We introduce space-time block coding, a new paradigm for communication over Rayleigh fading channels using multiple transmit antennas. Data is encoded using a space-time block code and the encoded data is split into n streams which 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. Maximum-likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection. This uses the orthogonal structure of the space-time block code and gives a maximum-likelihood decoding algorithm which is based only on linear processing at the receiver. Space-time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm. The classical mathematical framework of orthogonal designs is applied to construct space-time block codes. It is shown that space-time block codes constructed in this way only exist for few sporadic values of n. Subsequently, a generalization of orthogonal designs is shown to provide space-time block codes for both real and complex constellations for any number of transmit antennas. These codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as PAM. For an arbitrary complex constellation such as PSK and QAM, space-time block codes are designed that achieve 1/2 of the maximum possible transmission rate for any number of transmit antennas. For the specific cases of two, three, and four transmit antennas, space-time block codes are designed that achieve, respectively, all, 3/4, and 3/4 of maximum possible transmission rate using arbitrary complex constellations. The best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well.
Previous article Next article Polynomial Codes Over Certain Finite FieldsI. S. Reed and G. SolomonI. S. Reed and G. Solomonhttps://doi.org/10.1137/0108018PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. W. Hamming, Error detecting and error correcting codes, Bell System Tech. J., 29 (1950), 147–160 MR0035935 CrossrefISIGoogle Scholar[2] Irving S. Reed, A class of multiple-error-correcting codes and the decoding scheme, Trans. I.R.E., PGIT-4 (1954), 38–49, Prof. Group on Information Theory MR0089789 Google Scholar[3] Neal Zierler, Linear recurring sequences, J. Soc. Indust. Appl. Math., 7 (1959), 31–48 10.1137/0107003 MR0101979 0096.33804 LinkISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A General Family of MSRD Codes and PMDS Codes with Smaller Field Sizes from Extended Moore MatricesUmberto MartÍnez-Pen͂asSIAM Journal on Discrete Mathematics, Vol. 36, No. 3 | 16 August 2022AbstractPDF (562 KB)Algorithmic Fault Tolerance Using the Lanczos MethodSIAM Journal on Matrix Analysis and Applications, Vol. 13, No. 1 | 31 July 2006AbstractPDF (1962 KB)A New Class of Cyclic CodesSIAM Journal on Applied Mathematics, Vol. 16, No. 1 | 12 July 2006AbstractPDF (1580 KB)Multiple-Burst Error Correction with the Chinese Remainder TheoremJournal of the Society for Industrial and Applied Mathematics, Vol. 11, No. 1 | 13 July 2006AbstractPDF (758 KB)A Class of Error-Correcting Codes in $p^m $ SymbolsDaniel Gorenstein and Neal ZierlerJournal of the Society for Industrial and Applied Mathematics, Vol. 9, No. 2 | 10 July 2006AbstractPDF (691 KB) Volume 8, Issue 2| 1960Journal of the Society for Industrial and Applied Mathematics225-424 History Submitted:21 January 1959Published online:10 July 2006 InformationCopyright © 1960 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0108018Article page range:pp. 300-304ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics
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.
This paper presents a new family of convolutional codes, nicknamed turbo-codes, built from a particular concatenation of two recursive systematic codes, linked together by nonuniform interleaving. Decoding calls on iterative processing in which each component decoder takes advantage of the work of the other at the previous step, with the aid of the original concept of extrinsic information. For sufficiently large interleaving sizes, the correcting performance of turbo-codes, investigated by simulation, appears to be close to the theoretical limit predicted by Shannon.
A new class of convolutional codes called turbo-codes, whose performances in terms of bit error rate (BER) are close to the Shannon limit, is discussed. The turbo-code encoder is built using a parallel concatenation of two recursive systematic convolutional codes, and the associated decoder, using a feedback decoding rule, is implemented as P pipelined identical elementary decoders.>
The probability of error in decoding an optimal convolutional code transmitted over a memoryless channel is bounded from above and below as a function of the constraint length of the code. For all but pathological channels the bounds are asymptotically (exponentially) tight for rates aboveR_{0}, the computational cutoff rate of sequential decoding. As a function of constraint length the performance of optimal convolutional codes is shown to be superior to that of block codes of the same length, the relative improvement increasing with rate. The upper bound is obtained for a specific probabilistic nonsequential decoding algorithm which is shown to be asymptotically optimum for rates aboveR_{0}and whose performance bears certain similarities to that of sequential decoding algorithms.
We introduce LT codes, the first rateless erasure codes that are very efficient as the data length grows.
A low-density parity-check code is a code specified by a parity-check matrix with the following properties: each column contains a small fixed numberj \geq 3of l's and each row contains a small fixed numberk > jof l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixedj. When used with maximum likelihood decoding on a sufficiently quiet binary-input symmetric channel, the typical probability of decoding error decreases exponentially with block length for a fixed rate and fixedj. A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described. Both the equipment complexity and the data-handling capacity in bits per second of this decoder increase approximately linearly with block length. Forj > 3and a sufficiently low rate, the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length. Some experimental results show that the actual probability of decoding error is much smaller than this theoretical bound.
