Li, QL,2 L2 2 x 16K (N2 N, - NtN2 + 3 N3) 4864

3 QR,1 R1. QR,2 R2 2 x 16K (N2 N, - NTNj + 3 N3) 4864

4 Hj Fi H 1,2 8N2 N, + 4NTN2 + 2NTNj 696

5 HT,1 F2H2,2 8N2 N, + 4NTN2 + 2NTNj 696

6 Ci 2K (32N2N, + 8NTNj + 2N2 - 4N, + 3NT) 14,856

7 K ( 134 N3 - 2N2 + NT) 2826

8 detB2 4K N + N2 + 2NT) 3168

Total 33,530

times its real computation. According to [31], the required number of FLOPs of each matrix is described as follows:

1. Multiplication of m x n and n x p complex matrices: 8mnp — 2mp;

2. Multiplication of m x n and n x m complex matrices: 4nm x (m + 1);

3. SVD of an m x n(m < n) complex matrix where only £ is obtained: 32(mn2 — n3/3);

4. SVD of an m x n(m < n) complex matrix where only £ and A are obtained: 32(nm2 + 2m3);

5. SVD of an m x n(m < n) complex matrix where U, £, and A are obtained: 8(4n2m + 8nm2 + 9m3);

6. Inversion of an m x m real matrix using Gauss-Jordan elimination: 2m3 — 2m2 + m;

7. Cholesky factorization of an m x m complex matrix: 8m3/3;

8. QR or QL decomposition of an m x n conplex matrix 16 (n2m — nm2 + 3 m3).

For the RBD method [32], the authors consider a linear MU-MIMO precoding scheme for DL MIMO systems. For the non-regenerative MIMO relay systems [33], the authors investigate a precoding design for a three-node MIMO relay network. In [2], a relay-aided system based on a quasi-EVD channel is proposed. We

compare the required number of FOLPs of our proposed method with conventional precoding algorithms, such as the RBD, the non-regenerative MIMO relay system, and the CD-BD algorithm as shown in Tables 1, 2, 3, and 4, respectively, under the assumption that NT = Nr and Ni = Nt — Ni.

For instance, the (2,2,2) x 6 case denotes a system with three users (K = 3), where each user is equipped with two antennas (Ni = 2) and the total number of transmit antennas is six (NT = 2 x 3 = 6). The required number of FLOPs of the QL-QR algorithm, the RBD, the non-regenerative MIMO relay system, and the CD-BD algorithm are counted as 33,530, 40,824, 45,306, and 34,638, respectively. From these results, we can see that the reduction in the number of FLOPs of our proposed precoding method is 17.87, 25.99, and 3.20 % on an individual basis compared to the RBD, the non-regenerative MIMO relay systems, and the CD-BD algorithm. Thus, our proposed QL-QR algorithm exhibits lower complexity than conventional algorithms. In addition, the complexity reduces as Ni and NT increase with fixed K.

We summarize our calculation results of the required number of FLOPs of the alternative methods in Tables 1, 2, 3, and 4 and show them in Figs. 5 and 6. Figure 5 shows

Table 2 Computational complexity of the non- -regenerative MIMO relay system [33]

Step Operations FLOPS Case: (2,2,2) x 6

1 8K (4N2 N, + 8NtN2 + 9N3) 13,248

2 Uf^fAf 8K (4N2 N, + 8NtN2 + 9N3) 13,248

3 H,hH, 4KNiNT (N, + 1) 432

4 H^H 4KNiNT (N, + 1) 432

5 H,H [ff,2ff22(HF)HHF + I]-1 H, 2K (N3 + 8N,N2 + 4N2NT + 2N,NT - N2 + N,) 4212

6 VA AaVH 8K (4N2 N, + 8NTN2 + 9N3 + 1N 13,272

7 diag(G) K [4N,NT(N, + 1) + 2N3 - 2N2 + N,] 462

Total 45,306

Table 3 Computational complexity of the conventional RBD [32]

Step Operations FLOPS Case: (2,2,2) x 6

1 UaXaAaH 32K (NTN2 + 2N3) 21,504

2 ((Ef)T Ei + P2') 1 K (18NtN2 - 2N2) 336

3 VaDa 8KN3T 5184

K (8NTNj - 2Nf)

64K ( 9 N3 + NtN2 + 2 N2 N, )

