Multistage Decoding Research Articles

On the iterative decoding of multilevel codes

Iterative decoding of multilevel coded modulation is discussed. Despite its asymptotic optimality with proper design, the error correcting capability of multilevel codes may not be fully exploited for finite block length with conventional multistage decoding. This fact stems from the suboptimality of multistage decoding giving rise to increased error multiplicity at lower index stages and the associated error propagation to higher stages. Such problems can be overcome in many situations by introducing iterative decoding which often significantly compensates the suboptimality of a staged decoder. The class of multilevel codes achieving practically important bit-error performance near the Shannon limit becomes far wider with iterative decoding.

On multilevel codes and iterative multistage decoding

This paper develops an approach to iterative multistage decoding of multilevel codes. This involves passing reliability information to previous and subsequent decoders instead of only hard decisions to subsequent decoders. The paper also develops an adaptive version of the suboptimal soft output decoding algorithm of Picart and Pyndiah (1996). This adaptive algorithm provides a gain of approximately 0.24 dB at a bit error rate (BER) of 10/sup -5/ after four iterations and approximately 0.43 dB after ten iterations over the algorithm of Picart et al. If the adaptive algorithm is used in conjunction with iterative multistage decoding then a gain of approximately 0.62 dB is obtained at a BER of 10/sup -5/ after four iterations and approximately 0.9 dB after ten iterations over the algorithm of Picart et al.

Sphere-bound-achieving coset codes and multilevel coset codes

A simple sphere bound gives the best possible tradeoff between the volume per point of an infinite array L and its error probability on an additive white Gaussian noise (AWGN) channel. It is shown that the sphere bound can be approached by a large class of coset codes or multilevel coset codes with multistage decoding, including certain binary lattices. These codes have structure of the kind that has been found to be useful in practice. Capacity curves and design guidance for practical codes are given. Exponential error bounds for coset codes are developed, generalizing Poltyrev's (1994) bounds for lattices. These results are based on the channel coding theorems of information theory, rather than the Minkowski-Hlawka theorem of lattice theory.

Multilevel coded CPFSK systems for AWGN and fading channels

In this paper, we introduce multilevel coded continuous-phase frequency-shift keying (CPFSK) systems for both additive white Gaussian noise and fading channels based on multilevel coding and multistage decoding techniques. These schemes are designed under the constraint M/spl ges/2P for M-ary CPFSK with modulation index J/P. In order to maintain the phase continuity property after multilevel coding, we use some specific set partitioning rules. We construct examples to show that the proposed systems outperform the corresponding conventional one-level schemes. For the fading case, we consider an ideal fading channel where adjacent fading symbols are assumed to be independent and a correlated fading channel where bit interleavers are needed at each coding level. The computer simulation results show that the proposed systems have bit-error performance and decoder complexity advantages over the corresponding reference codes taken from the literature. It is also shown that there is only a slight degradation on the bit-error performance for correlated fading, compared to the ideal fading case.

Multilevel coded modulation for unequal error protection and multistage decoding .I. Symmetric constellations

In this paper, theoretical upper bounds and computer simulation results on the error performance of multilevel block coded modulations for unequal error protection (UEP) and multistage decoding are presented. It is shown that nonstandard signal set partitionings and multistage decoding provide excellent UEP capabilities beyond those achievable with conventional coded modulation. The coding scheme is designed in such a way that the most important information bits have a lower error rate than other information bits. The large effective error coefficients, normally associated with standard mapping by set partitioning, are reduced by considering nonstandard partitionings of the underlying signal set. The bits-to-signal mappings induced by these partitionings allow the use of soft-decision decoding of binary block codes. Moreover, parallel operation of some of the staged decoders is possible, to achieve high data rate transmission, so that there is no error propagation between these decoders. Hybrid partitionings are also considered that trade off increased intraset distances in the last partition levels with larger effective error coefficients in the middle partition levels. The error performance of specific examples of multilevel codes over 8-PSK and 64-QAM signal sets are simulated and compared with theoretical upper bounds on the error performance.

Design of optimum M-PSK codes for Rayleigh fading channel

A methodology for the design of M-PSK codes for the Rayleigh fading channel optimised for a specific target BER is proposed, using the framework of multilevel coded modulation. The principle is to equalise the BER at each stage of a multistage decoder. An 8-PSK code is designed which exceeds the performance of the code of Seshadri and Sundberg by 2.4 dB at BER = 10-3, as well as 16-PSK codes optimised for BERs of 10-3 and 10-6. No additional interleaver delay is introduced beyond that required by the channel.

