This paper deals with 2 ` --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 cut--off 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, dimensional...

Abstract — We present a bandwidth-efficient channel coding scheme that has an overall structure similar to binary turbo codes, but employs trellis-coded modulation (TCM) codes (including multidimensional codes) as component codes. The combination of turbo codes with powerful bandwidth-efficient component codes leads to a straightforward encoder structure, and allows iterative decoding in analogy to the binary turbo decoder. However, certain special conditions may need to be met at the encoder, and the iterative decoder needs to be adapted to the decoding of the component TCM codes. The scheme has been investigated for 8-PSK, 16-QAM, and 64-QAM modulation schemes with varying overall bandwidth efficiencies. A simple code choice based on the minimal distance of the punctured component code has also been performed. The interset distances of the partitioning tree can be used to fix the number of coded and uncoded bits. We derive the symbol-by-symbol MAP component decoder operating in the log domain, and apply methods of reducing decoder complexity. Simulation results are presented and compare the scheme with traditional TCM as well as turbo codes with Gray mapping. The results show that the novel scheme is very powerful, yet of modest complexity since simple component codes are used. Index Terms—Decoding, iterative methods, trellis-coded modulation. I.

This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint source-channel coding schemes. We first show some examples of sources and channels where no coding is optimal for all values of the signal-to-noise ratio (SNR). When the source bandwidth is greater than the channel bandwidth, joint coding schemes based on space-filling curves and other families of curves are proposed. For uniform sources and modulo channels, our coding scheme based on space-filling curves operates within 1.1 dB of Shannon’s rate-distortion bound. For Gaussian sources and additive white Gaussian noise (AWGN) channels, we can achieve within 0.9 dB of the rate-distortion bound. The second scheme is based on low-density parity-check (LDPC) codes. We first demonstrate that we can translate threshold values of an LDPC code between channels accurately using a simple mapping. We develop some models for density evolution

In this contribution, we provide an overview of the novel class of channel codes referred to as turbo codes, which have been shown to be capable of performing close to the Shannon Limit. We commence with a brief discussion on turbo encoding, and then move on to describing the form of the iterative decoder most commonly used to decode turbo codes. We then elaborate on various decoding algorithms that can be used in an iterative decoder, and give an example of the operation of such a decoder using the so-called Soft Output Viterbi Algorithm (SOVA). Lastly, the effect of a range of system parameters is investigated in a systematic fashion, in order to gauge their performance ramifications.

In this paper, we deal with the design of high-rate multilevel two-dimensional (2D) bar codes for the print-and-scan channel. Firstly, we introduce a framework for evaluating the performance limits of these codes by studying an inter-symbol interference (ISI) free, synchronous, and noiseless print-and-scan channel, where the input and output alphabets are finite and the printer device uses halftoning to simulate multiple gray levels. Secondly, we present a new model for the print-and-scan channel specifically adapted to the problem of communications via multilevel 2D bar codes. This model, inspired by our experimental work, assumes no ISI and perfect synchronization, but independence between the channel input and the noise is not supposed. We adapt the theory of multilevel coding with multistage decoding (MLC/MSD) to the print-and-scan channel. Finally, we present experimental results confirming the utility of our channel model, and showing that multilevel 2D bar codes using MLC/MSD can reliably achieve the high capacity storage requirements of many multimedia security and management applications.

In 1948 Shannon developed fundamental limits on the efficiency of communication over noisy channels. The coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero. Fifty years later, codes for the Gaussian channel have been discovered that come close to these fundamental limits. There is now a substantial algebraic theory of error-correcting codes with as many connections to mathematics as to engineering practice, and the last 20 years have seen the construction of algebraic-geometry codes that can be encoded and decoded in polynomial time, and that beat the Gilbert–Varshamov bound. Given the size of coding theory as a subject, this review is of necessity a personal perspective, and the focus is reliable communication, and not source coding or cryptography. The emphasis is on connecting coding theories for Hamming and Euclidean space and on future challenges, specifically in data networking, wireless communication, and quantum information theory.

The development of theory for multilevel coding during the last 20 years is reminded. Several design rules are compared and the capacity region of multilevel codes is outlined. We discuss the dimensionality of the constituent signal constellation and compare different labeling strategies by means of random coding exponent. Finally, by discussing bit interleaved coded modulation we show that the progress in theory now leads back to the origin of coded modulation. 1 Introduction and Historical Overview During the first 25 years after the foundation of information theory by C.E. Shannon 50 years ago research in channel coding was almost entirely restricted to coding for binary antipodal signalling like BPSK, QPSK with Gray mapping etc. It soon was recognized that a direct application of traditional forward error correcting codes (FEC) to bandwidth efficient modulation schemes with more than two signal points per dimension of a signal space, e.g. M ? 2--ary ASK, M ? 4--ary QAM, M ? 4--ar...

This is a tutorial paper meant to introduce the reader to the new concept of turbo codes. This is a new and very powerful error correction technique which outperforms all previous known coding schemes. It can be used in any communication system where a significant power saving is required or the operating signal--to--noise ratio (SNR) is very low. Deep space communications, mobile satellite/cellular communications, microwave links, paging, etc., are some of the possible applications of this revolutionary coding technique. Part I of the paper discussed the history of turbo codes, why they are different from traditional convolutional/block codes, the turbo encoder structures and issues related to the interleaver design. Part II addresses the decoder architecture, the achievable performance for turbo codes for a wide range of coding rates and modulation techniques and discusses delay and implementation issues. 2 1 Introduction The optimum decoding of turbo codes is the maximum likeli...

The stability behaviour of the iterative decoding (ID) of Turbo-Codes and its influence on the overall performance are discussed. Several criteria to characterize the ID are proposed and the most reliable ones are determined. These criteria are used to partition the transnitted blocks into stable and unstable ones. For unstable blocks the decoding step is determined which most likely produces a low number of decoding errors. These operations allow to lower the bit error rate (BER) significantly and to detect the erroneous blocks with high probability. Simulations show an improvement in BER by factor 10 and more. 1 Introduction For moderate reliability requirements Turbo-Codes (TC) ([1]) are currently the most powerful known error control codes. The coding scheme of TC's is a parallel concatenation of constituent codes. Decoding of these codes is performed via iterative decoding (ID) of the constituent codes. TC's allow to create very long codewords and to achieve a moderate BER at an ...

A self--adapting hierarchical channel coding scheme is discussed for the transmission of digital video data on time variant fading channels. Based on the cutoff rate for multilevel coding on fading channels we derive a procedure for the joint design of unequal error protection and a clustered signal constellation. The derived scheme is suitable for a data rate ratio of 1:1 between basic and enhancement data and is considerably superior to time sharing. Using binary 'Turbo' codes as component codes in a multilevel coding scheme simulation results show excellent agreement with the predicted performance. 1 Introduction To increase the compression rate all video source coding algorithms use variable length Huffman coding. This makes the coded data stream very sensitive to residual errors after the channel decoder. A single error can result in error propagation and may cause a serious degradation of the image quality. For this reason an error free reception of the video data is required. B...