13,248

40,824

4 H,Pa

5 UfEfVfH Total

the computational complexity where N. = 2 and a value of K varies. And Fig. 6 shows the computational complexity where K = 4 and a value of N. varies. For the RBD method, the orthogonal complementary vector Vk,o requires K times SVD operations. If only Vk,0 is obtained, it is not computationally efficient. In step 5, the efficient channel Heff = HiPia is decomposed by the SVD with a dimension Reff x NT, where Reff is the rank of Heff. In the non-regenerative MIMO relay method and the CD-BD algorithm, two SVD operations are performed for the channels from the source to the relay and from the relay to the destination, and then the efficient channel covariance matrix is measured. In the non-regenerative MIMO relay method, the authors compute A using the EVD, and then they diagonalize G. In the CD-BD algorithm, the authors calculate Va by the SVD of Hmse, and then they structure Vb by using the Choleskey decomposition.

In our proposed QL-QR algorithm, we take advantage of QL and QR decompositions instead of the SVD operation, and then we compute an efficient channel as well as decompose a noise covariance matrix by the Choleskey decomposition. Finally, we calculate the determinant of B2 to solve an optimization problem. Obviously, our proposed QL-QR algorithm outperforms conventional algorithms in the light of the computational complexity.

5 Simulation results

In this section, we study the performance of the proposed QL-QR algorithm for two-way MIMO relay networks. All the simulations are performed on the assumption

that all the channel estimates are the Rayleigh fading channels, and they are independent and identically distributed (i.i.d.) complex Gaussian random variables. The noise variances af are equally given as a2. All the simulation results are averaged over 1000 channel trials.

In Fig. 7, we compare the sum mutual information (SMI) of various MU-MIMO schemes where full CSI is known at each node. We set P1 = P2 = 10 dB and M = 1 with an equal power budget for the two relays. The negative SMI is adopted in [16] which can be defined as

MIsum = log2 |MSE11 + log2 IMSE2I . (62)

In our proposed method, the SMI shown in the simulation results is calculated as —2 log2 |B21 by using (45), (52), and (53). It can be observed that the proposed QL-QR algorithm has the same SMI performance as an optimal solution in [16].

Figure 8 shows the performance of our proposed SMI performance versus the number of the relays, T, which is even. We consider a practical scenario with different relay power constraints and set Pr = 30 dB. It is clear that, for different values of P1 and P2, a solution of our proposed QL-QR algorithm shows better performance than a maxpower solution.

Figure 9a, b exhibits the BER performance of the BD water filling, the RBD, the SVD-RBD, and our proposed QL-QR method, where the quadrature phase shift keying (QPSK) and 16 quadrature amplitude modulation (16-QAM) are made use of. As pointed out in [35], the

Table 4 Computational complexity of the CD-BD algorithm [2]

Step Operations FLOPS Case: (2,2,2) x 6

1 UHi ,1 A,,1 8K (4N2tN, + 8NtN2 + 9N3) 13,248

2 A,H2S,tU,,2 8K (4NT N , + 8NTN2 + 9N3) 13,248

3 H ,,TWH J K [8N,NT - 2N,NT + 4N,NT x (N, + 1 )] 2088

4 LHL, 2K (N, + 2NTN, x (N, + 1) + 4N3/3) 508

5 nmse 4N3/3 + 12NTNt - TNT - 2NTNR 2736

6 H,,,VaVb 8K [4NtNT - 4N3/3 + NT(N, + 1)] 2336

7 (Q,QH + a^,)-1 K [4NRN , x (N , + 1) + 3N, + 2N3 - 2N,T] 474

Total 34,638

Fig. 5 The complexity comparisons for required FLOPs versus the number of the users K

Fig. 7 The achieved SMI for N = 4,2

BER performance for a MIMO precoding system is actually determined by the energy of the transmitted signal. To simplify our discussion, we assume a = 0. In the RBD, det (hhh) = UT=i where H e CNxM, for M < N, is an equivalent channel matrix with its eigenvalues Xi. In our proposed QL-QR method, for source node Si, we have det (HiHH) = UT=i Sr• Under the stipulation that detF^,i = 1, we are able to easily obtain Xi = Si. Therefore, our proposed QL-QR method has the same BER performance as that of the SVD-RBD method.

