Low-complexity Maximum Likelihood Decoding Research Articles

Low complexity maximum likelihood decoding for orthogonal space-time block codes using non-square and rotated quadrature amplitude modulation

ABSTRACT High diversity gain can be achieved for orthogonal space-time block codes (OSTBCs) using rotated QAM constellations at a cost of high Maximum Likelihood (ML) decoding complexity. Thus far, low complexity ML decoding algorithms were proposed only for OSTBCs with square QAM constellations. In this paper, an algorithm has been proposed to reduce the ML decoding complexity of OSTBCs with non-square and rotated QAM constellations. The ML formulation using real-valued lattice representations is modified in such a way that the complexity depends upon the number of real values required to construct the alphabet of any constellation. Arithmetic progression and symmetry are the two elementary characteristics observed in the real numbers of the alphabets. Based on these characteristics, a new approach called Add and accumulate method has been proposed which drastically reduces the complexity to and without sacrificing the performance.

On Low Complexity ML Decoder for Quaternion Orthogonal Designs

Quaternion orthogonal designs (QODs) provide an efficient way of constructing orthogonal space time polarization block codes to achieve larger transmit diversity gains without substantial increase in antenna dimensions. However, their optimal decoding remains a big challenge. This letter introduces an important constraint, which has not been emphasized before, for decoupled decoding of QODs and proposes a generalized low complexity maximum-likelihood decoder for QODs based on symmetric-paired designs. The main contribution of this letter is to explore the benefits of QODs for smaller as well as higher order designs without compromising system performance. Results show that the proposed decoder yields optimal decoupled decoding with remarkably low complexity. This letter also compares the performance of quaternion coordinate interleaved orthogonal designs (QCIODs), which provide full diversity, with proposed symmetric-paired designs and shows that proposed designs significantly outperform QCIODs.

Time-Reversal Space–Time Codes in Asynchronous Two-Way Relay Networks

We consider an asynchronous two-way relay network in which a few distributed relays assist in the communication between two single-antenna terminals through analog network coding. The asynchronous transmission between relays and terminals causes symbol misalignments and results in diversity loss in space–time codes (STCs). We propose a family of zero-padded time-reversal STC that can achieve full diversity with low-complexity maximum likelihood (ML) decoding, given a bound for the path delay difference. With ML decoding, the proposed code is decomposed into several independent parts, which greatly facilitate the decoding process. For two-relay scenarios, three code designs based on Alamouti code, quasi-orthogonal space–time block code, and multigroup decodable STC are provided for each relay with one, two, and four antennas, respectively. For three-relay scenarios, one code design based on quasi-orthogonal space-time block code is provided for each relay with one antenna. Proof of full diversity is established, and the decoding complexity order is analyzed for all four designs. Simulations confirm the full diversity gain of all designs. The bit-error-rate performance in asynchronous scenarios is almost similar to that in synchronous scenarios.

Low-Complexity Detection Schemes for QOSTBC With Four-Transmit-Antenna

In this letter, we propose two improved decoding algorithms based on traditional low-complexity maximum likelihood (LC-ML) detection algorithm for the quasi-orthogonal space-time block code (QOSTBC) with four-transmit-antennas. The decoding complexities of the proposed algorithms are both lower than LC-ML decoder. By decomposing the channel matrix and its elementary transformation form with QR decomposition, the first algorithm can detect the four signals independently at the cost of some BER performance penalty whereas the second algorithm achieves almost the same BER performance with LC-ML decoder by using the interference cancellation method twice.

A Space-Time Block Coded Spatial Modulation From (n,k) Error Correcting Code

In this letter, a high-rate space-time block coded (STBC) spatial modulation (SM) scheme called STBC-SM (n,k) is presented, in which according to the codewords of the (n,k) error correcting code (ECC), different pairs of antennas are activated to transmit STBC. For nT = 2n transmit antennas, the STBC-SM (n,k) scheme can achieve the flexible transmission rates. Furthermore, two rotation angles are optimized to guarantee the transmit diversity of two for STBC-SM (n,k). Owing to the orthogonality of STBC, the new design enables a low-complexity maximum likelihood (ML) decoder. Simulation results are presented to verify the performance of the proposed scheme.

