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A generalized base station-relay-user equipment (BS-Relay-UE) beamforming design is investigated for a cooperative multiple-input multiple-output (MIMO) multirelay networks with imperfect channel state information (CSI). In order to minimize the worst-case mean square error (MSE) which is subject to a semi-infinite (SI) relay power constraints, a generalized optimal beamforming structure for the relay amplifying matrix is effectively proposed, and then the SI relay power constraints are converted into linear matrix inequalities (LMIs) version. In such conversion, the objective problem recasts as a decoupled biconvex semidefinite programming (SDP) one which can be efficiently solved by the proposed alternating algorithm. The system performance has been verified in terms of worst-case MSE using a set of qualitative analyses. The results show us that the proposed beamforming method outperforms the conventional schemes and can also effectively reduce the computational complexity when it is compared to the cutting-set schemes and also to the nonrobust ones.

Recently, cooperative multi-input multi-output (MIMO) relay network approaches are popularized to increase the system capacity and to improve the transmission reliability by leveraging spatial diversity. Since the channel estimation is important to the wireless communications, the authors in [

Considering inaccurate channel estimation and feedback delay, the perfect channel state information (CSI), which are proposed in the above works, is usually hard to obtain in practice. In order to circumvent these problems, by taking into account the channel uncertainties, the imperfect CSI scenario has been studied in [

Furthermore, in [

Since that, for the multiple relays scheme, not only the performance of the capacity outperforms that of the single relay one, but also the multiple relays scheme is more practical and challenging for the wireless communication scenarios. The downlink network considered here consists of a base station (BS), multiple relays, and user equipment (UE). We have named this particular AF system base BS-Relays-UE beamforming. The main contributions of this paper are summarized in the following:

We comprehensively investigate a generalized beamforming design for a cooperative MIMO multirelay networks with imperfect CSI. In the proposed scheme, the joint optimal design considers the precoding matrix at the BS, the beamforming matrix at the relays and the receiving beamforming matrix at the UE with multiple relays, which is general and practical.

Since the considered worst-case MSE optimization problem is not only nonconvex but also subject to the semi-infinite relay power constraints, a generalized optimal beamforming structure of the relay amplifying matrix is investigated. In addition, the SI relay power constraints are converted into linear matrix inequalities (LMIs). By this way, the objective problem recasts as a decoupled biconvex semidefinite programming (SDP) one, which can be efficiently solved by our proposed alternating algorithm.

By means of the numerical results, the proposed beamforming design significantly reduces the computational cost and improves the performance in terms of the worst-case MSE compared with the nonrobust and cutting-set cases.

In the following, a system model for the proposed cooperative MIMO multirelay networks is presented in Section

In this section, an amplify-and-forward relaying system model is described. In the proposed system, users are away from a base station and they are isolated and out of the BS coverage. In other words, the direct link between BS and UE does not exist so that UE can get a connection to the BS only via relays in order to achieve any signal receptions. To be more specific, the proposed multirelay network consists of one BS,

Cooperative MIMO multirelay networks.

Now, concerning the communication channels related to the proposed network, there are two of them. The one is the group of estimated communication channels

It is assumed that the BS transmits the signals to the UE using two consecutive time slots. At the first time slot, after being linearly processed by the matrix

At the second time slot, the relay node

By taking into account the estimation error, the CSI is considered partially known at each node. With this consideration, the actual channel coefficients of the links follow that

With this observation, the worst-case MSE at the UE node can be obtained as

With the fixed BS and UE beamforming matrices

Using the SVDs in (

The proof is similar to [

Without loss of generality,

Using

From (

For the relay power constraint, after introducing the slack variable

Thus, the objective problem

It is clear that the problem

By initializing the small

In this section, we examine the performance of the proposed robust scheme in terms of the worst-case MSE compared with cutting-set method in [

Figure

The worst-case MSE versus the number of iterations.

Figure

The worst-case MSE versus the transmission SNR.

So far, we have presented an efficient way to achieve signal transmission from a BS to an isolated, out-of-coverage user by realizing a generalized BS-R-UE beamforming. In our proposed cooperative MIMO multirelay networks with imperfect CSI; first, the optimal relay beamforming is derived as a means to improve the efficiency. Then, the semi-infinite objective problem is converted into a biconvex problem, which is subject to the LMI constraints. Furthermore, the converted, biconvex problem is efficiently solved by our alternating algorithm. The system performance in terms of worst-case MSE is verified using a set of qualitative analysis; the results show us that the proposed beamforming method outperforms the conventional schemes and can also effectively reduce the computational complexity when it is compared to the cutting-set schemes and also to the nonrobust ones. Our future concerns will be the generalized optimization problems on the two-way relay networks and cooperative relaying nonorthogonal multiple access (NOMA) systems.

The authors declare that they have no conflicts of interest.

The authors would gratefully acknowledge the grants from the National Natural Science Foundation of China (61371113, 61401241, 61401240, and 61771264), Nantong University-Nantong Joint Research Center for Intelligent Information Technology (KFKT2016B01 and KFKT2017B01), the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (no. 2015D02), and the Brain Korea BK21 plus.