Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 654865, 6 pages

http://dx.doi.org/10.1155/2015/654865

## Fast Maximum-Likelihood Decoder for Quasi-Orthogonal Space-Time Block Code

^{1}Department of Electrical Engineering, Shahid Bahonar University of Kerman, P.O. Box 76169-133, Kerman, Iran^{2}Advanced Communications Research Institute, Sharif University of Technology, P.O. Box 11365-9363, Tehran, Iran

Received 8 February 2015; Accepted 13 April 2015

Academic Editor: Sergio Preidikman

Copyright © 2015 Adel Ahmadi and Siamak Talebi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Motivated by the decompositions of sphere and QR-based methods, in this paper we present an extremely fast maximum-likelihood (ML) detection approach for quasi-orthogonal space-time block code (QOSTBC). The proposed algorithm with a relatively simple design exploits structure of quadrature amplitude modulation (QAM) constellations to achieve its goal and can be extended to any arbitrary constellation. Our decoder utilizes a new decomposition technique for ML metric which divides the metric into independent positive parts and a positive interference part. Search spaces of symbols are substantially reduced by employing the independent parts and statistics of noise. Symbols within the search spaces are successively evaluated until the metric is minimized. Simulation results confirm that the proposed decoder’s performance is superior to many of the recently published state-of-the-art solutions in terms of complexity level. More specifically, it was possible to verify that application of the new algorithms with 1024-QAM would decrease the computational complexity compared to state-of-the-art solution with 16-QAM.

#### 1. Introduction

Quasi-orthogonal space-time block codes (QOSTBCs) [1, 2] have attracted much attention lately due to their desired performances and pairwise detections. Given these advantages, in [3–5] investigation is carried out to optimize diversity and coding gain performances. One drawback of these codes, however, is that complexity of pairwise optimum maximum-likelihood (ML) decoder rises drastically when size of constellation is increased. A survey of related literature shows that much research has been underway to develop an approach that reduces complexity of QOSTBC decoder. In [6, 7], lattice reduction aided (LRA) method is combined with suboptimal detectors in order to improve performance of the suboptimal decoders in multiple-input multiple-output (MIMO) communications systems. In [8], the QOSTBC is employed to reduce gap between error ratio of ML and LRA methods. This work also compares the performances of spatial multiplexing and QOSTBC with those of LRA zero-forcing and ML decoders. Authors of [9] employ QR decomposition and sorting to simplify detection of QOSTBC with four transmit antennas. This method achieves ML error performance and offers low complexity. For rotated QOSTBC with four transmit antennas and quadrature amplitude modulation (QAM) constellation, [10] explores a fast scheme with ML error performance and reduced complexity. In [11], a different pairwise structure is proposed for QOSTBCs decoding with three or four transmit antennas. The algorithm in [11] initializes the first symbol by rounding its zero-forcing estimation to the nearest constellation point. Then, the other symbol of the pair is detected by minimizing its cost function and the first symbol is discovered by minimizing its cost function. The quality of the detected symbols is dependent upon the initial value which means that the detection procedure may have to be repeated a number of times if the distance between the selected value and equalized symbol is higher than a threshold. This method achieves a near-ML performance with low complexity decoder. A new 4 × 4 QOSTBC is reported in [12] that employs precoder and rotated symbols. The complexity of the decoder is low and a reasonable suboptimum error performance is obtained when rotation angles are optimized. The optimization procedure is related to signal-to-noise ratio (SNR) and utilized constellation. In [13], a suboptimum fast decoding algorithm is investigated for block diagonal QOSTBC with arbitrary transmit antennas. This decoder employs precomputed look-up tables and statistics of phases in order to prepare a limited search area. Authors of [14] introduce a 4 × 4 nonorthogonal space-time block code (STBC) with improved coding gain and a low complexity decoder. In [15], a STBC is proposed for four transmit antennas based on mapping of PAM and QAM constellations. This code offers a high coding gain and authors design a low complexity decoder for it. A new structure for fast group decodable codes is presented in [16] whose decoder offers the lowest worst-case complexity among STBCs.