A low-complexity maximum likelihood decoder for tail-biting trellis

Abstract Tail-biting trellises are defined on a circular time axis and can have a smaller number of states than equivalent conventional trellises. Existing circular Viterbi algorithms (CVAs) on the tail-biting trellis are non-convergent and sub-optimal. In this study, we show that the net path metric of each tail-biting path is lower-bounded during the decoding process of the CVA. This property can be applied to remove unnecessary iterations of the CVA and results in a convergent maximum likelihood (ML) decoder. Simulation results show that the proposed algorithm exhibits higher decoding efficiency compared with other existing ML decoders.

On-Line Construction of Irregular Repeat Accumulate Codes for Packet Erasure Channels

In many applications erasure correcting codes are used to recover packet losses at high protocol stack layers. The objects (e.g. files) to be transmitted often have variable sizes, resulting in a variable number of packets to be encoded by the packet-level encoder. In this paper, algorithms for the (on-line) flexible design of parity-check matrices for irregular-repeat-accumulate codes are investigated. The proposed algorithms allow designing in fast manner parity-check matrices that are suitable for low-complexity maximum-likelihood decoding. The code ensembles generated by the algorithms are analyzed via extrinsic information transfer charts. Numerical results show how the designed codes can attain codeword error rates as low as 10-5 without appreciable losses w.r.t. the performance of idealized maximum-distance separable codes. Finally, we apply the proposed codes to the upcoming aeronautical communication standard, showing large performance improvements and proving the efficiency and the flexibility of the developed method.

Full-diversity high-rate space-time block-coded systems using estimated channel state information for symbol detection

In this paper, the frame-error-rate (FER) performance of full-diversity high-rate space-time block-coded (STBC) system is presented for quadrature phase-shift keying (QPSK) digital modulation technique, which is using the numeric-variable-forgetting-factor (NVFF) least-squares (LS) algorithm based channel estimator. The STBC transmission data-rates with two-transmitter antennas and with four-transmitter antennas are achieved by maximizing the coding gain and by minimizing the peak-to-minimum-power-ratio while using the selective power scaling of the information symbols, which also guarantees full-diversity in case of the two-transmitter antennas. The imperfect channel estimation leads to degradation in the FER performance of the low complexity maximum likelihood decoding algorithms, though the optimum power scaling factor is incorporated along with the constellation rotation scheme. Simulation results are presented to demonstrate that the FER performance of high-rate STBC system under idealistic conditions is quite different from its performance under realistic wireless environment due to the channel estimation errors. Therefore, the tracking performance of the pilot channel estimator is found to play significant role in the overall performance evaluation of the proposed wireless systems under slowly time-varying channels also in case of the general M-ary phase-shift keying (M-PSK) constellations. Key words: Space-time block-code, coding gain, peak-to-minimum-power-ratio, numeric-variable-forgetting-factor, variable-forgetting-factor, least-squares algorithm.

Fountain Codes Over GF(q)

ABSTRACTBinary fountain codes such as Luby transform codes are a class of erasure codes which have demonstrated an asymptotic performance close to the Shannon limit when decoded with the belief propagation algorithm. When these codes are generalized to GF(q) for q > 2, their performance approaches the Shannon limit much faster than the usual binary fountain codes. In this paper, we extend binary fountain codes to GF(q). In particular, we generalize binary Luby transform codes to GF(q) to develop a low complexity maximum likelihood decoder. The proposed codes have numerous advantages, including low coding overhead, low encoding and decoding complexity, and good performance over various message block lengths, making them practical for real‐time applications. Copyright © 2011 John Wiley & Sons, Ltd.

Quasi-Orthogonal Space-Time Block Codes for Two Transmit Antennas and Three Time Slots

In this paper, a class of quasi-orthogonal space-time block codes (Q-STBC) is proposed for systems with two transmit antennas and three time slots, where the Alamouti code is not applicable due to the odd time slots. The proposed Q-STBC codes achieve rate one and full diversity with low complexity maximum likelihood decoding. The Q-STBC design also shows excellent properties in other practical aspects, such as the compatibility with the single antenna transmission mode, low power fluctuation, and low receiver decoding and transmitter encoding complexity.

