# Linear Precoding Scheme Design for MIMO Two-Way Relay Systems with Imperfect Channel State Information

Authors: Xiao-min Chen; Jun-xu Su; Qiu-ming Zhu; Xu-jun Hu; Zhu Fang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The aim of this paper is to investigate a linear precoding scheme design for a multiple-input multiple-output two-way relay system with imperfect channel state information. The scheme design is simplified as an optimal problem with precoding matrix variables, which is deduced with the maximum power constraint at the relay station based on the minimum mean square error criterion. With channel feedback delay at both ends of the channel and the channel estimation errors being taken into account, we propose a matrix decomposition scheme and a joint iterative scheme to minimize the average sum mean square error. The matrix decomposition method is used to derive the closed form of the relay matrix, and the joint iterative algorithm is used to optimize the precoding matrix and the processing matrix. According to numerical simulation results, the matrix decomposition scheme reduces the system bit error rate (BER) effectively and the joint iterative scheme achieves the best performance of BER against existing methods.


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## Body

## 1. Introduction

Multiple-input multiple-output (MIMO) technology has become the key technology owing to its ability to increase system capacity and improve spectral efficiency without increasing the bandwidth [1]. The relay technology is a common space diversity method that can effectively reduce the effects of channel fading. The combination of MIMO with relay technology can further improve the capacity and system performance of the relay network. There has been extensive theoretical research on the performance improvement of unidirectional MIMO relay system [2, 3]. In fact, the MIMO two-way relay system has received great deal of attention recently owing to its high spectrum efficiency [4, 5].Some existing studies have investigated for MIMO relay systems. In the actual communication system, the reliability of the communication link could be ensured by minimizing the mean square error (MSE) of the receiver. In references [6, 7] analysis under the condition of perfect channel state information (CSI), the precoding algorithm regards the MSE at two source nodes as the optimization target; the study [8] presents the complete CSI and the joint optimization method based on the criterion of MSE duality of two-way MIMO relay system. In [9], the precoding algorithm of amplify-and-forward MIMO two-way relay system based on channel capacity analysis is studied under limited channel feedback condition. However, the precoding scheme only considered the power constraints of the relay node without considering the power constraints of the transmitter. The paper [10] details the design of the combined precoding scheme for the two-way relay system, using the iterative method to optimize the combined precoding matrix. However, due to the time-varying wireless channel noise, the existence of the desired CSI is not realistic. References [11–13] researched for the amplify-and-forward two-way MIMO relay system, assuming that, in the relay terminal known source-relay, relay destination channel from the channel estimation errors and transmit antenna correlation proposed a joint transceiver design scheme based on the imperfect CSI. The study [14] presents the precoding algorithm with consideration channel feedback delay and channel estimation errors; however, it is only suitable for the MIMO system.Although some excellent works about precoding design have been done for MIMO two-way relay networks [15–17], most of them have not taken the channel estimation errors and feedback delay into account. However, in practical networks, due to the limitation of the channel estimation method and the existence of channel feedback delay, the ideal CSI is difficult to obtain. This motivates us to investigate the problem of optimal precoding design in MIMO two-way relay networks in the presence of channel estimation errors and feedback delay. In this paper, a MIMO two-way relay system is considered where the channel estimation error and feedback delay between relay node and the destination exist. At the relay node, the linear precoding design method is produced based on the minimum mean square error (MMSE) rule. Finally, the optimal precoding matrix and the destination linear matrix are derived using the theoretical proof.