Multilevel codes: theoretical concepts and practical design rules

This paper deals with 2/sup l/-ary transmission using multilevel coding (MLC) and multistage decoding (MSD). The known result that MLC and MSD suffice to approach capacity if the rates at each level are appropriately chosen is reviewed. Using multiuser information theory, it is shown that there is a large space of rate combinations such that MLC and full maximum-likelihood decoding (MLD) can approach capacity. It is noted that multilevel codes designed according to the traditional balanced distance rule tend to fall in the latter category and, therefore, require the huge complexity of MLD. The capacity rule, the balanced distances rules, and two other rules based on the random coding exponent and cutoff rate are compared and contrasted for practical design. Simulation results using multilevel binary turbo codes show that capacity can in fact be closely approached at high bandwidth efficiencies. Moreover, topics relevant in practical applications such as signal set labeling, dimensionality of the constituent constellation, and hard-decision decoding are emphasized. Bit interleaved coded modulation, proposed by Caire et al. (see ibid., vol.44, p.927-46, 1998), is reviewed in the context of MLC. Finally, the combination of signal shaping and coding is discussed. Significant shaping gains are achievable in practice only if these design rules are taken into account.

Constructions of generalized concatenated codes and their trellis-based decoding complexity

In this article, constructions of generalized concatenated (GC) codes with good rates and distances are presented. Some of the proposed GC codes have simpler trellis complexity than Euclidean geometry (EG), Reed-Muller (RM), or Bose-Chaudhuri-Hocquenghem (BCH) codes of approximately the same rates and minimum distances, and in addition can be decoded with trellis-based multistage decoding up to their minimum distances. Several codes of the same length, dimension, and minimum distance as the best linear codes known are constructed.

Multilevel concatenated-coded M-DPSK modulation schemes for the shadowed mobile satellite communication channel

This paper presents a bandwidth-efficient multilevel concatenated-coded modulation scheme for reliable data transmission over the shadowed mobile satellite communication (MSAT) channel. In this scheme, bandwidth-efficient block modulation codes are used as the inner codes, and Reed-Solomon codes of various error correcting capabilities are used as the outer codes. The inner and outer codes are concatenated in multiple levels. A general method for constructing multilevel concatenated modulation codes is presented, and a multistage closest coset decoding for these codes is proposed. Specific multilevel concatenated 8-PSK modulation codes have been constructed. These codes are designed to lower the bit error rate (BER) of the error floor caused by the large Doppler frequency shift due to the motion of vehicles. Simulation results show that these codes perform very well and achieve large coding gains over the uncoded reference modulation systems.

On the calculation of the error probability for a multilevel modulation scheme using QAM-signaling

We analyze the quadrature amplitude modulation (QAM) version of multilevel modulation with multistage decoding using a suboptimal metric, when transmission takes place over a memoryless Gaussian channel. The upper bounds for decoding error probabilities are functions of the Chernoff bounding parameter Z. We argue that the conventional approximation of Z is not adequate; new values of Z that tightens the error bounds without causing them to lose their validity are given. The capacity for this system is also calculated, and we conclude that the use of a suboptimal metric in multistage decoding causes very little degradation in capacity compared to when the optimal metric is used in each decoding stage.

Bit-error probability for maximum-likelihood decoding of linear block codes and related soft-decision decoding methods

In this correspondence, the bit-error probability P/sub b/ for maximum-likelihood decoding of binary linear block codes is investigated. The contribution P/sub b/(j) of each information bit j to P/sub b/ is considered and an upper bound on P/sub b/(j) is derived. For randomly generated codes, it is shown that the conventional approximation at high SNR P/sub b//spl ap/(d/sub H//N).P/sub s/, where P/sub s/ represents the block error probability, holds for systematic encoding only. Also systematic encoding provides the minimum P/sub b/ when the inverse mapping corresponding to the generator matrix of the code is used to retrieve the information sequence. The bit-error performances corresponding to other generator matrix forms are also evaluated. Although derived for codes with a generator matrix randomly generated, these results are shown to provide good approximations for codes used in practice. Finally, for soft-decision decoding methods which require a generator matrix with a particular structure such as trellis decoding, multistage decoding, or algebraic-based soft-decision decoding, equivalent schemes that reduce the bit-error probability are discussed. Although the gains achieved at practical bit-error rates are only a fraction of a decibel, they remain meaningful as they are of the same orders as the error performance differences between optimum and suboptimum decodings. Most importantly, these gains are free as they are achieved with no or little additional circuitry which is transparent to the conventional implementation.

Set partitioning for multilevel codes on a Rayleigh-fading channel

On a Rayleigh-fading channel, multilevel codes achieve larger Hamming distances than single-level codes at comparable decoder complexity and bandwidth efficiency by employing a multistage decoder. Typically, multilevel codes are designed using Ungerboeck's (1982) set partitioning. This results in a large path multiplicity that degrades the code performance significantly. In order to overcome the performance degradation, we propose a set partitioning method that greatly reduces the path multiplicity.

