Complexity Of 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.

Linear Time-Invariant Anytime Codes for Control Over Noisy Channels

The problem of stabilizing an unstable plant over a noisy communication link is an increasingly important one that arises in problems of distributed control and networked control systems. Although the work of Schulman, and Sahai and Mitter over the past two decades, and their development of the notions of “tree codes” and “anytime capacity” respectively, provides the theoretical framework for studying such problems, there has been scant practical progress in this area because explicit constructions of tree codes with efficient encoding and decoding did not exist. To stabilize an unstable plant driven by bounded noise over a noisy channel one often needs real-time encoding and real-time decoding and a reliability which increases exponentially with delay, which is what tree codes guarantee. We propose an ensemble of random causal linear codes with a time invariant structure and show that they are tree codes with probability one. For erasure channels, we show that the average complexity of maximum likelihood decoding is bounded by a constant for all time if the code rate is smaller than the computational cutoff rate. For rates larger than the computational cutoff rate, we present an alternate way to perform maximum likelihood decoding with a complexity that grows linearly with time. We give novel sufficient conditions on the rate and reliability required of the tree codes to stabilize vector plants and argue that they are asymptotically tight.

Low complexity and high diversity gain non-linear constellation precoded MIMO-OFDM system with subcarrier grouping

The maximum distance separable codes based non-linear signal space diversity technique provides flexibility in the design of diversity channels and diversity orders. Still, in actual practice, these diversity orders can only be truly augmented by increasing the constellation size. In this paper, we enhance the diversity order/gain by introducing multiple-input multiple-output (MIMO) to the existing non-linear constellation precoded orthogonal frequency division multiplexing (NCP-OFDM) system with subcarrier grouping. Though, it still suffers from the complexity of maximum likelihood (ML) decoding, which exponentially increases with the number of antennas. Assuming the channel is quasi-static in nature, we exploit it to reduce the complexity by introducing space-time block-codes (STBC). The ML decoding complexity of a 2×2 system is of O(224) whereas, by using STBC it reduces to O(213). In addition, we propose an algorithm for decoding, called the maximum scaling coefficient selection (MSCS) to further reduce the complexity to O(28). Simulation results show that there was only a marginal performance loss when using the MSCS algorithm based on ML decoding. Furthermore, the simulation results demonstrate how the diversity gain is maximized by adding additional antennas and by introducing STBC.

Iterated Space-Time Code Constructions From Cyclic Algebras

We propose a full-rate iterative space-time code construction, which uses an algebraic space-time code in order to design a new space-time code of double the code rank. We give a condition for determining whether the resulting codes satisfy the full diversity property, and study their maximum likelihood decoding complexity with respect to sphere decoding. Particular emphasis is given to the asymmetric multiple input double output codes. In the process, we derive an interesting way of obtaining division algebras, and study their centers and maximal subfields.

A Novel QO-STBC Scheme with Linear and Single-Symbol Decoding

In this paper, a quasi-orthogonal space time block coding (QO-STBC) scheme with maximum likelihood (ML) decoding via simple linear and single-complex-symbol detection is proposed. We present a fast and simple linear decode processing by embedding a transformed signal space. The method benefits from the reduced maximum likelihood decoding complexity via the pre-interleaving to QO-STBC. We also proposed codes with 3/4 of the full transmission rate for the specific cases of 8 transmit antennas. Finally, the numerical results show the proficiency of our design procedure, demonstrating that significant improvements in BER are achievable when the rate or receive antennas increases.

Two-stage decoding algorithm for unmodulated parallel combinatory high-compaction multicarrier modulation signals

Abstract A new decoding algorithm that consists of two decoding stages to reduce the computational complexity of maximum likelihood decoding for parallel combinatory high-compaction multicarrier modulation is proposed. The first decoding stage is responsible for a preliminary decision that serves to roughly find candidate messages using the QRD-M algorithm, and the second decoding stage is responsible for the final decision that reduces the error contained in the candidate and determines the message using the minimum value of the Hamming distances between the candidate and the replicas of the message. The complexity is considerably reduced by the proposed two-stage decoding algorithm at a cost of approximately 1.5 dB or less in E b /N 0 with respect to the bit-error rate of 10-3 under the given parameter settings.

Non‐square rate‐one space–time block codes for LTE‐advanced systems with optimised coding gain

Alamouti code is an excellent option for a two-transmitter and one-receiver network owing to its optimal performance and low detection complexity. Unfortunately, with the requirement of odd number of time slots in the frame structure for data transmission of long-term evolution (LTE)-advanced systems, this scheme is not applicable to such systems. In this correspondence, three new space–time block codes (STBCs) are proposed for LTE-advanced systems with two transmit antennas and a receive antenna with three-time-slots. The proposed designs achieve rate of one symbol per channel use and full transmission diversity. The main objective of designing the first and second codewords is to improve the coding gain of STBCs with three time slots. These STBCs are constructed using real rotation of quadrature amplitude modulation (QAM) information symbols in three dimensions. As will be seen later, the idea of real rotation of transmitted symbols using orthogonal matrices allows a decrease in the number of unknown parameters to be determined in order to maximise the coding gain. Using real rotation to encode the transmitted symbols, a new scheme is obtained which has the same minimum determinant (MD) as Alamouti code for 4, 16 and 64-QAM signal constellations. In designing the last codeword, the authors attempt to reduce the maximum-likelihood decoding complexity by decoupling detection of one encoded symbol from others. The simulation results confirm an improved performance of new schemes as compared to recently proposed codes.

