Belief propagation based decoding of large non-orthogonal STBCs

Abstract

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.

Low-Complexity MIMO Detection Based on Belief Propagation Over Pairwise Graphs

“…withp(x j |x i , y j|i ) given by (24). The above message passing is computed for all edges in both directions, and they are repeated by a pre-defined number or until the messages do not change any more.…”

“…withp(x j |x i , y j|i ) given by (24). After a pre-defined number of iterations, the belief is finally obtained by…”

“…To derive (24) and the Gaussian BP rule, (33)-(40), we use the properties of the Gaussian PDF in [21], some of which are as follows…”

“…On the other hand, in [20], noticing that BP may not work well for a loopy graph, the authors proposed a tree approximation on the basis of Kullback-Leibler distance (KLD) optimality criterion. In [24], the same authors have proposed using the potential functions obtained by two dimensional projection and the expectation-maximization (EM) algorithm based post detection. Bit-based probabilistic data association is another approach to low complexity MIMO detection especially for higher order QAM.…”

“…In this paper, we investigate a similar approach to the pair-wise MRF based MIMO detector, but with different formulation, i.e., instead of using the potential functions obtained from the direct decomposition of multivariate Gaussian function [20], [24] or from the two dimensional projection in [24], we propose using the functions obtained by marginalizing the posterior joint probability density under the Gaussian input assumption. As implicated in [23], [25], the corresponding bipartite graph has an edge degree of only two and the proposed scheme has much less complexity than that of ML/MAP.…”