## 2. System Model and Channel Model

### 2.1. System Model

Considering MIMO two-way relay system as shown in Figure1, which consists of two source nodes S1 and S2 with one relay node denoted by F, both S1 and S2 are equipped with ns antennas. Relay node F has nr antennas. At present, the research on the performance of MIMO two-way relay system is based on the model of time division duplex (TDD). During the first phase, source nodes S1 and S2 transmit their information to relay node F. In the second time slot, relay node multiplies its received signal by a linear precoding matrix F∈CNr×Nr and forwards it to S1 and S2, respectively. It is assumed that all nodes are operating in half-duplex mode due to the hardware complexity and the relay node transmits and receives in two orthogonal time slots.Figure 1
Diagram of MIMO two-way relay system.In the first time slot, source nodesSi(i=1,2) transmit information xi∈Cns×1 to relay node, xi is satisfied with covariance matrix Rxi=ε[xixiH], ε[∙] donates the statistical expectation, and then the signal vector ys∈Cnr×1 received at relay node is(1)ys=H1x1+H2x2+nr.Here Hi∈Cnr×ns represents the channel matrix of Si to F and nr∈Cnr×1 denotes the additive white Gaussian noise (AWGN) vector with covariance matrix Rnr=ε[nrnrH].In the second time slot, relay node multiplies its received signal and forwards it toS1 and S2, respectively. The received signal yi∈Cns×1 at Si can be expressed as(2)yi=HiHFHjxj+HiHFHixi+HiHFnr+ni,where j=2 if i=1 and j=1 if i=2. ni∈Cns×1 denotes the AWGN vector with covariance matrix Rni=ε[niniH].


### 2.2. Channel Model

Considering the channel estimation error and channel feedback delay, the channel matrix at time instantt can be expressed as [18, 19](3)Ht=H^t+Tt+Ξt,where Ξt denotes the channel estimation error matrix and Tt is the delay estimation error matrix. With a certain amount of estimation error the channel matrix Ht changes to be H^t. In addition, after experiencing a period of time τ for feedback delay, H^t turns to be ρH^t-τ. Then the channel matrix Ht can be written as [19](4)Ht=ρH^t-τ+Tt+Ξt,where ρ is the channel correlation coefficient formulated as in [19]. For Clarke’s fading spectrum, ρ could be obtained as ρ=J02(2πfdτ), and here J0(∙) is the zeroth-order Bessel function of the first kind and fd is the Doppler frequency [20].In this paper, we discuss the channel matrix of source-relay and relay-source at a certain time, and correlation coefficientρ is related to the delay coefficient τ. For the convenience of expression, Ht,i and H^t-τ,i could be written as Ht and H^i. Therefore, our channel matrix could be further expressed as [21](5)Hi=ρiH^i+Ti+Ξi.Ξt is independent of Ti and its elements are subjected to CN(0,σe2). CN(u,σ2) represents the complex Gauss distribution with a mean value of u and a variance of σ2. Let Σi=Ti+Ξt and Ht can be noted as (6)Hi=ρiH-i+Σi.Each element of Σi is subjected to CN(0,(1-ρi2)(1-σei2)+σei2). Let σΣi2=(1-ρi2)(1-σei2)+σei2. The receiver signal y~i at Si is(7)y~i=yi-ρiH-iHFH-ixi=HiHFHjxj+HiHFnr+ni+ki,where ki=(HiHFHi-ρiH-iHFH-i)xi is the residual self-interference (SI) with covariance matrix Rki=ε[kikiH]. When the detection matrix Qi is employed by Si, the mean squared error (MSE) of the detected signal at Si can be expressed as below: (8)MSEiF,Qi=εQiy~i-xj2.


## 2.1. System Model

Considering MIMO two-way relay system as shown in Figure1, which consists of two source nodes S1 and S2 with one relay node denoted by F, both S1 and S2 are equipped with ns antennas. Relay node F has nr antennas. At present, the research on the performance of MIMO two-way relay system is based on the model of time division duplex (TDD). During the first phase, source nodes S1 and S2 transmit their information to relay node F. In the second time slot, relay node multiplies its received signal by a linear precoding matrix F∈CNr×Nr and forwards it to S1 and S2, respectively. It is assumed that all nodes are operating in half-duplex mode due to the hardware complexity and the relay node transmits and receives in two orthogonal time slots.Figure 1
Diagram of MIMO two-way relay system.In the first time slot, source nodesSi(i=1,2) transmit information xi∈Cns×1 to relay node, xi is satisfied with covariance matrix Rxi=ε[xixiH], ε[∙] donates the statistical expectation, and then the signal vector ys∈Cnr×1 received at relay node is(1)ys=H1x1+H2x2+nr.Here Hi∈Cnr×ns represents the channel matrix of Si to F and nr∈Cnr×1 denotes the additive white Gaussian noise (AWGN) vector with covariance matrix Rnr=ε[nrnrH].In the second time slot, relay node multiplies its received signal and forwards it toS1 and S2, respectively. The received signal yi∈Cns×1 at Si can be expressed as(2)yi=HiHFHjxj+HiHFHixi+HiHFnr+ni,where j=2 if i=1 and j=1 if i=2. ni∈Cns×1 denotes the AWGN vector with covariance matrix Rni=ε[niniH].