The Treewidth of MDS and Reed–Muller Codes

The constraint complexity of a graphical realization of a linear code is the maximum dimension of the local constraint codes in the realization. The treewidth of a linear code is the least constraint complexity of any of its cycle-free graphical realizations. This notion provides a useful parameterization of the maximum-likelihood decoding complexity for linear codes. In this paper, we show the surprising fact that for maximum distance separable codes and Reed-Muller codes, treewidth equals trelliswidth, which, for a code, is defined to be the least constraint complexity (or branch complexity) of any of its trellis realizations. From this, we obtain exact expressions for the treewidth of these codes, which constitute the only known explicit expressions for the treewidth of algebraic codes.

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.

On the Performance of Energy-Division Multiple Access with Regular Constellations

In this paper we describe a multiple-access protocol in which different users are assumed to share the same bandwidth and the same pulse. Users employ the same modulation (binary-phase shift keying, quadrature-phase shift keying, and rectangular-phase shift keying are considered) with different transmitted magnitude, and are discriminated on the basis of the corresponding magnitude at receiver location. Conditions for user discrimination are analyzed. The proposed receiver uses successive decoding in order to avoid exponential complexity of maximum-likelihood decoding. Such a scheme, compared to orthogonal multiaccess schemes (e.g. time- or frequency-division multiple access) allows to achieve larger normalized throughput for systems operating in large signal-to-noise ratio range, and may be jointly applied with classical protocols in personal-area networks. Analytical and numerical results, in terms of bit error rate and normalized throughput, are derived for performance evaluation on additive white Gaussian noise channels.

Complexity of Decoding Positive-Rate Primitive Reed–Solomon Codes

It has been proved that the maximum likelihood decoding problem of Reed-Solomon codes is NP-hard. However, the length of the code in the proof is at most polylogarithmic in the size of the alphabet. For the complexity of maximum likelihood decoding of the primitive Reed-Solomon code, whose length is one less than the size of alphabet, the only known result states that it is at least as hard as the discrete logarithm in some cases where the information rate unfortunately goes to zero. In this paper, it is proved under a well known cryptography hardness assumption that: 1) There does not exist a randomized polynomial time maximum likelihood decoder for the Reed-Solomon code family [q, k(q)] <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , where k(x) is any function in Z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> → Z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> computable in time x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1)</sup> satisfying √x ≤ k(x) ≤ x - √x. 2) There does not exist a randomized polynomial time bounded-distance decoder for primitive Reed-Solomon codes at distance 2/3 + ϵ of the minimum distance for any constant 0 <; ϵ <; 1/3. In particular, this rules out the possibility of a polynomial time algorithm for maximum likelihood decoding problem of primitive Reed-Solomon codes of any rate under the assumption.

A* Based Algorithm for Reduced Complexity ML Decoding of Tailbiting Codes

The A* algorithm is a graph search algorithm which has shown good results in terms of computational complexity for Maximum Likelihood (ML) decoding of tailbiting convolutional codes. The decoding of tailbiting codes with this algorithm is performed in two phases. In the first phase, a typical Viterbi decoding is employed to collect information regarding the trellis. The A* algorithm is then applied in the second phase, using the information obtained in the first one to calculate the heuristic function. The improvements proposed in this work decrease the computational complexity of the A* algorithm using further information from the first phase of the algorithm. This information is used for obtaining a more accurate heuristic function and finding early terminating conditions for the A* algorithm. Simulation results show that the proposed modifications decrease the complexity of ML decoding with the A* algorithm in terms of the performed number of operations.

Achieving a near‐optimum erasure correction performance with low‐complexity LDPC codes

AbstractLow‐density parity‐check (LDPC) codes are shown to tightly approach the performance of idealized maximum distance separable (MDS) codes over memoryless erasure channels, under maximum likelihood (ML) decoding. This is possible down to low error rates and even for small and moderate block sizes. The decoding complexity of ML decoding is kept low thanks to a class of decoding algorithms, which exploit the sparseness of the parity‐check matrix to reduce the complexity of Gaussian elimination. ML decoding of LDPC codes is reviewed at first. A performance comparison among various classes of LDPC codes is then carried out, including a comparison with fixed‐rate Raptor codes for the same parameters. The results confirm that a judicious LDPC code design allows achieving a near‐optimum performance over the erasure channel, with very low error floors. Furthermore, it is shown that LDPC and Raptor codes, under ML decoding, provide almost identical performance in terms of decoding failure probability vs. overhead. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Guest Editorial: IJSCN Special Issue on ASMS2008

