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We introduce the superimposed training strategy into the multiple-input multiple-output (MIMO) amplify-and-forward (AF) one-way relay network (OWRN) to perform the individual channel estimation at the destination. Through the superposition of a group of additional training vectors at the relay subject to power allocation, the separated estimates of the source-relay and relay-destination channels can be obtained directly at the destination, and the accordance with the two-hop AF strategy can be guaranteed at the same time. The closed-form Bayesian Cramér-Rao lower bound (CRLB) is derived for the estimation of two sets of flat-fading MIMO channel under random channel parameters and further exploited to design the optimal training vectors. A specific suboptimal channel estimation algorithm is applied in the MIMO AF OWRN using the optimal training sequences, and the normalized mean square error performance for the estimation is provided to verify the Bayesian CRLB results.

In recent years, the use of relays has gained significant interest for the advantage of enhancing link reliability and increasing channel capacity in a wireless network. By transmitting the signal through one or more relays located at geographically separated nodes, the effect of signal fading due to multipath propagation and strong shadowing can be compensated through the exploitation of spatial diversity provided by the network nodes, as well as the efficient use of power resources, which can be achieved by a scheme that simply receives and forwards a given information, yet designed under certain optimality criterion. In particular, the source can cooperate with the relay to forward the replicas of the source signal to the destination thereby providing additional diversity and improved signal quality at the destination. There has been several relaying strategies, of which the most dominant are amplify-and-forward (AF) and decode-and-forward (DF).

Multiple-input multiple-output (MIMO) technology, by providing significant improvements in terms of spectral efficiency and link reliability, has been introduced to achieve high data rates required by the next-generation wireless communication systems. Since MIMO systems are able to support high-data rates by combating fading and interference, coupling relay networks with MIMO techniques is a natural extension to the state of the art. The basic idea is to introduce relays that forward the data to the destination, which is otherwise out of the reach of the source. With two-hop relaying, we can increase the rank and consequently the capacity of ill-conditioned (rank deficient) MIMO channels. Relaying information on two-hops also decreases the need for high power at the transmitter. For MIMO relay channels where every terminal in the wireless network can be deployed with multiple antennas, studies are mainly concentrated on spatial multiplexing (SM) systems. The potentiality of MIMO relay channels in wireless networks has been discussed and several results on capacity bounds have been found in previous works [

Before enjoying all the benefits brought by the MIMO relay network, an accurate channel state information (CSI) is required at the destination (for AF) or at both relay and destination (for DF). However, most of the existing works are based on the assumption that all nodes have perfect CSI, which is actually the perfect knowledge of backward (source-relay) and forward (relay-destination) channels. Recently, the estimation algorithms for the cascaded channel of source-relay-destination link have been proposed in [

However, only the cascaded channel information is not enough in some other applications, and the individual information of both the backward and forward channels is needed for the destination to perform certain operations. For example, in relay beamforming schemes [

Conventionally, the backward and forward channels can be estimated directly at the relay and destination, respectively, and then the relay spends additional time slots to send the backward channel estimation to the destination through the feedback channel. Apart from the additional energy and time slots consumed on the feedback transmission, the feedback information are subject to further distortion such as the quantization errors of the channel estimates and the errors in the communication through the feedback channel. An alternate way to estimate the individual channels at the destination is to apply a three-phase training strategy, that is, after the two phase training as in the estimation of the cascaded channel, the relay sends its own training to the destination so that the individual channels can be estimated [

Therefore, in a practical scenario, it is important to design a MIMO relay scheme that provides both in accordance with the two-hop data transmission and the estimates of individual channels at the destination. In [

Inspired by the superimposed training in point-to-point communications, [

The rest of the paper is organized as follows. The system model of superimposed training in the MIMO AF OWRN is given in Section

Consider a nonregenerative MIMO relay system as in Figure

A MIMO relay system.