## 2.2. Channel Model

Considering the channel estimation error and channel feedback delay, the channel matrix at time instantt can be expressed as [18, 19](3)Ht=H^t+Tt+Ξt,where Ξt denotes the channel estimation error matrix and Tt is the delay estimation error matrix. With a certain amount of estimation error the channel matrix Ht changes to be H^t. In addition, after experiencing a period of time τ for feedback delay, H^t turns to be ρH^t-τ. Then the channel matrix Ht can be written as [19](4)Ht=ρH^t-τ+Tt+Ξt,where ρ is the channel correlation coefficient formulated as in [19]. For Clarke’s fading spectrum, ρ could be obtained as ρ=J02(2πfdτ), and here J0(∙) is the zeroth-order Bessel function of the first kind and fd is the Doppler frequency [20].In this paper, we discuss the channel matrix of source-relay and relay-source at a certain time, and correlation coefficientρ is related to the delay coefficient τ. For the convenience of expression, Ht,i and H^t-τ,i could be written as Ht and H^i. Therefore, our channel matrix could be further expressed as [21](5)Hi=ρiH^i+Ti+Ξi.Ξt is independent of Ti and its elements are subjected to CN(0,σe2). CN(u,σ2) represents the complex Gauss distribution with a mean value of u and a variance of σ2. Let Σi=Ti+Ξt and Ht can be noted as (6)Hi=ρiH-i+Σi.Each element of Σi is subjected to CN(0,(1-ρi2)(1-σei2)+σei2). Let σΣi2=(1-ρi2)(1-σei2)+σei2. The receiver signal y~i at Si is(7)y~i=yi-ρiH-iHFH-ixi=HiHFHjxj+HiHFnr+ni+ki,where ki=(HiHFHi-ρiH-iHFH-i)xi is the residual self-interference (SI) with covariance matrix Rki=ε[kikiH]. When the detection matrix Qi is employed by Si, the mean squared error (MSE) of the detected signal at Si can be expressed as below: (8)MSEiF,Qi=εQiy~i-xj2.


## 3. Problem Formulation

In this paper, a joint design of the relay precoding matrixF and the detection matrixes Q1 and Q2 is proposed with the maximum transmission power constraint based on the MMSE criterion [22]. Taking the channel feedback delay and estimation errors into account, the MSE expression of Si can be further represented as [23](9)MSEiF,Qi=TrεΣiQiy~i-xj2=TrεΣ1,Σ2QiHiHFHjRxjQiHiHFHjH+TrεΣiQiHiHFRnrFHHiQiH-Tr2ReεΣ1,Σ2QiHiHFHjRxi+TrεΣiQiRkiQiH+TrRxj+TrQiRniQiH,where Tr[∙]is the trace of the matrix and Re[∙] denotes the real part. It is assumed that each of the elements in a random matrix x∈Cn×n is an independent and identically distributed zero mean complex Gauss random variable, x is satisfied with Tr(xAxH)=σx2In, and here In is n×n unit matrix. We assume the variance of Σi is much smaller than one. Hence, we have(10)TrεΣiQiRkiQiH=ρiTrεΣiQiH-iHFΣiRxiΣiHFHH-i+ΣiHFH-iRxiH-iHFHΣi+H-iHFΣiRxiH-iHFHΣi+H-iHFΣiRxiH-iHFHΣQiH=ρiTrQiH-iHFZiFHH-iQiHTrRxi+TrFH-iRxiH-iHFHQiZiQiH.Here Zi=σΣi2Ins. In the same way, some simplifications can be written as(11)TrεΣ1,Σ2QiHiHFHjRxjQiHiHFHjH=ρiρjTrQiH-iHFH-jRxjH-jHFHH-iQiH+ρiTrQiH-iHFZiFHH-iQiHTrRxi+ρjTrQiQiHTrFH-jRxjH-jHFHZi+ρiTrQiH-iHH-iQiHTrFZiFHTrRxi=ρiTrQiH-iHFΠjFHH-iQiH+TrQiZiQiHTrFΠjFH,TrεΣiQiHiHFRnrFHHiQiH=TrεΣiQiρiH-iH+ΣiHFRnrFHρiH-i+ΣiQiH=ρiTrQiH-iHFRnrFHH-iQiH+TrQiZiQiHTrFRnrFH,Tr2ReεΣ1,Σ2QiHiHFHjRxi=2ρiρjTrReQiH-iHFH-jRxi,where Πj=H-jRxjH-jH+Tr(Rxj)Zi; substituting (10)~(11) into (9), the original expression (9) can be simplified as(12)MSEiF,Qi=ρiTrQiH-iHFΘiFHH-iQiH+TrRsj+TrQiΦiQiH-2ρiρjReTrQiH-iHFH-jRxi,where(13)Θi=Πj+Rnr+TrRxiZi,Φi=TrFΠj+Rnr+ρiH-iRxiH-iHFHZi+Rni.We are now ready to design detection matrix Qi and precoding matrix F to minimize the average sum MSE of Si. The problem can be formulated as (14)minQ1,Q2,FMSE=MSE1F,Q1+MSE2F,Q2s.t.TrFRrFH≤Prand here(15)Rr=εΣiysysH=Π1+Π2+Rnr.


