Abstract
The aim of this paper is to investigate a linear precoding scheme design for a multipleinput multipleoutput twoway 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.
1. Introduction
Multipleinput multipleoutput (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 twoway 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 twoway MIMO relay system. In [9], the precoding algorithm of amplifyandforward MIMO twoway 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 twoway relay system, using the iterative method to optimize the combined precoding matrix. However, due to the timevarying wireless channel noise, the existence of the desired CSI is not realistic. References [11–13] researched for the amplifyandforward twoway MIMO relay system, assuming that, in the relay terminal known sourcerelay, 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 twoway 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 twoway relay networks in the presence of channel estimation errors and feedback delay. In this paper, a MIMO twoway 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 twoway relay system as shown in Figure 1, which consists of two source nodes and with one relay node denoted by , both and are equipped with antennas. Relay node has antennas. At present, the research on the performance of MIMO twoway relay system is based on the model of time division duplex (TDD). During the first phase, source nodes and transmit their information to relay node . In the second time slot, relay node multiplies its received signal by a linear precoding matrix and forwards it to and , respectively. It is assumed that all nodes are operating in halfduplex mode due to the hardware complexity and the relay node transmits and receives in two orthogonal time slots.
In the first time slot, source nodes transmit information to relay node, is satisfied with covariance matrix , donates the statistical expectation, and then the signal vector received at relay node isHere represents the channel matrix of to and denotes the additive white Gaussian noise (AWGN) vector with covariance matrix .
In the second time slot, relay node multiplies its received signal and forwards it to and , respectively. The received signal at can be expressed aswhere if and if . denotes the AWGN vector with covariance matrix .
2.2. Channel Model
Considering the channel estimation error and channel feedback delay, the channel matrix at time instant can be expressed as [18, 19]where denotes the channel estimation error matrix and is the delay estimation error matrix. With a certain amount of estimation error the channel matrix changes to be . In addition, after experiencing a period of time for feedback delay, turns to be . Then the channel matrix can be written as [19]where is the channel correlation coefficient formulated as in [19]. For Clarke’s fading spectrum, could be obtained as , and here is the zerothorder Bessel function of the first kind and is the Doppler frequency [20].
In this paper, we discuss the channel matrix of sourcerelay and relaysource at a certain time, and correlation coefficient is related to the delay coefficient . For the convenience of expression, and could be written as and . Therefore, our channel matrix could be further expressed as [21] is independent of and its elements are subjected to . represents the complex Gauss distribution with a mean value of and a variance of . Let and can be noted as Each element of is subjected to . Let . The receiver signal at iswhere is the residual selfinterference (SI) with covariance matrix . When the detection matrix is employed by , the mean squared error (MSE) of the detected signal at can be expressed as below:
3. Problem Formulation
In this paper, a joint design of the relay precoding matrix and the detection matrixes and 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 can be further represented as [23]where is the trace of the matrix and denotes the real part. It is assumed that each of the elements in a random matrix is an independent and identically distributed zero mean complex Gauss random variable, is satisfied with , and here is unit matrix. We assume the variance of is much smaller than one. Hence, we haveHere . In the same way, some simplifications can be written aswhere ; substituting (10)~(11) into (9), the original expression (9) can be simplified aswhereWe are now ready to design detection matrix and precoding matrix to minimize the average sum MSE of . The problem can be formulated as and here
4. The MMSEBased 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 is determined, the optimal linear detection matrix can be uniquely determined.
From the following condition , can be written asSubstituting (16) into (12), then (12) can be expressed as whereObserving that only the term is related to , the optimization problem in (14) is equivalent to minimizing . Assuming by contraction method, whereand here , hence the original optimization problem (14) can be simplified asNext, using matrix decomposition method to simplify the structure of , it can be rewritten asand here and . can be expressed as by EVD decomposition, is unitary matrix, and is diagonal matrix. Expression (21) can be described asand here . We propose the following structure:Here, is unitary, is diagonal, and is the nonsingular matrix obtained by the generalized singular value decomposition (GSVD) from and . can be written aswhere is unitary and is diagonal. is obtained from the following singular value decomposition (SVD):and here is diagonal, from (25), and . Substituting (23)~(25) into (22) givesand here . In order to achieve some simple solutions, we further relax the problem by only considering the main diagonal of . Finally, the problem in (20) is simplified to the following problem:where and ; the optimization problem (27) can be rewritten in scalar form aswhere , , , and are the diagonal entries of , , , and . 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, is the Lagrange multiplier, and the Lagrange function is constructed for gives is in accord withand then should be chosen from bisection algorithm and satisfied with , could be obtained by substituting into (30), and then , , , and will be gotten as follows.
4.2. Joint Iterative Algorithm Design
When the detection matrix is determined, the optimization problem of (14) can be translated into a convex optimization problem of relay precoding matrix . The Lagrangian function of (14) is formulated asSeeking partial derivative of where ; together with the complementary slackness and the relay station transmit power constraint, we obtain the KKT conditions asand from (34) and (35) one haswhereThe Lagrange multiplier is determined by (36) and (37). It satisfied [26] and here denotes Kronecker product, represents the maximum eigenvalue of the matrix , and can be obtained by bisection algorithm with its upper and lower bounds. By (17) and (38), we can see that when the signal matrix 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.

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 nodes and are equipped with antennas. Relay node is equipped with antennas. The source node transmits independent uncoded QPSK symbol streams to the corresponding receiver. The covariance of the transmit signal at source nodes and is , and here is the transmit power at each source node; let . The covariance of the AWGN is satisfied with and . We investigate the scenarios with various while fixing . and are the average signaltonoise ratio (SNR) for the first phase and the second phase of the twoway 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) Amplifyandforward (AF) relay approach [27] is
(ii) Zeroforcing (ZF) relay approach [28] isFigures 2 and 3 are given the performance comparison of BER while fixing estimated errors and delay coefficients . Simulation parameters are as follows: iterations and convergence threshold .
Figure 4 is given the performance comparison of BER with different iteration numbers. Simulation parameters are as follows: , , and convergence threshold .
Figures 5 and 6 are the BER performance comparisons in the two kinds of SNR while fixing feedback delay coefficient and estimation error variance , respectively. The simulation parameters are as follows: iterations and convergence threshold .
From the above simulations, the following conclusions can be drawn:(1)From Figures 2 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 , , and , 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 Figure 4, the system BER performance could be improved by 12 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 Figures 5 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 twoway 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.
Conflicts of Interest
The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents conflicts of interest in connection with the work submitted.
Acknowledgments
This work is jointly supported by the Fundamental Research Funds for the Central Universities (no. NS2016044).