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Noisy-channel Coding Theorem
In information theory, the noisy-channel coding theorem (sometimes Shannon's theorem or Shannon's limit), establishes that for any given degree of noise contamination of a communication channel, it is possible to communicate discrete data (digital information) nearly error-free up to a computable maximum rate through the channel. This result was presented by Claude Shannon in 1948 and was based in part on earlier work and ideas of Harry Nyquist and Ralph Hartley. The Shannon limit or Shannon capacity of a communication channel refers to the maximum rate of error-free data that can theoretically be transferred over the channel if the link is subject to random data transmission errors, for a particular noise level. It was first described by Shannon (1948), and shortly after published in a book by Shannon and Warren Weaver entitled '' The Mathematical Theory of Communication'' (1949). This founded the modern discipline of information theory. Overview Stated by Claude Sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sampling Theorem
Sampling may refer to: * Sampling (signal processing), converting a continuous signal into a discrete signal * Sampling (graphics), converting continuous colors into discrete color components * Sampling (music), the reuse of a sound recording in another recording ** Sampler (musical instrument), an electronic musical instrument used to record and play back samples * Sampling (statistics), selection of observations to acquire some knowledge of a statistical population *Sampling (case studies), selection of cases for single or multiple case studies * Sampling (audit), application of audit procedures to less than 100% of population to be audited *Sampling (medicine), gathering of matter from the body to aid in the process of a medical diagnosis and/or evaluation of an indication for treatment, further medical tests or other procedures. * Sampling (occupational hygiene), detection of hazardous materials in the workplace *Sampling (for testing or analysis), taking a representative portio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Low-density Parity-check Code
In information theory, a low-density parity-check (LDPC) code is a linear error correcting code, a method of transmitting a message over a noisy transmission channel. An LDPC code is constructed using a sparse Tanner graph (subclass of the bipartite graph). LDPC codes are capacity-approaching codes, which means that practical constructions exist that allow the noise threshold to be set very close to the theoretical maximum (the Shannon limit) for a symmetric memoryless channel. The noise threshold defines an upper bound for the channel noise, up to which the probability of lost information can be made as small as desired. Using iterative belief propagation techniques, LDPC codes can be decoded in time linear to their block length. LDPC codes are finding increasing use in applications requiring reliable and highly efficient information transfer over bandwidth-constrained or return-channel-constrained links in the presence of corrupting noise. Implementation of LDPC codes has l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Exponent
In information theory, the error exponent of a channel code or source code over the block length of the code is the rate at which the error probability decays exponentially with the block length of the code. Formally, it is defined as the limiting ratio of the negative logarithm of the error probability to the block length of the code for large block lengths. For example, if the probability of error P_ of a decoder drops as e^, where n is the block length, the error exponent is \alpha. In this example, \frac approaches \alpha for large n. Many of the information-theoretic theorems are of asymptotic nature, for example, the channel coding theorem states that for any rate less than the channel capacity, the probability of the error of the channel code can be made to go to zero as the block length goes to infinity. In practical situations, there are limitations to the delay of the communication and the block length must be finite. Therefore, it is important to study how the probabi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Equipartition Property
In information theory, the asymptotic equipartition property (AEP) is a general property of the output samples of a stochastic source. It is fundamental to the concept of typical set used in theories of data compression. Roughly speaking, the theorem states that although there are many series of results that may be produced by a random process, the one actually produced is most probably from a loosely defined set of outcomes that all have approximately the same chance of being the one actually realized. (This is a consequence of the law of large numbers and ergodic theory.) Although there are individual outcomes which have a higher probability than any outcome in this set, the vast number of outcomes in the set almost guarantees that the outcome will come from the set. One way of intuitively understanding the property is through Cramér's large deviation theorem, which states that the probability of a large deviation from mean decays exponentially with the number of samples. Such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Entropy Function
In information theory, the binary entropy function, denoted \operatorname H(p) or \operatorname H_\text(p), is defined as the entropy of a Bernoulli process with probability p of one of two values. It is a special case of \Eta(X), the entropy function. Mathematically, the Bernoulli trial is modelled as a random variable X that can take on only two values: 0 and 1, which are mutually exclusive and exhaustive. If \operatorname(X=1) = p, then \operatorname(X=0) = 1-p and the entropy of X (in shannons) is given by :\operatorname H(X) = \operatorname H_\text(p) = -p \log_2 p - (1 - p) \log_2 (1 - p), where 0 \log_2 0 is taken to be 0. The logarithms in this formula are usually taken (as shown in the graph) to the base 2. See ''binary logarithm''. When p=\tfrac 1 2, the binary entropy function attains its maximum value. This is the case of an unbiased coin flip. \operatorname H(p) is distinguished from the entropy function \Eta(X) in that the former takes a single real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutual Information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such as shannons ( bits), nats or hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable. Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair (X,Y) is from the product of the marginal distributions of X and Y. MI is the expected value of the pointwise mutual information (PMI). The quantity was defined and analyzed by Claude Shannon in his landmark paper " A Mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE Communications Letters
''IEEE Communications Letters'' is a peer-reviewed scientific journal published monthly by the IEEE Communications Society since 1997 and covering communications technology. The editor-in-chief is Marco Di Renzo (Laboratory of Signals and Systems Paris-Saclay University CNRS--CentraleSupelec--University Paris-Sud-Paris, France). According to the ''Journal Citation Reports'', it has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 3.457. References External links * Communications Letters Engineering journals Publications established in 1997 English-language journals Monthly journals {{Engineering-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rüdiger Urbanke
Rüdiger Leo Urbanke (born 1966) is an Austrian computer scientist and professor at the Ecole polytechnique fédérale de Lausanne (EPFL). Life Urbanke studied at the Technical University of Vienna with the diploma as an electrical engineer in 1988 and at the Washington University in St. Louis with the master's degree in 1992 and his doctorate in 1995. He then worked at Bell Laboratories. Career From 2000 to 2004 he was an Associate Editor of the IEEE Transactions on Information Theory. From 2009 till 2012 he was the head of the I&C Doctoral School and in 2013 he served as a Dean of I&C. Distinctions Urbanke is a co-recipient of the 2002 and the 2013 IEEE Information Theory Society Best Paper Award, a recipient of the 2011 IEEE Koji Kobayashi Computers and Communications Award, the 2014 IEEE Richard W. Hamming Medal and the 2023 Claude E. Shannon Award The Claude E. Shannon Award of the IEEE Information Theory Society was created to honor consistent and profound c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas J
Clarence Thomas (born June 23, 1948) is an American jurist who serves as an associate justice of the Supreme Court of the United States. He was nominated by President George H. W. Bush to succeed Thurgood Marshall and has served since 1991. After Marshall, Thomas is the second African American to serve on the Court and its longest-serving member since Anthony Kennedy's retirement in 2018. Thomas was born in Pin Point, Georgia. After his father abandoned the family, he was raised by his grandfather in a poor Gullah community near Savannah. Growing up as a devout Catholic, Thomas originally intended to be a priest in the Catholic Church but was frustrated over the church's insufficient attempts to combat racism. He abandoned his aspiration of becoming a clergyman to attend the College of the Holy Cross and, later, Yale Law School, where he was influenced by a number of conservative authors, notably Thomas Sowell, who dramatically shifted his worldview from progressiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