## 4. The MMSE-Based Joint Precoding Scheme Design

### 4.1. Matrix Decomposition Algorithm Design

Since the joint iterative algorithm is being proposed, this paper introduces a kind of matrix decomposition algorithm with relatively low computational complexity. For (12), when the precoding matrix F is determined, the optimal linear detection matrix Qi can be uniquely determined.From the following condition∂MSE/∂QQi=Qi∗=0, Qi can be written as(16)Qi=ρjRxiH-jHFHH-jρiH-iHFΘiFHH-i+Φi-1.Substituting (16) into (12), then (12) can be expressed as (17)MSEiF,Qi=TrRxj-ρiρjTrRxjH-jHFHH-iρiH-iHFΘiFHH-i+Φi-1H-iHFH-jRxj=TrRxj-ρjTrRxjH-jHΘi-1H-jRxj+piF,where(18)piF=ρjTrRxjH-jHΘi-1/2ρiQi1/2FHH-iΦi-1H-iHFQi1/2+Ins-1Θi-1/2H-jRxj.Observing that only the term pi(F) is related to F, the optimization problem in (14) is equivalent to minimizing p1(F)+p2(F). Assuming pi(F)≤p-i(F) by contraction method, where(19)p-iF=ρjTrΘi-1/2H-jRxj2H-jHΘi-1/2ρiΘi1/2FHH-iΦ-i-1H-iHFΘi1/2+Ins-1and here Φ-i=PrZi+Rni, hence the original optimization problem (14) can be simplified as(20)minFMSE=p-1F+p-2Fs.t.TrFRrFH≤Pr.Next, using matrix decomposition method to simplify the structure of p-i(F), it can be rewritten as(21)p-iF=TrXiΘi1/2FHΞiFΘi1/2+Ins-1and here Ξ=H-iHΦ-i-1H-i and Xi=Θi-1/2H-jRxj2H-jHΘi-1/2. Xi can be expressed as Xi=UXiΛXiUXiH by EVD decomposition, UXi is unitary matrix, and ΛXi is diagonal matrix. Expression (21) can be described as(22)p-iF=ρjTrΛXiUXiHΘi1/2FHΞiH/2Ξi1/2FΘi1/2+Ins-1UXi=ρjTrΛXiYiHFHΞiH/2Ξi1/2FYi+Ins-1and here Yi=Θi1/2UXi. We propose the following structure:(23)F=UΞΛFUY-1.Here, UΞ is unitary, ΛF is nr×nr diagonal, and UY is the nonsingular matrix obtained by the generalized singular value decomposition (GSVD) from Y1 and Y2. YiHcan be written as(24)YiH=VYiΛYiUYH,where VYi is unitary and ΛYi is diagonal. UΞH is obtained from the following singular value decomposition (SVD):(25)Ξi1/2=VΞiV~ΞiΛΞ0UΞHand here ΛΞ is nr×nr diagonal, from (25), and Ξi1/2=VΞiΛΞUΞH. Substituting (23)~(25) into (22) gives(26)p-iF=ρjTrAiρiVΞiHVΞiΛΞ2ΛF2ΛYi2+Ins-1and here Ai=VYiHΛXiVYi. In order to achieve some simple solutions, we further relax the problem by only considering the main diagonal of VΞiHVΞi. Finally, the problem in (20) is simplified to the following problem:(27)minΛFρjTrAiIns+UΞiΛΞ2ΛBi2ΛF2-1s.t.TrDΛF2≤Pr,where UΞi=diag(VΞiHVΞi) and D=UB-1RrUB-H; the optimization problem (27) can be rewritten in scalar form as(28)minη1,η2,…,ηns∑i=12∑n=1nsρiρjϖin1+μinηns.t.∑n=1nsdnηn≤Pr,where ηn, μin, ϖin, and dn are the diagonal entries of ΛF2, UΞiΛΞ2ΛBi2, Ai, and D. It is easy to recognize that (28) is a convex optimization problem [24, 25]; translating the nonconvex problem of formula (28) into a convex optimization problem by Lagrange multiplier method, λ is the Lagrange multiplier, and the Lagrange function is constructed for(29)Lηn,λ=∑i=12∑n=1Nsρiρjϖin1+μinηn+λ∑n=1Ndnηn-Pr.∂L/∂ηn=0 gives(30)ηn=-2λdnμin+4λ2dn2μin2-4λdnμin2λdn-∑i=12ρiρjμinϖin2λdnμin2=-1∑i=12μin+∑i=12ρiρjϖinλdnμin.λ is in accord with(31)λ=∑i=12ρiρjμinϖindn1+μinηn2≤∑i=12ρiρjμinϖindn≤max⁡∑i=12ρiρjμinϖindn,∀=1,2,…,nsand then(32)0≤λ≤max⁡∑i=12ρiρjμinϖindn,∀=1,2,…,ns.λ should be chosen from bisection algorithm and satisfied with ∑n=1nsdnηn=Pr, ηn could be obtained by substituting λ into (30), and then ΛF2, F, Q1, and Q2 will be gotten as follows.


