Multilevel codes and multistage decoding
TL;DR: The author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets.
Abstract: H. Imai and S. Hirakawa have proposed (1977) a multilevel coding method based on binary block codes that admits a staged decoding procedure. The author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets. The multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels. It also allows the use of suboptimal multistage decoding procedures that have performance/complexity advantages over maximum-likelihood decoding. >
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TL;DR: In this paper, the authors consider the design of channel codes for improving the data rate and/or the reliability of communications over fading channels using multiple transmit antennas and derive performance criteria for designing such codes under the assumption that the fading is slow and frequency nonselective.
Abstract: We consider the design of channel codes for improving the data rate and/or the reliability of communications over fading channels using multiple transmit antennas. Data is encoded by a channel code and the encoded data is split into n streams that are simultaneously transmitted using n transmit antennas. The received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise. We derive performance criteria for designing such codes under the assumption that the fading is slow and frequency nonselective. Performance is shown to be determined by matrices constructed from pairs of distinct code sequences. The minimum rank among these matrices quantifies the diversity gain, while the minimum determinant of these matrices quantifies the coding gain. The results are then extended to fast fading channels. The design criteria are used to design trellis codes for high data rate wireless communication. The encoding/decoding complexity of these codes is comparable to trellis codes employed in practice over Gaussian channels. The codes constructed here provide the best tradeoff between data rate, diversity advantage, and trellis complexity. Simulation results are provided for 4 and 8 PSK signal sets with data rates of 2 and 3 bits/symbol, demonstrating excellent performance that is within 2-3 dB of the outage capacity for these channels using only 64 state encoders.
7,105 citations
TL;DR: This paper deals with 2/sup l/-ary transmission using multilevel coding (MLC) and multistage decoding (MSD) and shows that capacity can in fact be closely approached at high bandwidth efficiencies.
Abstract: 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.
1,030 citations
Cites methods from "Multilevel codes and multistage dec..."
..., [48], [26], [27], [51], and [52]), the balanced distances rule was used for rate design although multistage decoding was applied....
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TL;DR: New fundamental code constructions for both quasi-static and time-varying channels are developed, perhaps the first general constructions-other than delay diversity schemes-that guarantee full spatial diversity for an arbitrary number of transmit antennas.
Abstract: The design of space-time codes to achieve full spatial diversity over fading channels has largely been addressed by handcrafting example codes using computer search methods and only for small numbers of antennas. The lack of more general designs is in part due to the fact that the diversity advantage of a code is the minimum rank among the complex baseband differences between modulated codewords, which is difficult to relate to traditional code designs over finite fields and rings. We present general binary design criteria for PSK-modulated space-time codes. For linear BPSK/QPSK codes, the rank of (binary projections of) the unmodulated codewords, as binary matrices over the binary field, is a sufficient design criterion: full binary rank guarantees full spatial diversity. This criterion accounts for much of what is currently known about PSK-modulated space-time codes. We develop new fundamental code constructions for both quasi-static and time-varying channels. These are perhaps the first general constructions-other than delay diversity schemes-that guarantee full spatial diversity for an arbitrary number of transmit antennas.
391 citations
TL;DR: The optimized irregular LDPC codes at each level of MLC with multistage decoding (MSD) are able to perform well at signal-to-noise ratios (SNR) very close to the capacity of the additive white Gaussian noise (AWGN) channel.
Abstract: We design multilevel coding (MLC) and bit-interleaved coded modulation (BICM) schemes based on low-density parity-check (LDPC) codes. The analysis and optimization of the LDPC component codes for the MLC and BICM schemes are complicated because, in general, the equivalent binary-input component channels are not necessarily symmetric. To overcome this obstacle, we deploy two different approaches: one based on independent and identically distributed (i.i.d.) channel adapters and the other based on coset codes. By incorporating i.i.d. channel adapters, we can force the symmetry of each binary-input component channel. By considering coset codes, we extend the concentration theorem based on previous work by Richardson et al. ( see ibid., vol.47, p.599-618, Feb. 2001) and Kavc/spl caron/ic/spl acute/ et al.(see ibid., vol.49, p.1636-52, July 2003) We also discuss the relation between the systems based on the two approaches and show that they indeed have the same expected decoder behavior. Next, we jointly optimize the code rates and degree distribution pairs of the LDPC component codes for the MLC scheme. The optimized irregular LDPC codes at each level of MLC with multistage decoding (MSD) are able to perform well at signal-to-noise ratios (SNR) very close to the capacity of the additive white Gaussian noise (AWGN) channel. We also show that the optimized BICM scheme can approach the parallel independent decoding (PID) capacity as closely as does the MLC/PID scheme. Simulations with very large codeword length verify the accuracy of the analytical results. Finally, we compare the simulated performance of these coded modulation schemes at finite codeword lengths, and consider the results from the perspective of a random coding exponent analysis.
316 citations
Cites background from "Multilevel codes and multistage dec..."
...However, past research has primarily focused on the maximization of minimum Euclidean distance and asymptotic gains [3], [7]....
