- Research Article
- Open Access
Joint Linear Filter Design in Multiuser Cooperative Nonregenerative MIMO Relay Systems
© Gen Li et al. 2009
- Received: 30 November 2008
- Accepted: 1 July 2009
- Published: 19 August 2009
This paper addresses the filter design issues for multiuser cooperative nonregenerative MIMO relay systems in both downlink and uplink scenarios. Based on the formulated signal model, the filter matrix optimization is first performed for direct path and relay path respectively, aiming to minimize the mean squared error (MSE). To be more specific, for the relay path, we derive the local optimal filter scheme at the base station and the relay station jointly in the downlink scenario along with a more practical suboptimal scheme, and then a closed-form joint local optimal solution in the uplink scenario is exploited. Furthermore, the optimal filter for the direct path is also presented by using the exiting results of conventional MIMO link. After that, several schemes are proposed for cooperative scenario to combine the signals from both paths. Numerical results show that the proposed schemes can reduce the bit error rate (BER) significantly.
- Mean Square Error
- Direct Path
- Minimum Mean Square Error
- Filter Design
- Uplink System
Wireless relays are essential to provide reliable transmission, high-throughput, and broad coverage for next-generation wireless networks . Deploying a relay between a source and a destination cannot only overcome shadowing due to obstacles but also reduce the required transmitted power from the source and hence interference to neighboring nodes. Relays can be regenerative  or nonregenerative . The former employs decode-and-forward scheme and regenerates the original information from the source. The latter employs amplify-and-forward scheme, which only performs linear processing for the received signal and then transmits to the destinations. As a result of the above difference, a nonregenerative relay generally causes smaller delay than a regenerative relay.
MIMO techniques are well studied to promise significant improvements in terms of spectral efficiency and link reliability. In [4, 5], the capacity of point-to-point MIMO channel is investigated and extensive work on multi-user MIMO has been done for a decade . Therefore, combined with the above two technologies, a novel system called MIMO-relay emerges to accommodate users with high data rate requests and extend the network coverage. Recently, there is a vigorous body of work on MIMO-relay systems [7–15]. For example, [7, 8] derives upper bounds and lower bounds for the capacity of MIMO-relay channels. In , the optimal design of non-regenerative MIMO relays is investigated. Assuming relays and receivers with multiple antennas, the optimal relay matrix that maximizes the capacity between the source and destination is developed when a direct link is not considered or is negligible. The same problem is studied in , and  extends the work to partial channel state information (CSI) scenario.
Despite significant research efforts and advances on MIMO relay systems, most of the aforementioned research is based on a point-to-point scenario with a single user equipped with multiple antennas. In practical systems, however, each relay will need to support multiple users. This motivates us to study multiuser MIMO-relay systems, where the relay forwards data to multiple users. The most different feature between the researches on the single-user (with multiantenna) and multiuser (with single antenna) system is that the signals of the multiple users cannot be cooperatively pretransformed (e.g., uplink of a cellular system) or posttransformed (e.g., downlink of a cellular system). While single-user MIMO-relay systems have been a primary focus of prior research, a few researchers begin to pay attention to multiuser scenario as well. In , the optimal design of nonregenerative relays for multiuser MIMO-relay systems based on sum rate is investigated. Assuming zero-forcing dirty paper coding at the base station and linear operations at the relay station , it proposes upper and lower bounds on the achievable sum rate, neglecting the direct links from the BS to the users.
In this paper, we consider the problem of joint linear optimization for both downlink and uplink in multiuser cooperative nonregenerative MIMO-relay systems based on MSE criterion, which is different from the sum rate criterion in . The MSE criterion is motivated by robustness to channel estimation errors and a lower implementation complexity. Then our main contributions are as follows.
(i)We derive the optimal joint design of the BS and RS filter matrices that achieves the minimum mean squared error (MMSE) for both downlink and uplink of the multiuser MIMO relay systems at the absence of direct path.
(ii)We propose several schemes for the design of the BS and RS filter matrices based on MSE criterion in the presence of direct path, which is called cooperative scenario in this paper.
(iii)We compare different schemes for direct-path-only scenario, relay-path-only scenario and cooperative scenario, and the numerical results are provided to show the effectiveness joint filter design and cooperative combine operation.
The rest of this paper is organized as follows. Sections 2 and 3 formulate the system model and propose the joint filter design schemes for downlink and uplink of multiuser MIMO relay systems, respectively. Numerical results are given in Section 4. Finally, Section 5 concludes this paper.
Boldface capital letters and boldface small letters denote matrices and vectors, respectively. Superscripts , and stand for the conjugate, transpose, and complex conjugated transpose operation, respectively., while and represent inversion and pseudoinversion of matrices. Also, and denote the expectation and trace operation, respectively, and, finally, is the identity matrix.
The bit error rates (BER) of the proposed schemes in the previous sections are evaluated by applying them to a -user MIMO-relay system with antennas at the BS and antennas at the RS. We obtain the BER plots of OJ-MMSE/RP (Section 2.3.1), SOJ-MMSE/RP (Section 2.3.2), MMSB/DP (Section 2.2), CS1-MMSE/RDP (Section 2.4) and CS2-MMSE/RDP (Section 2.4) in downlink systems, together with OJ-MMSE/RP (Section 3.3), MMSB/DP (Section 3.2), CS1-MMSE/RDP (Section 3.4), and CS2-MMSE/RDP (Section 3.4) in uplink systems. Note that RP and DP denote direct path and relay path, respectively, while RDP represents the cooperative scenario with both the paths. In addition, we also evaluate the following two schemes as a reference for downlink and uplink systems, respectively.
where is given to meet the power constraint at the relay, and hence the BS filter matrix and the scalar are obtained by substituting (80) into (29).
which is similar with that in conventional MIMO systems by regarding and as equivalent channel matrix and noise vector. Then the received MMSE filter can be obtained via the Linear MMSE receiver in .
4.1. BER versus SNR
4.2. BER versus the Number of Antennas per Node
4.3. BER versus the Relevant Path Loss of Direct Path
Computational complexity of the proposed schemes in downlink systems.
Computational complexity ofthe proposed schemes in uplink systems.
4.5. Convergence of Iterative Algorithm
In this paper, the local optimal MSE-based joint (BS and RS) filters have been proposed for a multiuser cooperative nonregenerative MIMO-relay system. Both uplink and downlink are considered. It is clear that the cooperative system can be divided into two paths, that is, the direct path and the relay path. As the optimal filter for the direct path can be obtained by using the exiting results of conventional MIMO link, we focus on the optimization for the relay path first. To be more specific, we propose the joint local optimal filter scheme, which involves an iterative algorithm in downlink scenario. Thus a simpler suboptimal scheme is derived for practical use. Then, in uplink scenario a closed-form optimal solution is exploited based on matrix analysis theory. The proposed optimal scheme firstly transform the MIMO relay channel into parallel sub-channels and then the optimal power allocation among the sub-channels has been found to follow a water-filling pattern. Furthermore, based on the results for direct path and relay path, two schemes are proposed for downlink systems and uplink systems with different combination methods, respectively. Numerical results and analysis show that joint filter design and cooperative operation can offer significant performance gain in terms of BER.
This work was supported in part by Ericsson Company, Beijing Science and Technology Committee under project no. 2007B053, National Natural Science Foundation of China (NSFC) under no. 60772112, National 973 Program under no. 2009CB320406, National 863 Program under no. 2009AA011802 and no. 2009AA01Z262.
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