### 4.2. Joint Iterative Algorithm Design

When the detection matrixQi is determined, the optimization problem of (14) can be translated into a convex optimization problem of relay precoding matrix F. The Lagrangian function of (14) is formulated as(33)LF,λ=minQ1,Q2,FMSE+λTrFRrFH-Pr.Seeking partial derivative of F∗(34)∂L∂FF=F∗=∑i=12ρiH-iQiHQiH-iHFΘi+∑i=12TrQiZiQiHFΩj-∑i=12ρiρjH-iQiHRxjH-jH+λFRr,where Ωj=Πj+H-iRxiH-iH+Rnr; together with the complementary slackness and the relay station transmit power constraint, we obtain the KKT conditions as(35)∂L∂FF=F∗=0,(36)λTrFRrFH-Pr=0,(37)TrFRrFH≤Prand from (34) and (35) one has(38)vecF=Γ-1vecQ,where(39)Γ=∑i=12ΘiT⊗ρiH-iQiHQiH-iH+∑i=12ΩjT⊗TrQiZiQiH+λRrT⊗Ins,Β=∑i=12ρiρjH-iQiHRxjH-jH.The Lagrange multiplier is determined by (36) and (37). It satisfied [26] (40)0≤λ≤TrRrT⊗Ins-1λ1vecΒvecHΒλ1RrPrand here ⊗ denotes Kronecker product, λ1(A) represents the maximum eigenvalue of the matrix A, and λ can be obtained by bisection algorithm with its upper and lower bounds. By (17) and (38), we can see that when the signal matrix x of the source node is certain, the relay matrix and the detection matrix are correlated. A joint iterative design algorithm is proposed as shown in Algorithm 1.Algorithm 1:Joint optimization of iterative algorithm.
Step of Joint Optimization of Iterative Algorithm
Step 1. Initialize xi(0)=Ins, F(0)=Pr/Tr(Rr)Ins, ζ>0, n=0, Itermax
Step 2. Repeat
Step 3. Increment counter n←n+1
Step 4. Optimize and update Φ1(n), Φ2(n)
Step 5. Optimize and update Q1(n), Q2(n)
Step 6. Optimize and update F(n)
Step 7. Until minQ1(n),Q2(n),F(n)⁡MSE-minQ1(n-1),Q2(n-1),F(n-1)⁡MSEF2≤ζ  or  n>Itermax
Step 8. Compute B and λ, F(n)←Fopt  and  Q1(n)←Q1,opt, Q2(n)←Q2,opt


