Abstract-In this paper, we deal with low-complexity near-optimal detection/equalization in large-dimension multiple-input multiple-output inter-symbol interference (MIMO-ISI) channels using message passing on graphical models. A key contribution in the paper is the demonstration that near-optimal performance in MIMO-ISI channels with large dimensions can be achieved at low complexities through simple yet effective simplifications/approximations, although the graphical models that represent MIMO-ISI channels are fully/densely connected (loopy graphs). These include 1) use of Markov random field (MRF)-based graphical model with pairwise interaction, in conjunction with message damping, and 2) use of factor graph (FG)-based graphical model with Gaussian approximation of interference (GAI). The per-symbol complexities are ( 2 2 ) and ( ) for the MRF and the FG with GAI approaches, respectively, where and denote the number of channel uses per frame, and number of transmit antennas, respectively. These low-complexities are quite attractive for large dimensions, i.e., for large . From a performance perspective, these algorithms are even more interesting in large-dimensions since they achieve increasingly closer to optimum detection performance for increasing . Also, we show that these message passing algorithms can be used in an iterative manner with local neighborhood search algorithms to improve the reliability/performance of -QAM symbol detection.Index Terms-Factor graphs, graphical models, large dimensions, low-complexity detection, Markov random fields, multiple-input multiple-output inter-symbol interference (MIMO-ISI) channels, pairwise interaction, severe delay spreads.

In this paper, we consider the application of belief propagation (BP) to achieve near-optimal signal detection in lar ge multiple-input mUltiple-output (MIMO) systems at low com plexities. Large-MIMO architectures based on spatial multi plexing (V-BLAST) as well as non-orthogonal space-time block codes (STBC) from cyclic division algebra (CDA) are consid ered. We adopt graphical models based on Markov random fields (MRF) and factor graphs (FG). In the MRF based ap proach, we use pairwise compatibility functions although the graphical models of MIMO systems are fully/densely connected.In the FG approach, we employ a Gaussian approximation (GA) of the multi-antenna interference, which significantly reduces the complexity while achieving very good performance for large dimensions. We show that i) both MRF and FG based BP ap proaches exhibit large-system behavior, where increasingly closer to optimal performance is achieved with increasing number of dimensions, and ii) damping of messages/beliefs significantly im proves the bit error performance.

Abstract-In this paper, we present a belief propagation (BP) based algorithm for decoding non-orthogonal space-time block codes (STBC) from cyclic division algebras (CDA) having large dimensions. The proposed approach involves message passing on Markov random field (MRF) representation of the STBC MIMO system. Adoption of BP approach to decode non-orthogonal STBCs of large dimensions has not been reported so far. Our simulation results show that the proposed BP-based decoding achieves increasingly closer to SISO AWGN performance for increased number of dimensions. In addition, it also achieves near-capacity turbo coded BER performance; for e.g., with BP decoding of 24 × 24 STBC from CDA using BPSK (i.e., 576 real dimensions) and rate-1/2 turbo code (i.e., 12 bps/Hz spectral efficiency), coded BER performance close to within just about 2.5 dB from the theoretical MIMO capacity is achieved. Keywords -Non-orthogonal STBCs, large dimensions, low-complexity decoding, belief propagation, Markov random fields, high spectral efficiencies.I. INTRODUCTION Use of multiple antennas at the transmitter can offer the benefits of transmit diversity (e.g., using space-time coding) and high data rates (e.g., using spatial multiplexing) [1]. MIMO systems that employ non-orthogonal space-time block codes (STBC) from cyclic division algebras (CDA) for arbitrary number of transmit antennas, N t , are particularly attractive because they can simultaneously provide both full-rate (i.e., N t complex symbols per channel use, which is same as in V-BLAST) as well as full transmit diversity [2],[3]. The 2 × 2 Golden code is a well known non-orthogonal STBC from CDA for 2 transmit antennas [4]. High spectral efficiencies of the order of tens of bps/Hz can be achieved using large non-orthogonal STBCs. For e.g., a 16 × 16 STBC from CDA has 256 complex symbols in it with 512 real dimensions; with 16-QAM and rate-3/4 turbo code, this system offers a high spectral efficiency of 48 bps/Hz. Decoding of non-orthogonal STBCs with such large dimensions, however, has been a challenge. Sphere decoder and its low-complexity variants are prohibitively complex for decoding such STBCs with hundreds of dimensions. Recently, we proposed a low-complexity near-ML achieving algorithm to decode large non-orthogonal STBCs from CDA; this algorithm, which is based on bitflipping approach, is termed as likelihood ascent search (LAS) algorithm [5]- [7]. Our new contribution in this paper is that we present a belief propagation (BP) based approach to decoding of non-orthogonal STBCs with large dimensions, and report very good uncoded BER and near-capacity performance. To our knowledge, BP has not been reported for decoding non-orthogonal STBCs from CDA having large dimensions.

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Abstract-In this letter, we analyze the end-to-end average bit error probability (ABEP) of space shift keying (SSK) in cooperative relaying with decode-and-forward (DF) protocol, considering multiple relays with a threshold based best relay selection, and selection combining of direct and relayed paths at the destination. We derive an exact analytical expression for the end-to-end ABEP in closed-form for binary SSK, where analytical results agree with simulation results. For non-binary SSK, approximate analytical and simulation results are presented.