An Optimal Temporal-and-Spatial Equalizer for Two-Hop MIMO Relay Networks with Backward CSIs Keshav Singh ∗ , Meng-Lin Ku † , and Jia-Chin Lin ‡ Department of Communication Engineering, National Central University, Taiwan, R.O.C. 995403601@cc.ncu.edu.tw * , mlku@ce.ncu.edu.tw † , jiachin@ce.ncu.edu.tw ‡ Abstract—This paper considers an equalize-and-forward (EF) strategy for two-hop multiple-input multiple-output (MIMO) re- lay networks in multipath fading channels, where the relay nodes and the destination only know its respective backward channel state information (CSI) knowledge, and each node is equipped with multiple antennas for transmitting, receiving or forwarding signals. For such a relay network, the inter-symbol interference (ISI) and multiple-antenna interference (MAI) are two detrimen- tal effect to degrade the bit error rate (BER) performance. In order to compensate for the interference problem, we design temporal-and-spatial (TS) equalizers to assist in forwarding and decoding the signals at the relay nodes and the destination node, respectively. Based on the minimum mean square error (MMSE) criterion with a total power constraint, an optimization framework is formulated to find the optimal TS equalizers with the backward CSI knowledge, and an iterative algorithm using the Karush-Kuhn-Tucker (K.K.T.) conditions is investigated to achieve the optimal solution. With these optimal TS equalizers, the MIMO relay network can not only effectively mitigate the interference but also provide both the spatial and multipath diversity gains. Simulation results indicate the effectiveness of the proposed algorithm, yielding a significant improvement on the BER performance. I. I NTRODUCTION In recent years, relay-assisted networks have gained con- siderable attention from the research community because of its substantial benefits. Some of the benefits resulting from the use of relays in wireless systems are coverage expansion, capacity boost, improvement in link quality and reliability, etc. The main idea behind the relay networks is the compensation for signal degradation due to channel propagation loss, shad- owing and multipath fading effect through the deployment of intermediate relay nodes between the source and destination nodes [1][2]. With multiple antenna techniques, multiple-input multiple-output (MIMO) relays are capable of simultaneously forwarding multiple streams to the destination node, thereby enhancing the data rate and the link reliability [3][4]. However, the inter-symbol interference (ISI) and the multiple-antenna interference (MAI) will severely deteriorate the system per- formance of an MIMO relay network in frequency selective channels, and there is a need for an efficient relaying strategy to appropriately handle these two phenomena. In relay communications, relaying schemes are major fac- tors to determine the achievable capacity and outage perfor- mance. There have been several relaying protocols defining retransmission strategies at the relays, and most popular ones are amplify-and-forward (AF), decode-and-forward (DF) and equalize-and-forward (EF) [5][6]. The EF strategy considers more sophisticated signal processing to filter the received signals at the relay before retransmission. Although the EF strategy requires higher computational complexity than the AF strategy, it is more likely to be an effective remedy to mitigate the interference for an MIMO relay network in multipath channels. Recently, studies on incorporating the MIMO technology in relay networks for further performance enhancement have been reported in the literature [3], [7]-[14]. In [3], the capacity bound for single-relay MIMO channels has been investigated. In [7], a relay precoder was designed to maximize the capacity of a non-regenerative relay network with a single MIMO source-destination pair and an MIMO relay node. The precoder design problem was further extended to an MIMO point-to- multipoint network in [8] with the aid of an MIMO relay station. Apart from the capacity maximization, the minimum mean square error (MMSE) was adopted as an alternative criterion to design relay precoders in [12]. Based on the MMSE criterion, source and relay precoders were jointly investigated for multi-carrier MIMO relay networks in [9]. More recently, the joint source and relay precoder designs for capacity maximization and mean square error (MSE) minimization were studied in [10] and [13], respectively. In these existing works [7]-[14], precoders were designed under the assumption that the channel state information (CSI) is fully available at all the nodes. In practice, it is unlikely to acquire the full CSI knowledge at the source and the relay nodes since it necessitates a large amount of signalling overhead to exchange the instantaneous CSIs among multiple communication nodes. Moreover, most of the previous works can only be operated in frequency flat fading channels under the assumption of perfect synchronization among relay nodes. However, in practical scenarios the channels are likely to be frequency selective. Although there have been few works concerning the ISI effect in multipath fading channels [15], they are incompetent to tackle the interference effect in MIMO relay networks. In fact, the ISI and MAI can make it difficult 978-1-4673-5939-9/13/$31.00 ©2013 IEEE 978-1-4673-5939-9/13/$31.00 ©2013 IEEE 2013 IEEE Wireless Communications and Networking Conference (WCNC): PHY 2013 IEEE Wireless Communications and Networking Conference (WCNC): PHY 3242