## 4.1. Matrix Decomposition Algorithm Design

Since the joint iterative algorithm is being proposed, this paper introduces a kind of matrix decomposition algorithm with relatively low computational complexity. For (12), when the precoding matrix F is determined, the optimal linear detection matrix Qi can be uniquely determined.From the following condition∂MSE/∂QQi=Qi∗=0, Qi can be written as(16)Qi=ρjRxiH-jHFHH-jρiH-iHFΘiFHH-i+Φi-1.Substituting (16) into (12), then (12) can be expressed as (17)MSEiF,Qi=TrRxj-ρiρjTrRxjH-jHFHH-iρiH-iHFΘiFHH-i+Φi-1H-iHFH-jRxj=TrRxj-ρjTrRxjH-jHΘi-1H-jRxj+piF,where(18)piF=ρjTrRxjH-jHΘi-1/2ρiQi1/2FHH-iΦi-1H-iHFQi1/2+Ins-1Θi-1/2H-jRxj.Observing that only the term pi(F) is related to F, the optimization problem in (14) is equivalent to minimizing p1(F)+p2(F). Assuming pi(F)≤p-i(F) by contraction method, where(19)p-iF=ρjTrΘi-1/2H-jRxj2H-jHΘi-1/2ρiΘi1/2FHH-iΦ-i-1H-iHFΘi1/2+Ins-1and here Φ-i=PrZi+Rni, hence the original optimization problem (14) can be simplified as(20)minFMSE=p-1F+p-2Fs.t.TrFRrFH≤Pr.Next, using matrix decomposition method to simplify the structure of p-i(F), it can be rewritten as(21)p-iF=TrXiΘi1/2FHΞiFΘi1/2+Ins-1and here Ξ=H-iHΦ-i-1H-i and Xi=Θi-1/2H-jRxj2H-jHΘi-1/2. Xi can be expressed as Xi=UXiΛXiUXiH by EVD decomposition, UXi is unitary matrix, and ΛXi is diagonal matrix. Expression (21) can be described as(22)p-iF=ρjTrΛXiUXiHΘi1/2FHΞiH/2Ξi1/2FΘi1/2+Ins-1UXi=ρjTrΛXiYiHFHΞiH/2Ξi1/2FYi+Ins-1and here Yi=Θi1/2UXi. We propose the following structure:(23)F=UΞΛFUY-1.Here, UΞ is unitary, ΛF is nr×nr diagonal, and UY is the nonsingular matrix obtained by the generalized singular value decomposition (GSVD) from Y1 and Y2. YiHcan be written as(24)YiH=VYiΛYiUYH,where VYi is unitary and ΛYi is diagonal. UΞH is obtained from the following singular value decomposition (SVD):(25)Ξi1/2=VΞiV~ΞiΛΞ0UΞHand here ΛΞ is nr×nr diagonal, from (25), and Ξi1/2=VΞiΛΞUΞH. Substituting (23)~(25) into (22) gives(26)p-iF=ρjTrAiρiVΞiHVΞiΛΞ2ΛF2ΛYi2+Ins-1and here Ai=VYiHΛXiVYi. In order to achieve some simple solutions, we further relax the problem by only considering the main diagonal of VΞiHVΞi. Finally, the problem in (20) is simplified to the following problem:(27)minΛFρjTrAiIns+UΞiΛΞ2ΛBi2ΛF2-1s.t.TrDΛF2≤Pr,where UΞi=diag(VΞiHVΞi) and D=UB-1RrUB-H; the optimization problem (27) can be rewritten in scalar form as(28)minη1,η2,…,ηns∑i=12∑n=1nsρiρjϖin1+μinηns.t.∑n=1nsdnηn≤Pr,where ηn, μin, ϖin, and dn are the diagonal entries of ΛF2, UΞiΛΞ2ΛBi2, Ai, and D. It is easy to recognize that (28) is a convex optimization problem [24, 25]; translating the nonconvex problem of formula (28) into a convex optimization problem by Lagrange multiplier method, λ is the Lagrange multiplier, and the Lagrange function is constructed for(29)Lηn,λ=∑i=12∑n=1Nsρiρjϖin1+μinηn+λ∑n=1Ndnηn-Pr.∂L/∂ηn=0 gives(30)ηn=-2λdnμin+4λ2dn2μin2-4λdnμin2λdn-∑i=12ρiρjμinϖin2λdnμin2=-1∑i=12μin+∑i=12ρiρjϖinλdnμin.λ is in accord with(31)λ=∑i=12ρiρjμinϖindn1+μinηn2≤∑i=12ρiρjμinϖindn≤max⁡∑i=12ρiρjμinϖindn,∀=1,2,…,nsand then(32)0≤λ≤max⁡∑i=12ρiρjμinϖindn,∀=1,2,…,ns.λ should be chosen from bisection algorithm and satisfied with ∑n=1nsdnηn=Pr, ηn could be obtained by substituting λ into (30), and then ΛF2, F, Q1, and Q2 will be gotten as follows.