6 Conclusions

This paper studies a joint optimization problem of an AF based on the MIMO TWRC, where two source nodes exchange their messages with two relay nodes. A relay filter is designed, which is able to efficiently join the source and the relay nodes. Our main contribution is that the optimal beamforming vectors can efficiently be computed using determinant maximization techniques through an iterative QL-QR algorithm based on a MSE balancing method. Our proposed QL-QR algorithm can significantly reduce the computational complexity and has an equivalent BER performance to that of the SVD-BD algorithm.

Fig. 6 The complexity comparisons for required FLOPs versus the number of the receive antennas N, for each user

Fig. 8 The SMI versus the number of the relays T

10 1 ■ ---1 or 1 --

0 10 20 30 0 10 20 30

(a)SNR(dB) (b)SNR(dB)

Fig. 9 a BER performance on the Rayleigh fading channel with QPSK. b BER performance on the Rayleigh fading channel with 16-QAM

Endnote

1For example: For Si, the equivalent channel can be written as Hi = H^FiHu + H^ F2H2,2 = H^Fl,iFd,iFr,iHi,2 + H^Fl,2Fd,2Fr,2H2,2. For S2, the equivalent channel can be written as

H2 = HT2 Fi Hi,i + Hj,2 F2 H2,i =

Hl,2 FR,i FD,i FL,i Hi,i + hT,1 FR,2 FD,2 FL,2 H2,2-Competing interests

The authors declare that they have no competing interests. Acknowledgements

This work was supported by MEST 2015R1A2A1A05000977, NRF, South Korea, National Nature Science Foundation of China (61201249,61359153, 61272495), the Brain Korea 21 PLUS Project, National Research Foundation of Korea, DHU Distinguished Young Professor Program (15D210402), Natural Science Foundation of Guangdong Province (S2011040004068), China, and the scientific research foundation for returned overseas Chinese Scholars, State Education Ministry.

Author details

1 Division of Electronic and Information Engineering, Chonbuk National University, Jeonju-si, South Korea. 2School of Information Science and Technology, Donghua University, Shanghai, China. 3School of Information Science and Engineering, Central South University, Changsha, China. 4School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou, China.

Received: 15 February 2015 Accepted: 1 November 2015 Published online: 19 November 2015

References

1. JN Laneman, DNC Tse, GW Wornell, Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inf. Theory. 50(12), 3062-3080 (2004)

2. W Duan, W Song, SS Song, MH Lee, Precoding method interference management for quasi-EVD channel. Sci. World J. 2014 (2014). Article ID 678578

3. TM Cover, AAE Gamal, Capacity theorems for the relay channel. IEEE Trans. Inf. Theory. 25(5), 572-584 (1979)

4. Z Zhao, M Peng, Z Ding, W Wang, H-H Chen, Denoise-and-forward network coding for two-way relay MIMO systems. IEEE Trans. Veh. Technol. 63(2), 775-788 (2014)

5. S Simoens, O Munoz-Medina, J Vidal, Coso AD, Compress-and-forward cooperative MIMO relaying with full channel state information. IEEE Trans. Signal Pross. 58(2), 781-791 (2010)

6. B Rankov, A Wittneben, Spectral efficient protocols for half-duplex fading relay channels. IEEE J. Sel. Areas Commun. 25(2), 379-389 (2007)

7. T Koike-Akino, P Popovski, VTarokh, Optimized constellations for two-way wireless relaying with physical network coding. IEEE J. Sel. Areas Commun. 27(5), 773-787 (2009)

8. S Xu,Y Hua, Optimal design of spatial source-and-relay matrices for a non-regenerative two-way MIMO relay system. IEEE Trans. Wireless Commun. 5(10), 1645-1655 (2011)

9. Y Rong, Joint source and relay optimization for two-way linear nonregenerative MIMO relay communications. IEEE Trans. Signal Process. 60(12), 6533-6546 (2012)

10. K-J Lee, H Sung, E Park, I Lee, Joint optimization for one and two-way MIMO AF multiple-relay systems. IEEE Trans. Wireless Commun. 9(12), 3671-3681 (2010)

11. H Bolcskei, RU Nabar, O Oyman, AJ Paulraj, Capacity scaling laws in MIMO relay networks. IEEE Trans. Wireless Commun. 5(6), 1433-1444 (2006)

12. K-J Lee, KW Lee, H Sung, I Lzee, in Proc. IEEE Veh. Technol. Conf. Sum-rate maximization for two-way MIMO amplify-and-forward relaying systems (Barcelona, Spain, 2009), pp. 1-5