This Special Issue of the Wiley International Journal of Satellite Communications and Networking hosts aselection of papers from the fourth Advanced Satellite Mobile Systems (ASMS) Conference, held inBologna, Italy, on 26–28 August 2008. The conference was organized through a joint effort of theUniversity of Bologna, the Institute of Communications and Navigation of DLR (German AerospaceCenter) and the ISI (Integral SatCom Initiative) European Technology Platform, in cooperation with theSatellite Communications Network of Excellence (SatNEx) and the sISI Specific Support Action,supported by the European Commission FP6 and FP7 programmes, respectively.The ASMS conference has become a recognized event for industry and research institutions to exchangeup-to-date information about recent advances and emerging technologies in the field of mobile satellitecommunication systems. In addition, following the successful path opened by the 2006 edition, the scope ofthe conference has been widened to include broadband and broadcast satellite systems. This is veryimportant not only because the major part of the satcom market belongs to broadcasting and broadbandaccess operators but also fundamentally because the convergence of broadcast, mobile and fixed satellitecommunications is essential to offer seamless connectivity anywhere at anytime, which is recognized as thekey element for the successful deployment of future satellite systems.This Issue collects the extendedversions of eight of the best papers presented at the ASMS 2008 Conference. The papers have been selectedwith the aim of providing an insight in the developments and findings in this exciting field. The first papershows that Low Density Parity Check (LDPC) codes can tightly approach the performance of idealmaximum distance separable codes over memoryless erasure channels, under Maximum Likelihood (ML)decoding [1]. The decoding complexity of ML decoding can be kept low thanks to a class of decodingalgorithms which exploit the sparseness of the parity-check matrix to reduce the complexity of Gaussianelimination. The second paper includes a review and characterization of possible data link technologies foraeronautical communications covering a vast range of terrestrial (direct air-to-ground) and satellite-basedsolutions, with the final goal to identify suitable candidates for a future integrated aeronautical networkbased on IPv6 protocol, supporting both cockpit and cabin communications [2]. A new method to enableIPsec authentication and integrity services at the network layer is proposed in the thirdpaper [3]. Target scenarios include satellite multicast for large-scale delivery of data and video content.In these scenarios, a cross-layer architecture can mitigate the effects of physical layer impairments andlarge latency. Stemming from the three-state land mobile satellite channel model used for physical layersimulation in the course of the DVB-SH standardization process, an improved model is proposed in thefourth paper [4]. The new model includes two major modifications: a reduction in the number of states and

Quasi-Orthogonal Space-Frequency and Space-Time-Frequency Block Codes for MIMO OFDM Channels

In this paper, we propose a novel class of Space-Frequency and Space-Time-Frequency block codes based on Quasi-Orthogonal designs, over a frequency selective Rayleigh fading channel. The proposed Space-Frequency code is able to achieve rate-one and full space and multipath diversity gains available in the MIMO-OFDM channel. As simulation results demonstrate, the code outperforms the existing Space-Frequency block codes in terms of bit error rate performance. By coding across the three dimension of space, time and frequency, we propose a Quasi-Orthogonal Space-Time-Frequency code that is capable of achieving rate-one and exploiting all of the spatial, multipath and temporal diversity gains offered by the channel. In case of a channel which is quasi-static over adjacent OFDM symbol durations, we propose a Space-Time-Frequency code that benefits from a reduced maximum likelihood decoding complexity.

Linear Decoupled and Quasi-Decoupled Space–Time Codes

Space-time transmit diversity results in coupling of transmitted symbols across different antennas, which increases the complexity of maximum-likelihood decoding. Symbol coupling can be completely or partially avoided if the space-time code (STC) satisfies specific decoupling conditions; examples of such codes are orthogonal space-time block codes and quasi-orthogonal codes. In this letter, we study decoupling conditions for a linear full-diversity STC. Quasi-decoupled codes are proposed as a partially decoupled full-diversity STC of any rate for any number of transmit antennas with minimum decoding delay. By optimizing the coding gain of quasi-decoupled codes, it is shown that quasi-orthogonal codes have competitive performance with respect to the Alamouti code, and the more-recent threaded algebraic space-time (TAST) codes and ABBA codes. A general full-diversity decoupling condition is considered, and the general solution to this case, which also encompasses previously known orthogonal STCs, is derived

Optimising quasi-orthogonal STBC through group-constrained linear transformation

The generic algebraic structure of a quasi-orthogonal space-time block code (QO-STBC) has been derived, and group-constrained linear transformation (GCLT) as a mean to optimise the diversity and coding gains of a QO-STBC with square or rectangular QAM constellations has been proposed. Compared with QO-STBC with constellation rotation (CR), QO-STBC with GCLT requires only half the number of symbols for joint detection, hence lower maximum-likelihood decoding complexity. The optimum GCLT parameters for QO-STBC with square QAM constellation have been derived analytically. The optimised QO-STBCs with GCLT are able to achieve full-transmit diversity and have negligible performance loss compared with QO-STBCs with CR at the same code rate.