## 4.2. Joint Iterative Algorithm Design

When the detection matrixQi is determined, the optimization problem of (14) can be translated into a convex optimization problem of relay precoding matrix F. The Lagrangian function of (14) is formulated as(33)LF,λ=minQ1,Q2,FMSE+λTrFRrFH-Pr.Seeking partial derivative of F∗(34)∂L∂FF=F∗=∑i=12ρiH-iQiHQiH-iHFΘi+∑i=12TrQiZiQiHFΩj-∑i=12ρiρjH-iQiHRxjH-jH+λFRr,where Ωj=Πj+H-iRxiH-iH+Rnr; together with the complementary slackness and the relay station transmit power constraint, we obtain the KKT conditions as(35)∂L∂FF=F∗=0,(36)λTrFRrFH-Pr=0,(37)TrFRrFH≤Prand from (34) and (35) one has(38)vecF=Γ-1vecQ,where(39)Γ=∑i=12ΘiT⊗ρiH-iQiHQiH-iH+∑i=12ΩjT⊗TrQiZiQiH+λRrT⊗Ins,Β=∑i=12ρiρjH-iQiHRxjH-jH.The Lagrange multiplier is determined by (36) and (37). It satisfied [26] (40)0≤λ≤TrRrT⊗Ins-1λ1vecΒvecHΒλ1RrPrand here ⊗ denotes Kronecker product, λ1(A) represents the maximum eigenvalue of the matrix A, and λ can be obtained by bisection algorithm with its upper and lower bounds. By (17) and (38), we can see that when the signal matrix x of the source node is certain, the relay matrix and the detection matrix are correlated. A joint iterative design algorithm is proposed as shown in Algorithm 1.Algorithm 1:Joint optimization of iterative algorithm.
Step of Joint Optimization of Iterative Algorithm
Step 1. Initialize xi(0)=Ins, F(0)=Pr/Tr(Rr)Ins, ζ>0, n=0, Itermax
Step 2. Repeat
Step 3. Increment counter n←n+1
Step 4. Optimize and update Φ1(n), Φ2(n)
Step 5. Optimize and update Q1(n), Q2(n)
Step 6. Optimize and update F(n)
Step 7. Until minQ1(n),Q2(n),F(n)⁡MSE-minQ1(n-1),Q2(n-1),F(n-1)⁡MSEF2≤ζ  or  n>Itermax
Step 8. Compute B and λ, F(n)←Fopt  and  Q1(n)←Q1,opt, Q2(n)←Q2,opt


