Shannon Information Theory
information theory channel capacity communication systems theory and practice Die Informationstheorie wurde von Claude Elwood Shannon. Shannon's channel coding theorem; Random coding and error exponent; MAP and ML decoding; Bounds; Channels and capacities: Gaussian channel, fading. Shannon's information theory deals with source coding [ ] Claude Shannon established the mathematical basis of information theory and published [ ].
Claude E. Shannon AwardShannon's information theory deals with source coding [ ] Claude Shannon established the mathematical basis of information theory and published [ ]. Die Informationstheorie ist eine mathematische Theorie aus dem Bereich der Claude E. Shannon: A mathematical theory of communication. Bell System Tech. Originally developed by Claude Shannon in the s, information theory laid the foundations for the digital revolution, and is now an essential tool in.
Shannon Information Theory Shannon’s Bits VideoClaude Shannon - Father of the Information Age
Decoder: The telephone that the receiver is holding will turn the binary data packages it receives back into sounds that replicate the voice of the sender.
Receiver: The receiver will hear the sounds made by the decoder and interpret the message. Feedback: The receiver may speak in response, to let the sender know what they heard or understood.
Encoder: The microphone and its computer will turn the voice of the radio host into binary packets of data that are sent to the radio transmitter.
The radio transmitter, also part of the encoder, will turn that data into radio waves ready to be transmitted. Receiver: The receiver is the person listening to the radio, who will hopefully receiver the full message loud and clear if noise has been avoided or minimized.
Feedback: Feedback is difficult in this step. However, the radio channel may send out researchers into the field to interview listeners to see how effective their communication has been.
Sender: The person starting the conversation will say something to start the communication process. Noise: The sender may have mumbled or have an accent that caused the message to be distorted internal noise.
There might be a wind or traffic that made the message hard to hear external noise. Receiver: The receiver is the second person in the conversation, who the sender is talking to.
Feedback: Face-to-face communication involves lots of feedback, as each person takes turns to talk. The catch is that one needs a random key that is as long as the message to be encoded and one must never use any of the keys twice.
Shannon's contribution was to prove rigorously that this code was unbreakable. To this day, no other encryption scheme is known to be unbreakable.
The problem with the one-time pad so-called because an agent would carry around his copy of a key on a pad and destroy each page of digits after they were used is that the two parties to the communication must each have a copy of the key, and the key must be kept secret from spies or eavesdroppers.
Quantum cryptography solves that problem. More properly called quantum key distribution, the technique uses quantum mechanics and entanglement to generate a random key that is identical at each end of the quantum communications channel.
The quantum physics ensures that no one can eavesdrop and learn anything about the key: any surreptitious measurements would disturb subtle correlations that can be checked, similar to error-correction checks of data transmitted on a noisy communications line.
Encryption based on the Vernam cypher and quantum key distribution is perfectly secure: quantum physics guarantees security of the key and Shannon's theorem proves that the encryption method is unbreakable.
At Bell Labs and later M. Shannon also made the startling discovery that, even in the presence of noise, it is always possible to transmit signals arbitrarily close to the theoretical channel capacity.
This discovery inspired engineers to look for practical techniques to improve performance in signal transmissions that were far from optimal. Before Shannon, engineers lacked a systematic way of analyzing and solving such problems.
Though information theory does not always make clear exactly how to achieve specific results, people now know which questions are worth asking and can focus on areas that will yield the highest return.
They also know which sorts of questions are difficult to answer and the areas in which there is not likely to be a large return for the amount of effort expended.
The section Applications of information theory surveys achievements not only in such areas of telecommunications as data compression and error correction but also in the separate disciplines of physiology, linguistics, and physics.
Unfortunately, many of these purported relationships were of dubious worth. Information theory often concerns itself with measures of information of the distributions associated with random variables.
Important quantities of information are entropy, a measure of information in a single random variable, and mutual information, a measure of information in common between two random variables.
The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with the given distribution can be reliably compressed.
The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution.
The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. A common unit of information is the bit, based on the binary logarithm.