The innovative method developed in this letter decomposes the received vector into two pairs of symbols which are detected independently. To do this, the ML metric minimization of each pair is transformed into a sum of independent positive parts and an interference part. It should be noted that independency of the positive parts facilitates detection by allowing us to considerably limit the search spaces of the symbols. The candidates placed in relevant partial search areas are gradually evaluated and then the transmitted symbols are estimated by computing interference between them. If the search areas are small and no symbol is detected, then they are extended and evaluation is repeated till the transmitted symbols are detected. The ML metric decomposition studied in this letter boasts two important features, namely, decomposition is not a highly complex process, and more significantly, most of the decomposed parts are independent of each other and therefore the search space can be efficiently limited. Based on these features the proposed method offers the desired ML performance at very low complexity.

Compared with the proposed method, Sphere [17] and QR [9] decoders are more complex because their search spaces cannot effectively be reduced, and they both require significant precomputation steps. As for the fast ML decoder in [10], authors utilize a simplified quadratic ML decoding statistic for QAM constellations, and comparison of the simulation results indicates that the average complexity of this work is higher than that of the proposed method confirming that the latter can be extended to any arbitrary constellations. On the other hand, the nonoptimum detectors in [12, 13] both require rotation angle optimization and significant memory locations for look-up tables, respectively, whereas the proposed solution does not require any optimization or look-up tables.

#### 2. System Model

QOSTBC for transmit antennas has the following form:where , , , and the data symbols belong to constellation with unit average energy. The rotation angle can be selected such that full diversity and high coding gain are attained. For example, is optimum for QAM [10]. A space-time communication system with four transmit antennas and receive antennas that transmits four symbols over four time slots can be represented by an equivalent receive antenna aswhere SNR is indicated by and is the transmitted vector. The equivalent received vector and equivalent additive white Gaussian noise (AWGN) vector are denoted by and , respectively. Also, and denote the equivalent received signal and its relative noise at the th time slot of the th receive antenna, respectively. The AWGN has complex Gaussian distribution with zero mean and unit variance. The equivalent channel matrix for the th receive antenna is defined aswhere stands for channel fade between the th transmit antenna and the th receive antenna such that

The complete received vectors can be concatenated as:where , and .

#### 3. Fast Maximum-Likelihood Decoder

##### 3.1. Proposed ML Detection Method

The ML decoder should minimize the following norm in order to estimate the transmitted symbols:where norm is defined as and indicates conjugate transpose of . The above minimization can be rewritten aswhere is defined as

The matrix can be decomposed intowhere , , , and

By employing (9) and doing some math operations, we are able to compute through fewer mathematic operations aswhere and

In addition, can be decomposed into two independent parts by utilizing (9), which results inwhere and for and . By defining and , the matrix can be represented as

Based on (13), for and 2, the ML decoder can independently detect byFor the remaining part of this section, we focus on detection of noting that the other pair can be detectable by applying a similar approach. Furthermore, for the sake of simplicity and clarity, let us consider a square -QAM constellation, although the solution can also be straightforwardly extended to other constellations. For and 3, and can be represented by their real and imaginary parts as and , respectively. Under this scenario, are independent and belong to the set with real members. The decoder searches within to find the best choices for the real and imaginary parts of symbols which minimizes the ML metric.

The ML minimization (15) can be reformulated aswhere , , , , , and for . Based on (12), and we can rewrite minimization of (16) as minimization of which is decomposable into sum of positive parts:where and stands for the signum function. In (17), the parts are independent of each other which allows us to reduce the search space effectively, and the purpose of the third part is to represent interference between the other parts.

By assuming that the minimum of (17) is smaller than , we deduce that for . Therefore, the output of the ML decoder has to be located within corresponding intervals which reduces the search spaces effectively. The ML metric (16) for the remaining members is incrementally evaluated piece by piece to remove more inappropriate members and finally we obtain symbols and which minimize the ML metric. These steps for an -QAM constellation are demonstrated as a pseudocode in Pseudocode 1. In this pseudocode, cardinality of set is indicated by , the th member of this set is denoted by , and for .