13. F Roemer, M Haardt, Algebraic norm-maximizing (ANOMAX) transmit strategy for two-way relaying with MIMO amplify and forward relays. IEEE Signal Process. Lett. 16(10), 909-912 (2009)

14. T Unger, A Klein, Duplex schemes in multiple antenna two-hop relaying. EURASIP J. Adv. Signal Process. 2008 (2008). Article ID 128592

15. N Jindal, Z Luo, in Proceedings of the IEEE International Symposium on Information Theory (ISIT06). Capacity limits of multiple antenna multicast (Seattle, WA, USA, 2006), pp. 1841-1845

16. R Zhang, Y-C Liang, CC Chai, S Cui, Optimal beamforming for two-way multi-antenna relay channel with analogue network coding. IEEE J. Selected Areas Commun. 27(5), 699-712 (2009)

17. R Wang, M Tao, Joint source and relay precoding designs for MIMO two-way relaying based on MSE criterion. IEEE. Trans. Signal Process. 60(3), 1352-1365 (2012)

18. H Wang, L Li, L Song, X Gao, A linear precoding scheme for downlink multiuser MIMO precoding systems. IEEE Commun. Lett. 15(6), 653-655 (2011)

19. H Park, HJ Yang, J Chun, R Adve, A closed-form power allocation and signal alignment for a diagonalized MIMO two-way relay channel with linear receivers. IEEE Trans. Signal Process. 60(11), 5948-5962 (2012)

20. V Havary-Nassab, S Shahbazpanahi, A Grami, Optimal distributed beamforming for two-way relay networks. IEEE Trans. Signal Process. 58(3), 1238-1250 (2010)

21. P Viswanath, V Anantharam, DNC Tse, Optimal sequences, power control, (1999) and user capacity of synchronous CDMA systems with linear MMSE multiuser receivers. IEEE Trans. Inf. Theory. 45(6), 1968-1983

22. SM Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. (Prentice Hall, Englewood Cilffs, NJ, 1993)

23. DC Lay, Linear Algebra and Its Applications, 4th edition. (University of Maryland Press, College Park, USA, 2012)

24. R Zhang, Y-C Liang, CC Chai, Cui S, Optimal beamforming for two-way multi-antenna relay channel with analogue network coding. IEEE J. Sel. Areas Commun. 27(5), 699-712 (2009)

25. C Lameiro, J Va, I Santamara, Amplify-and-forward strategies in the two-way relay channel with analog Tx-Rx beamforming. IEEE Trans. Veh. Technol. 62(2), 642-654 (2013)

26. Y Rong, Simplified Algorithms for optimizing multiuser multi-hop MIMO relay systems. IEEE Trans. Commun. 59(10), 2896-2904 (2011)

27. S Boyd, L Vandenberghe, Convex Optimization. (Cambridge Univ. Press, Cambridge, UK, 2004)

28. Y Huang, D Palomar, S Zhang, Lorentz-positive maps and quadratic matrix inequalities with applications to robust MISO transmit beamforming. IEEE Trans. Signal Process. 61(5), 1121-1130 (2013)

29. A Benavoli, L Chisci, A Farina, Estimation of constrained parameters with guaranteed MSE improvement. IEEE Trans. Signal Process. 55(4), 1264-1274 (2007)

30. X Gao, L Li, P Zhang, L Song, G Zhang, Q Wang, H Tian, Low-complexity RBD precoding method for downlink of multi-user MIMO system. Electronics Lett. 46(23), 1574-1576 (2010)

31. GH Golub, CF Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, 3rd edition. (Johns Hopkins University Press, Baltimore, Md, USA, 1996)

32. H Wang, L Li, L Song, X Gao, A linear precoding scheme for downlink multiuser MIMO precoding systems. IEEE Commun. Lett. 15(6), 653-655 (2011)

33. R Mo, YH Chew, Precoder design for non-regenerative MIMO relay systems. IEEE Trans. Wireless Commun. 8(10), 5041-5049 (2009)

34. S Vishwanath, N Jindal, A Goldsmith, in Proceedings of the International Conference on Communications (ICC 02). On the capacity of multiple input multiple output broadcast channels (NY, USA, 2002), pp. 1444-1450

35. CB Peel, BM Hochwald, AL Swindlehurst, A vector-perturbation technique for near capacity multiantenna multiuser communication—part I: channel inversion and regularization. IEEE Trans. Commun. 52(1), 195-202 (2005)

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