Multilevel block codes with interdependency among component codes for Rayleigh fading channels

In this paper, in order to improve error correction capacity, the following method is proposed for constructing multilevel codes with interdependencies among the levels. First, one of the component codes of a basic multilevel code is divided into 2i (i = 1, 2, …) subcodes. Then, the number of codewords of the other component code is expanded by a factor of 2i and partitioned into 2i subcode groups. One-to-one assignments among the levels are made between these subcode groups, a code array is constructed by using only the codewords contained in these subcodes, and an interdependency among the levels is introduced. In QPSK and 8PSK modulation, multilevel codes that have superior performance in fading channels are determined through a computer search. Furthermore, the bit error rate performance in the Rayleigh fading channel is determined by computer simulation. Then, although the decoder complexity of the multistage decoding method is increased after the introduction of the interdependencies, the bit error rate becomes equivalent to that of maximum likelihood decoding. Compared with previously published results, it is demonstrated that codes can be constructed with higher coding gain. © 1998 Scripta Technica. Electron Comm Jpn Pt 3, 81(1): 91–100, 1998

Multilevel coded 16-QAM schemes for Rayleigh fadingchannels

Multilevel codes show better performance compared with trellis codes on Rayleigh fading channels at comparable decoder complexity and bandwidth. However, they suffer from performance degradation due to error propagation in the multistage decoder. The authors, with a view to minimising the error propagation, compare three multilevel coded 16-QAM schemes which are four-level codes, I/Q separated two-level codes and I/Q separated two-level codes with a new set partitioning.

On the spectrum of distances of a multilevel code, decoded by a multistage decoder

In this correspondence we are concerned with the error rates contributed by component codes in a multilevel code, decoded by a multistage decoder. These error rates are required for estimating the performance of the structure, but more importantly, they can serve as a design tool for achieving high coding gain. The latter objective can be reached by approaching a balance among these error rates. For evaluating the error rate contributed by a component code, it is essential to calculate the corresponding spectrum of Euclidean distances. A method for computing this spectrum is developed in this correspondence. The method is general and applicable to any signal constellation, and convolutional as well as block component codes. The error coefficient, available in the literature for some multilevel codes with an infinite size signal constellation, is shown to be significantly different from that of a practical finite constellation. Note that the complete spectrum is essential when codewords (of a component code) with Hamming distance beyond the minimum distance affect the error rate (see, e.g., Turbo codes).

Some decomposable codes: the |a+x|b+x|a+b+x| construction

Codes with decomposable structure allow the use of multistage decoding procedures to achieve suboptimum bounded-distance error performance with reduced decoding complexity. This correspondence presents some new decomposable codes, including a class of distance-8 codes, that are constructed based on the |a+x|b+x|a+b+x| construction method. Some existing best codes are shown to be decomposable and hence can be decoded with multistage decoding.

Coding for a channel with quantization in the presence of an estimable interference

The encoding and decoding schemes presented are aimed at enabling the transfer of data through a channel in which two types of interference are added to the transmitted signal and the sum is quantized. One of these interferences is known (or can be estimated), whereas the second is an additive white Gaussian noise (AWGN). Since the input of the quantizer is not accessible, the known interference can not be removed from the received signal. We show that the error rate for an uncoded transmission through this channel is unacceptably large, even for low noise levels and linear quantization. It is also shown that the problem becomes even more severe when a nonlinear quantization is present. Therefore, coding is essential and a huge coding gain is achievable in this application. An upper-bound on the error rate, contributed by the component codes of a multilevel code, has been developed for multistage decoding. Results of computer simulations of a practical case with optimal and suboptimal decoding algorithms, both developed in this paper, are presented.

Multidimensional trellis coded phase modulation using a multilevel concatenation approach. I. Code design

The first part of this paper presents a simple and systematic technique for constructing multidimensional M-ary phase shift keying (MPSK) trellis coded modulation (TCM) codes. The construction is based on a multilevel concatenation approach. In which binary convolutional codes with good free branch distances are used as the outer codes and block MPSK modulation codes are used as the inner codes (or the signal spaces). Conditions on phase invariance of these codes are derived and a multistage decoding scheme for these codes is proposed. The proposed technique can be used to construct good codes for both the additive white Gaussian noise (AWGN) and fading channels as is shown in the second part of this paper.

Two-level codes for frequency/phase modulation

Trellis codes for frequency/phase modulation have larger asymptotic coding gain than trellis codes for phase modulation. Multilevel codes which enable multistage decoding have the advantage of larger asymptotic coding gain and lower decoder complexity than trellis codes. Combining these results, new two-level codes for frequency/phase modulation are presented.

The error probability of M-ary PSK block coded modulation schemes

A tight upper bound on the decoding error probability is derived for block-coded modulation structures where an M-ary phase shift keying (M-PSK) signal constellation is employed. This bound, called a tangential sphere bound, is tight for very low (as well as for high) signal-to-noise ratios (SNRs). Berlekamp's tangential union bound, previously derived for binary codes, can be derived for an M-PSK block coded modulation structure as well. However, it is proven that our tangential sphere bound is tighter than Berlekamp's (1980) tangential bound. For particular schemes, it is shown that for low SNRs our bound is considerably tighter than the tangential bound. As one of the examples, a multistage decoder is considered.