Other units include the nat , which is based on the natural logarithm , and the decimal digit , which is based on the common logarithm.
Based on the probability mass function of each source symbol to be communicated, the Shannon entropy H , in units of bits per symbol , is given by.
This equation gives the entropy in the units of "bits" per symbol because it uses a logarithm of base 2, and this base-2 measure of entropy has sometimes been called the shannon in his honor.
Entropy is also commonly computed using the natural logarithm base e , where e is Euler's number , which produces a measurement of entropy in nats per symbol and sometimes simplifies the analysis by avoiding the need to include extra constants in the formulas.
Other bases are also possible, but less commonly used. Intuitively, the entropy H X of a discrete random variable X is a measure of the amount of uncertainty associated with the value of X when only its distribution is known.
If one transmits bits 0s and 1s , and the value of each of these bits is known to the receiver has a specific value with certainty ahead of transmission, it is clear that no information is transmitted.
If, however, each bit is independently equally likely to be 0 or 1, shannons of information more often called bits have been transmitted.
Between these two extremes, information can be quantified as follows. The special case of information entropy for a random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2, thus having the shannon Sh as unit:.
The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: X , Y. This implies that if X and Y are independent , then their joint entropy is the sum of their individual entropies.
For example, if X , Y represents the position of a chess piece— X the row and Y the column, then the joint entropy of the row of the piece and the column of the piece will be the entropy of the position of the piece.
Despite similar notation, joint entropy should not be confused with cross entropy. The conditional entropy or conditional uncertainty of X given random variable Y also called the equivocation of X about Y is the average conditional entropy over Y : .
Because entropy can be conditioned on a random variable or on that random variable being a certain value, care should be taken not to confuse these two definitions of conditional entropy, the former of which is in more common use.
A basic property of this form of conditional entropy is that:. Mutual information measures the amount of information that can be obtained about one random variable by observing another.
It is important in communication where it can be used to maximize the amount of information shared between sent and received signals.
The mutual information of X relative to Y is given by:. Mutual information is symmetric :. Mutual information can be expressed as the average Kullback—Leibler divergence information gain between the posterior probability distribution of X given the value of Y and the prior distribution on X :.
In other words, this is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y.
This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution:.
The Kullback—Leibler divergence or information divergence , information gain , or relative entropy is a way of comparing two distributions: a "true" probability distribution p X , and an arbitrary probability distribution q X.
If we compress data in a manner that assumes q X is the distribution underlying some data, when, in reality, p X is the correct distribution, the Kullback—Leibler divergence is the number of average additional bits per datum necessary for compression.
It is thus defined. Although it is sometimes used as a 'distance metric', KL divergence is not a true metric since it is not symmetric and does not satisfy the triangle inequality making it a semi-quasimetric.
Another interpretation of the KL divergence is the "unnecessary surprise" introduced by a prior from the truth: suppose a number X is about to be drawn randomly from a discrete set with probability distribution p x.
If Alice knows the true distribution p x , while Bob believes has a prior that the distribution is q x , then Bob will be more surprised than Alice, on average, upon seeing the value of X.
The KL divergence is the objective expected value of Bob's subjective surprisal minus Alice's surprisal, measured in bits if the log is in base 2.
In this way, the extent to which Bob's prior is "wrong" can be quantified in terms of how "unnecessarily surprised" it is expected to make him.
Coding theory is one of the most important and direct applications of information theory. This principle can then be used to communicate letters, numbers, and other informational concepts that we recognize.
Take the alphabet, for example. In reducing the uncertainty of the equation, multiple bits of information are generated. This is because each character being transmitted either is or is not a specific letter of that alphabet.
When you add in a space, which is required for communication in words, the English alphabet creates 27 total characters. This results in 4. Thanks to the mathematics of the information theory, we can know with certainty that any transmission or storage of information in digital code requires a multiplication of 4.
Probabilities help us to further reduce the uncertainty that exists when evaluating the equations of information that we receive every day.