## 5. Simulation Results and Analysis

In this section, simulation results are provided to evaluate the performance of the proposed approaches in a flat Rayleigh fading environment. We consider that the source nodesS1 and S2 are equipped with ns=4 antennas. Relay node F is equipped with nr=4 antennas. The source node transmits independent uncoded QPSK symbol streams to the corresponding receiver. The covariance of the transmit signal at source nodes S1 and S2 is Rx1=Rx2=(Ps/ns)Ins, and here Ps is the transmit power at each source node; let Ps=ns. The covariance of the AWGN is satisfied with Rnr=σr2Inr and Rn1=Rn2=σs2Inr. We investigate the scenarios with various SNR1=Ps/(nsσs2) while fixing SNR2=Pr/(nrσr2). SNR1 and SNR2 are the averagesignal-to-noise ratio (SNR) for the first phase and the second phase of the two-way communication protocol. The results are averaged over 5000 independent channel realizations, and in each realization 2000 QPSK symbols are transmitted for each data stream. In order to verify the superiority of the proposed algorithm, the proposed methods are compared with the existing methods as follows:(i) Amplify-and-forward (AF) relay approach [27] is(41)FAF∝PrTrRrInr.(ii) Zero-forcing (ZF) relay approach [28] is(42)FZF=∑i=12ρiH-iQiHQiH-iHΘi+λRr-1∑i=12ρiρjH-iQiHRxjH-jH,Qi∝Inr.Figures 2 and 3 are given the performance comparison of BER while fixing estimated errors σΣ2=σΣ12=σΣ22 and delay coefficients fdτ1=fdτ2=fdτ. Simulation parameters are as follows: iterations Itermax=30 and convergence threshold ζ=10-4.Figure 2
BER versusSNR with σΣ2=0.01.Figure 3
BER versusSNR with fdτ=0.05.Figure4 is given the performance comparison of BER with different iteration numbers. Simulation parameters are as follows: σΣ2=0.002, fdτ=0.02, and convergence threshold ζ=10-4.Figure 4
BER performance comparisons with different iterations.Figures5 and 6 are the BER performance comparisons in the two kinds of SNR while fixing feedback delay coefficient fdτ=0.02 and estimation error variance σΣ2=0.02, respectively. The simulation parameters are as follows: iterations Itermax=30 and convergence threshold ζ=10-4.Figure 5
BER performance comparisons with differentσΣ2.Figure 6
BER performance comparisons with differentfdτ.From the above simulations, the following conclusions can be drawn:(1)
From Figures2 and 3, the joint iterative algorithm achieved 3.2 dB and 4 dB performance gain compared to the ZF scheme and AF scheme while fixing fdτ=0.05, σΣ2=0.005, and BER=0.05, respectively. The matrix decomposition algorithm could obtain 2 dB and 3 dB performance gain compared to the ZF scheme and AF scheme. And it is obvious that matrix decomposition algorithm could get the optimal system BER performance when the channel estimation error and feedback delay coefficient are certain in low SNR conditions. As the SNR increases to 10 dB, the joint iterative algorithm obtains the best BER performance. Compared to the AF relay approach and ZF relay approach, the proposed two schemes have obvious advantages. Simulation results show that the proposed algorithm significantly improves the BER performance compared with the existing approaches. It is because the used AF relay scheme only amplifies and forwards the signal, while the existing ZF scheme ignores the interference signal of the relay node and performs the weighting processing on signal only. Moreover, the proposed method not only considers the channel estimation errors and feedback delay at each node but also can design each node matrix jointly. Thus, it can obtain obvious performance superiority over the other two methods.(2)
As shown in Figure4, the system BER performance could be improved by 1-2 dB with the increase of the number of iterations. However, this increase is limited. When the iteration number is more than 30 times, BER remains almost constant. By defining the threshold value of iterations as 30, the joint iterative method can obtain the best performance.(3)
From Figures5 and 6, we find that the system BER performance has a downward trend with the increase of the feedback delay coefficient or the channel estimation error; however, the proposed scheme achieves the best BER performance because it has a certain extent compensated with feedback delay coefficient and estimation error.


## 6. Conclusion

This paper investigates a linear precoding scheme design for the MIMO two-way relay system with imperfect CSI. The transceiver design is simplified as an optimal problem with precoding matrix variables, which is deduced with the maximum power constraint at the relay station based on the MMSE criterion. With channel feedback delay at both ends of the channel and the channel estimation errors being taken into account, a matrix decomposition scheme and a joint iterative scheme are proposed to minimize the BER. Numerical simulation results show that the matrix decomposition scheme can improve the system BER effectively and the joint iterative scheme can achieve the best BER performance with the increase of SNR against existing methods.



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*Source: 5164681-2017-04-12.xml*