The general problem of estimating the a posteriori probabilities of the states and transitions of a Markov source observed through a discrete memoryless channel is considered. The decoding of linear block and convolutional codes to minimize symbol error probability is shown to be a special case of this problem. An optimal decoding algorithm is derived.
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An optimum method of coding an ensemble of messages consisting of a finite number of members is developed. A minimum-redundancy code is one constructed in such a way that the average number of coding digits per message is minimized.
The author was led to the study given in this paper from a consideration of large scale computing machines in which a large number of operations must be performed without a single error in the end result. This problem of “doing things right” on a large scale is not essentially new; in a telephone central office, for example, a very large number of operations are performed while the errors leading to wrong numbers are kept well under control, though they have not been completely eliminated. This has been achieved, in part, through the use of self-checking circuits. The occasional failure that escapes routine checking is still detected by the customer and will, if it persists, result in customer complaint, while if it is transient it will produce only occasional wrong numbers. At the same time the rest of the central office functions satisfactorily. In a digital computer, on the other hand, a single failure usually means the complete failure, in the sense that if it is detected no more computing can be done until the failure is located and corrected, while if it escapes detection then it invalidates all subsequent operations of the machine. Put in other words, in a telephone central office there are a number of parallel paths which are more or less independent of each other; in a digital machine there is usually a single long path which passes through the same piece of equipment many, many times before the answer is obtained.
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AINSWORTH, B. E,., W. L. HASKELL, M. C. WHITT, M. L. IRWIN, A. M. SWARTZ, S .J. STRATH, W. L. O’BRIEN, D. R. BASSETT, JR., K. H. SCHMITZ, P. O EMPLAINCOURT, D. R. JACOBS, JR., and A. S. LEON. Compendium of physical activities: an update of activity codes and MET intensities. Med. Sci. Sports Exerc., Vol. 32, No. 9, Suppl., pp. S498–S516, 2000. We provide an updated version of the Compendium of Physical Activities, a coding scheme that classifies specific physical activity (PA) by rate of energy expenditure. It was developed to enhance the comparability of results across studies using self-reports of PA. The Compendium coding scheme links a five-digit code that describes physical activities by major headings (e.g., occupation, transportation, etc.) and specific activities within each major heading with its intensity, defined as the ratio of work metabolic rate to a standard resting metabolic rate (MET). Energy expenditure in MET-minutes, MET-hours, kcal, or kcal per kilogram body weight can be estimated for specific activities by type or MET intensity. Additions to the Compendium were obtained from studies describing daily PA patterns of adults and studies measuring the energy cost of specific physical activities in field settings. The updated version includes two new major headings of volunteer and religious activities, extends the number of specific activities from 477 to 605, and provides updated MET intensity levels for selected activities.
1. Coding for Reliable Digital Transmission and Storage. 2. Introduction to Algebra. 3. Linear Block Codes. 4. Important Linear Block Codes. 5. Cyclic Codes. 6. Binary BCH Codes. 7. Nonbinary BCH Codes, Reed-Solomon Codes, and Decoding Algorithms. 8. Majority-Logic Decodable Codes. 9. Trellises for Linear Block Codes. 10. Reliability-Based Soft-Decision Decoding Algorithms for Linear Block Codes. 11. Convolutional Codes. 12. Trellis-Based Decoding Algorithms for Convolutional Codes. 13. Sequential and Threshold Decoding of Convolutional Codes. 14. Trellis-Based Soft-Decision Algorithms for Linear Block Codes. 15. Concatenated Coding, Code Decomposition ad Multistage Decoding. 16. Turbo Coding. 17. Low Density Parity Check Codes. 18. Trellis Coded Modulation. 19. Block Coded Modulation. 20. Burst-Error-Correcting Codes. 21. Automatic-Repeat-Request Strategies.
While supervised learning has enabled great progress in many applications, unsupervised learning has not seen such widespread adoption, and remains an important and challenging endeavor for artificial intelligence. In this work, we propose a universal unsupervised learning approach to extract useful representations from high-dimensional data, which we call Contrastive Predictive Coding. The key insight of our model is to learn such representations by predicting the future in latent space by using powerful autoregressive models. We use a probabilistic contrastive loss which induces the latent space to capture information that is maximally useful to predict future samples. It also makes the model tractable by using negative sampling. While most prior work has focused on evaluating representations for a particular modality, we demonstrate that our approach is able to learn useful representations achieving strong performance on four distinct domains: speech, images, text and reinforcement learning in 3D environments.
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.
Comprend des références bibliographiques et un index.
The Search for the Codable Moment A Way of Seeing Developing Themes and Codes Deciding on Units of Analysis and Units of Coding as Issues of Sampling Developing Themes and a Code Using the Inductive Method An Example Using Life Stories Developing Themes Using the Theory-Driven and Prior-Research-Driven Method and Then Applying the Code An Example Using a Critical Incident Interview Scoring, Scaling and Clustering Themes Reliability Is Consistency of Judgment Don't Go Breaking My Heart Challenges in Using Thematic Analysis