## Entropie (Informationstheorie)

provides the first comprehensive treatment of the theory of I-Measure, network coding theory, Shannon and non-Shannon type information inequalities, and a. Shannon's channel coding theorem; Random coding and error exponent; MAP and ML decoding; Bounds; Channels and capacities: Gaussian channel, fading. Die Informationstheorie ist eine mathematische Theorie aus dem Bereich der Claude E. Shannon: A mathematical theory of communication. Bell System Tech.## Shannon Information Theory Navigation menu Video

Introduction to Complexity: Shannon Information Part 1 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. Claude Shannon first proposed the information theory in The goal was to find the fundamental limits of communication operations and signal processing through an operation like data compression. It is a theory that has been extrapolated into thermal physics, quantum computing, linguistics, and even plagiarism detection. 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. Information theory studies the quantification, storage, and communication of information. It was originally proposed by Claude Shannon in to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper titled "A Mathematical Theory of Communication". The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in inform. Dies führt zum Konzept der Quanteninformation. Important properties of codes and fundamental decoding strategies will be explained. Mahjjong Faz Informationskonzepte versagen teilweise in quantenmechanischen Systemen. Ansichten Lesen Bearbeiten Quelltext Marcel Lüske Versionsgeschichte.In the American inventor Samuel F. Morse built a telegraph line between Washington, D. Morse encountered many electrical problems when he sent signals through buried transmission lines, but inexplicably he encountered fewer problems when the lines were suspended on poles.

This attracted the attention of many distinguished physicists, most notably the Scotsman William Thomson Baron Kelvin. Much of their work was done using Fourier analysis , a technique described later in this article, but in all of these cases the analysis was dedicated to solving the practical engineering problems of communication systems.

This view is in sharp contrast with the common conception of information, in which meaning has an essential role. Shannon also realized that the amount of knowledge conveyed by a signal is not directly related to the size of the message.

Similarly, a long, complete message in perfect French would convey little useful knowledge to someone who could understand only English. Shannon thus wisely realized that a useful theory of information would first have to concentrate on the problems associated with sending and receiving messages, and it would have to leave questions involving any intrinsic meaning of a message—known as the semantic problem—for later investigators.

Clearly, if the technical problem could not be solved—that is, if a message could not be transmitted correctly—then the semantic problem was not likely ever to be solved satisfactorily.

Solving the technical problem was therefore the first step in developing a reliable communication system. The Shannon model was designed originally to explain communication through means such as telephone and computers which encode our words using codes like binary digits or radio waves.

However, the encoder can also be a person that turns an idea into spoken words, written words, or sign language to communicate an idea to someone.

Examples: The encoder might be a telephone, which converts our voice into binary 1s and 0s to be sent down the telephone lines the channel.

Another encode might be a radio station, which converts voice into waves to be sent via radio to someone. The channel of communication is the infrastructure that gets information from the sender and transmitter through to the decoder and receiver.

Examples: A person sending an email is using the world wide web internet as a medium. A person talking on a landline phone is using cables and electrical wires as their channel.

There are two types of noise: internal and external. Internal noise happens when a sender makes a mistake encoding a message or a receiver makes a mistake decoding the message.

External noise happens when something external not in the control of sender or receiver impedes the message.

So, external noise happens:. One of the key goals for people who use this theory is to identify the causes of noise and try to minimize them to improve the quality of the message.

Examples: Examples of external noise may include the crackling of a poorly tuned radio, a lost letter in the post, an interruption in a television broadcast, or a failed internet connection.

Decoding is the exact opposite of encoding. Shannon and Weaver made this model in reference to communication that happens through devices like telephones.

So, in this model, there usually needs to be a device that decodes a message from binary digits or waves back into a format that can be understood by the receiver.

For example, you might need to decode a secret message, turn written words into something that makes sense in your mind by reading them out loud, or you may need to interpret decode the meaning behind a picture that was sent to you.

Examples: Decoders can include computers that turn binary packets of 1s and 0s into pixels on a screen that make words, a telephone that turns signals such as digits or waves back into sounds, and cell phones that also turn bits of data into readable and listenable messages.

Examples: Examples of a receiver might be: the person on the other end of a telephone, the person reading an email you sent them, an automated payments system online that has received credit card details for payment, etc.

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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 : [10].

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. It can be subdivided into source coding theory and channel coding theory.

Using a statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the information entropy of the source.

This division of coding theory into compression and transmission is justified by the information transmission theorems, or source—channel separation theorems that justify the use of bits as the universal currency for information in many contexts.

However, these theorems only hold in the situation where one transmitting user wishes to communicate to one receiving user.

In scenarios with more than one transmitter the multiple-access channel , more than one receiver the broadcast channel or intermediary "helpers" the relay channel , or more general networks , compression followed by transmission may no longer be optimal.

Network information theory refers to these multi-agent communication models. Any process that generates successive messages can be considered a source of information.

A memoryless source is one in which each message is an independent identically distributed random variable , whereas the properties of ergodicity and stationarity impose less restrictive constraints.

All such sources are stochastic. These terms are well studied in their own right outside information theory.

Information rate is the average entropy per symbol. For memoryless sources, this is merely the entropy of each symbol, while, in the case of a stationary stochastic process, it is.

For the more general case of a process that is not necessarily stationary, the average rate is. For stationary sources, these two expressions give the same result.

It is common in information theory to speak of the "rate" or "entropy" of a language. This is appropriate, for example, when the source of information is English prose.

The rate of a source of information is related to its redundancy and how well it can be compressed, the subject of source coding.

Communications over a channel—such as an ethernet cable—is the primary motivation of information theory. However, such channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality.

Consider the communications process over a discrete channel. A simple model of the process is shown below:. Here X represents the space of messages transmitted, and Y the space of messages received during a unit time over our channel.

Let p y x be the conditional probability distribution function of Y given X. We will consider p y x to be an inherent fixed property of our communications channel representing the nature of the noise of our channel.

Then the joint distribution of X and Y is completely determined by our channel and by our choice of f x , the marginal distribution of messages we choose to send over the channel.

Under these constraints, we would like to maximize the rate of information, or the signal , we can communicate over the channel. The appropriate measure for this is the mutual information, and this maximum mutual information is called the channel capacity and is given by:.

This capacity has the following property related to communicating at information rate R where R is usually bits per symbol. Channel coding is concerned with finding such nearly optimal codes that can be used to transmit data over a noisy channel with a small coding error at a rate near the channel capacity.

Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban , was used in the Ultra project, breaking the German Enigma machine code and hastening the end of World War II in Europe.

Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext , it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability.

Information theory leads us to believe it is much more difficult to keep secrets than it might first appear.

A brute force attack can break systems based on asymmetric key algorithms or on most commonly used methods of symmetric key algorithms sometimes called secret key algorithms , such as block ciphers.

The security of all such methods currently comes from the assumption that no known attack can break them in a practical amount of time. Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute force attacks.

In such cases, the positive conditional mutual information between the plaintext and ciphertext conditioned on the key can ensure proper transmission, while the unconditional mutual information between the plaintext and ciphertext remains zero, resulting in absolutely secure communications.

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Erweiterte Suche. **Shannon Information Theory**external noises only which affect the messages or signals from external sources. Contents hide. This perturbation is called noise. Hi Jeff!

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