We restrict our discussion to the case of a slow, frequency-flat block fading model. The data transmission takes place in two time slots using two-hops. During the first time slot, the source transmits the signal to the relay and yields the received signal as

The relay processes the received signal and superimposes a new training vector

The received signal at the destination is represented as

For channel estimation purposes, we assume that two sets of known training sequences

Then the data model during one training period can be expressed as

For many practical estimation problems, popular estimators such as the maximum likelihood (ML) estimator or the maximum a posteriori (MAP) estimator are infeasible, so one has to resort to suboptimal estimators, which are typically evaluated by determining mean square error (MSE) through simulations and by comparing this error to theoretical performance bounds. In particular, the family of Cramér-Rao bound (CRB) has been shown to give tight estimation lower bounds in a number of practical scenarios.

The CRB for the estimation of deterministic parameters is given by the inverse of the FIM, and Van Trees derived an analogous bound to the CRB for random variables, referred to as “Bayesian CRB” (BCRB). Unlike the standard and modified CRBs, the statistical dependence is naturally considered within the BCRB framework. With the assist of BCRB, the performance of the suboptimal estimators in the MIMO OWRN can be assessed, and the optimal training design can be obtained. Given the parameter vector

The Bayesian FIM is computed as

See Appendix

Before proceeding, we give the Bayesian FIM results for two special cases:

The results for

As for

The results for

Since

For the purpose of minimizing the MSEs in the channel estimation, the two optimizations on the training sequences in the MIMO AF OWRN are formulated as

The training sequences satisfying

See Appendix

The BCRBs become diagonal matrices with the optimal training in

Using the optimal training vectors, we refer to the suboptimal estimators to verify the Bayesian Cramér-Rao Lower Bound (CRLB) results in the MIMO AF OWRN, since the channel statistics are assumed known. A linear minimum mean square error (LMMSE) estimator is designed to estimate the two sets of MIMO channels. The corresponding covariances

Recalling

The LMMSE estimate of

Similarly, the LMMSE estimate of

With

Other estimation algorithms can always be applied to achieve better MSE performance with additional computational complexity accompanied, and here we give an example of modified LMMSE algorithm. Since

The modified estimation process is as follows, after obtaining the LMMSE estimate

In this section, we provide numerical results to verify our studies. For the sake of simplicity and without loss of generality, we set

The theoretical Bayesian CRLBs are then calculated for

Bayesian CRLBs versus

Bayesian CRLBs versus SNR.

We then examine the NMSE performance of the suboptimal estimator, as shown together with the theoretical Bayesian CRLBs in Figures

Channel estimation NMSEs versus SNR for

Channel estimation NMSEs versus SNR for

Channel estimation NMSEs versus SNR for

We also provide the simulation results considering different numbers of antennas at the relay and destination, for example, same

Channel estimation NMSEs versus

The NMSE performance of the modified LMMSE algorithm is depicted in Figure

Modified LMMSE channel estimation NMSEs versus SNR for

The simulation results verify our proposal that with a fraction of relay power devoted to the superimposed training in the MIMO AF OWRN, the individual channel information of

In this paper, we have investigated the superimposed training strategy in MIMO AF TWRN, which superimposes a new set of training vectors at the relay and provides the individual channel information at the destination to accomplish and simplify the channel estimation. The closed-form Bayesian CRLB has been derived and then used to guide the optimal training design in MIMO AF OWRN. The simulation results have been provided to verify the Bayesian CRLB results by the practical NMSE performance of a suboptimal channel estimator. With the individual channel information, the joint optimization of the throughout, as well as the tradeoff between rate and detection, can be achieved at the destination.

The FIM can be modified as

Since

Denote

Then the received signal at the destination after vectorizing can be rewritten as

For computational simplicity, we further define

The log-likelihood function of the received signal at the destination can be written as

It can be obtained that

Note that

It is assumed that there is no spatial correlation at the relay. According to the widely used Kronecker correlation model [

From the above properties, it can then be calculated that

We first derive the optimal solutions for

Similarly we can obtain the optimal solution for

Combing both proofs yields the conditions of

This research was supported by the National Science Foundation of China (Grant no. NFSC