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TL;DR: Two approaches for providing unequal error protection to different classes of data are presented and can provide error protection for the important data to an extent that can hardly be achieved using conventional coded modulation with equal error protection.
Abstract: It is always desirable to maintain communications in difficult situations, even though fewer messages can get across. The author has developed such capabilities for one-way broadcast media, such as the envisioned terrestrial broadcasting of digital high-definition television signals. In this television broadcasting, the data from video source encoders are not equally important. It is desirable that the important data be recovered by each receiver even under poor receiving conditions. Two approaches for providing such unequal error protection to different classes of data are presented. Power-efficient and bandwidth-efficient coded modulation is used in both approaches. The first approach is based on novel signal constellations with nonuniformly spaced signal points. The second uses time division multiplexing of different conventional coded modulation schemes. Both approaches can provide error protection for the important data to an extent that can hardly be achieved using conventional coded modulation with equal error protection. For modest amounts of important data, the first approach has, additionally, the potential of providing immunity from impulse noise through simple bit or signal-point interleaving. >
298 citations
References
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Book•
01 Jan 1977
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
Abstract: Linear Codes. Nonlinear Codes, Hadamard Matrices, Designs and the Golay Code. An Introduction to BCH Codes and Finite Fields. Finite Fields. Dual Codes and Their Weight Distribution. Codes, Designs and Perfect Codes. Cyclic Codes. Cyclic Codes: Idempotents and Mattson-Solomon Polynomials. BCH Codes. Reed-Solomon and Justesen Codes. MDS Codes. Alternant, Goppa and Other Generalized BCH Codes. Reed-Muller Codes. First-Order Reed-Muller Codes. Second-Order Reed-Muller, Kerdock and Preparata Codes. Quadratic-Residue Codes. Bounds on the Size of a Code. Methods for Combining Codes. Self-dual Codes and Invariant Theory. The Golay Codes. Association Schemes. Appendix A. Tables of the Best Codes Known. Appendix B. Finite Geometries. Bibliography. Index.
10,083 citations
IBM1
TL;DR: A coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth by channel coding with expanded sets of multilevel/phase signals in a manner which increases free Euclidean distance.
Abstract: A coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth. This is achieved by channel coding with expanded sets of multilevel/phase signals in a manner which increases free Euclidean distance. Soft maximum--likelihood (ML) decoding using the Viterbi algorithm is assumed. Following a discussion of channel capacity, simple hand-designed trellis codes are presented for 8 phase-shift keying (PSK) and 16 quadrature amplitude-shift keying (QASK) modulation. These simple codes achieve coding gains in the order of 3-4 dB. It is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals, which we call "mapping by set partitioning." Based on a new distance measure between binary code sequences which efficiently lower-bounds the Euclidean distance between the corresponding channel signal sequences, a search procedure for more powerful codes is developed. Codes with coding gains up to 6 dB are obtained for a variety of multilevel/phase modulation schemes. Simulation results are presented and an example of carrier-phase tracking is discussed.
4,091 citations
TL;DR: It is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst-error correction abilities.
Abstract: A method is described for constructing long error-correcting codes from one or more shorter error-correcting codes, referred to as subcodes, and a bipartite graph. A graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes. Lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes. Both the encoders and decoders proposed are shown to take advantage of the code's explicit decomposition into subcodes to decompose and simplify the associated computational processes. Bounds on the performance of two specific decoding algorithms are established, and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed. The proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver, e.g., reliability information, without greatly increasing the number of computations required. It is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst-error correction abilities. The construction principles
3,246 citations
TL;DR: A new multilevel coding method that uses several error-correcting codes that makes effective use of soft-decisions to improve the performance of decoding and is superior to other multileVEL coding systems.
Abstract: A new multilevel coding method that uses several error-correcting codes is proposed. The transmission symbols are constructed by combining symbols of codewords of these codes. Usually, these codes are binary error-correcting codes and have different error-correcting capabilities. For various channels, efficient systems can be obtained by choosing these codes appropriately. Encoding and decoding procedures for this method are relatively simple compared with those of other multilevel coding methods. In addition, this method makes effective use of soft-decisions to improve the performance of decoding. The decoding error probability is analyzed for multiphase modulation, and numerical comparisons to other multilevel coding systems are made. When equally complex systems are compared, the new system is superior to other multilevel coding systems.
1,070 citations
TL;DR: This paper attempts to present a comprehensive tutorial survey of the development of efficient modulation techniques for bandlimited channels, such as telephone channels, with principal emphasis on coded modulation techniques, in which there is an explosion of current interest.
Abstract: This paper attempts to present a comprehensive tutorial survey of the development of efficient modulation techniques for bandlimited channels, such as telephone channels. After a history of advances in commercial high-speed modems and a discussion of theoretical limits, it reviews efforts to optimize two-dimensional signal constellations and presents further elaborations of uncoded modulation. Its principal emphasis, however, is on coded modulation techniques, in which there is an explosion of current interest, both for research and for practical application. Both block-coded and trellis-coded modulation are covered, in a common framework. A few new techniques are presented.
770 citations