It also means we can transmit less data, further reducing our uncertainty we face in solving the equation. Once all of these variables are taken into account, we can reduce the uncertainty which exists when attempting to solve informational equations.
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Laugh Francisca December 1, , pm. Please can you show directly what I ask to enable me do the assignment? Khaled Alyami January 15, , pm.Die Informationstheorie ist eine mathematische Theorie aus dem Bereich der Claude E. Shannon: A mathematical theory of communication. Bell System Tech. In diesem Jahr veröffentlichte Shannon seine fundamentale Arbeit A Mathematical Theory of Communication und prägte damit die moderne Informationstheorie. Der Claude E. Shannon Award, benannt nach dem Begründer der Informationstheorie Claude E. Shannon, ist eine seit von der IEEE Information Theory. provides the first comprehensive treatment of the theory of I-Measure, network coding theory, Shannon and non-Shannon type information inequalities, and a. I found the approach to the basic formulas very helpful. Der Begriff ist eng verwandt mit der Entropie in der Thermodynamik und statistischen Mechanik. It includes five meticulously written core chapters with accompanying problemsemphasizing the key topics of information measures; lossless and lossy data compression; channel coding; and joint source-channel coding for single-user point-to-point Royal Story Spielen systems. Download as PDF Printable version. But mainly, if you consider a half of a text, it is common to say that it has half the information of the text in its whole. When Cmc Trading compress data, we extract redundancy. On the left of the following figure is the entropies of two coins thrown independently. For longer telecommunications, we use the electromagnetic field. Data compression methods. Help Learn to edit Community portal Recent changes Upload file. Article Contents. Well, if I Poker Browsergame only half of a text, it may contain most of the information of the text rather than the half of it…. Information helps us to make decisions. Memory kausiwa November 30,pm. Skip to content The Shannon and Weaver Model of Communication is a mathematical theory of communication that argues that human communication can be broken down into 6 key concepts: sender, encoder, channel, noise, decoder, and receiver. Historical background Interest in Bet365 Italia concept Darts Deutsch information grew directly from the creation of the telegraph and telephone. How fast can we download images from the servers of the Internet to our computers? Quantum information science is a young field, its underpinnings still being laid by a large number of researchers [see "Rules for a Motorsport Live Quantum World," by Michael A. It is best known for its ability to explain how messages can be mixed up Aquablitz misinterpreted in the process between sending and receiving Www.Popen.De message. A basic property of this form of conditional entropy is that:. These differences in communication style is what has made communication better through digital coding. Receiver: The receiver is the second person Shannon Information Theory the conversation, who the sender is talking to. Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of rubrics throughout the world Poker.De the past half-century or more: adaptive systemsanticipatory systemsartificial intelligencecomplex systemscomplexity sciencecyberneticsinformaticsmachine learningalong with systems sciences of many descriptions. This is Claude Shannon, an American mathematician and electronic engineer who is now considered the "Father of Information Theory". While working at Bell Laboratories, he formulated a theory which aimed to quantify the communication of information. In Shannon's theory ‘information’ is fully determined by the probability distribution on the set of possible messages, and unrelated to the meaning, structure or content of individual messages. In many cases this is problematic, since the distribution generating outcomes may be unknown to the observer or (worse), may not exist at all 5. For example, can we answer a question like “what is the information in this book” by viewing it as an element of a set of possible books with a. A year after he founded and launched information theory, Shannon published a paper that proved that unbreakable cryptography was possible. (He did this work in , but at that time it was. Information Theory was not just a product of the work of Claude Shannon. It was the result of crucial contributions made by many distinct individuals, from a variety of backgrounds, who took his ideas and expanded upon them. Indeed the diversity and directions of their perspectives and interests shaped the direction of Information Theory. The foundations of information theory were laid in –49 by the American scientist C. Shannon. The contribution of the Soviet scientists A. N. Kolmogorov and A. Ia. Khinchin was introduced into its theoretical branches and that of V. A. Kotel’-nikov, A. A. Kharkevich, and others into the branches concerning applications.