Space-Time Block Coded Spatial Modulation

A novel multiple-input multiple-output (MIMO) transmission scheme, called space-time block coded spatial modulation (STBC-SM), is proposed. It combines spatial modulation (SM) and space-time block coding (STBC) to take advantage of the benefits of both while avoiding their drawbacks. In the STBC-SM scheme, the transmitted information symbols are expanded not only to the space and time domains but also to the spatial (antenna) domain which corresponds to the on/off status of the transmit antennas available at the space domain, and therefore both core STBC and antenna indices carry information. A general technique is presented for the design of the STBC-SM scheme for any number of transmit antennas. Besides the high spectral efficiency advantage provided by the antenna domain, the proposed scheme is also optimized by deriving its diversity and coding gains to exploit the diversity advantage of STBC. A low-complexity maximum likelihood (ML) decoder is given for the new scheme which profits from the orthogonality of the core STBC. The performance advantages of the STBC-SM over simple SM and over V-BLAST are shown by simulation results for various spectral efficiencies and are supported by the derivation of a closed form expression for the union bound on the bit error probability.

On Low-Complexity Maximum-Likelihood Decoding of Convolutional Codes

This letter considers the average complexity of maximum-likelihood (ML) decoding of convolutional codes. ML decoding can be modeled as finding the most probable path taken through a Markov graph. Integrated with the Viterbi algorithm (VA), complexity reduction methods often use the sum log likelihood (SLL) of a Markov path as a bound to disprove the optimality of other Markov path sets and to consequently avoid exhaustive path search. In this letter, it is shown that SLL-based optimality tests are inefficient if one fixes the coding memory and takes the codeword length to infinity. Alternatively, optimality of a source symbol at a given time index can be testified using bounds derived from log likelihoods of the neighboring symbols. It is demonstrated that such neighboring log likelihood (NLL)-based optimality tests, whose efficiency does not depend on the codeword length, can bring significant complexity reduction. The results are generalized to ML sequence detection in a class of discrete-time hidden Markov systems.

Block-Differential Modulation Over Doubly Selective Wireless Fading Channels

Differential encoding is known to simplify receiver implementation because it by-passes channel estimation. However, over rapidly fading wireless channels, extra transceiver modules are necessary to enable differential transmission. Relying on a basis expansion model for time and frequency selective (doubly selective) channels, we derive such a generalized block-differential (BD) codec and prove that it achieves maximum Doppler and multipath diversity gains, while affording low-complexity maximum-likelihood decoding. We further show that existing BD systems over frequency-selective or time-selective channels follow as special cases of our novel system. Simulations using the widely accepted Jakes' model corroborate our theoretical analysis.

Low-complexity maximum-likelihood decoding of shortened enumerative permutation codes for holographic storage

Volume holographic memories (VHM) are page-oriented optical storage systems whose pages commonly contain on the order of one million pixels. Typically, each stored data page is composed of an equal number of binary pixels in either a low-contrast ("off") state or a high-contrast ("on") state. By increasing the number of "off" pixels and decreasing the number of "on" pixels per page, there is an associated gain in VHM system storage capacity. When grayscale pixels are used, a further gain is possible by similarly controlling the fraction of pixels at each gray level. This paper introduces a constant-weight, nonbinary, shortened enumerative permutation modulation block code to produce pages that exploit the proposed capacity advantage. In addition to the code description, we present an encoder and a low-complexity maximum-likelihood (ML) decoder for the shortened permutation code. A proof verifies our claim of ML decoding. Applying this class of code to VHMs predicts a 49% increase in storage capacity when recording modulation coded 3-bit (eight gray level) pixels compared with a VHM using a binary signaling alphabet and equal-probable (unbiased) data.

Collaborative coding and decoding techniques for multiple access channel

It is highly desirable to use simple and effective multiple access coding and decoding techniques which are capable of multiple access function and error control. The collaborative coding multiple access (CCMA) techniques potentially permit efficient simultaneous transmission by several users sharing a common channel, without subdivision in time, frequency or orthogonal codes. The authors investigate the performance of uniquely decodable CCMA schemes employing hard decision and maximum likelihood decoding techniques. A low complexity maximum likelihood decoding technique is presented. The reliability performance of various coding schemes employing these decoding techniques are carried out in the presence of AWGN conditions. The simulation results are presented in the form of symbol and codeword error rates as a function of signal to noise ratios. It is shown that uniquely decodable CCMA schemes permit the multiple access function to be combined with that of forward error correction.