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Signal transmission using electronic signal processing. Transducers convert signals from other physical waveforms to electrical current or voltage waveforms, which then are processed, transmitted as electromagnetic waves, received and converted by another transducer to final form.

Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time. Signals of interest can include sound, images, time-varying measurement values and sensor data, for example biological data such as electrocardiograms, control system signals, telecommunication transmission signals, and many others. Signals are analog or digital electrical representations of time-varying or spatial-varying physical quantities.

Typical operations and applications[link]

Processing of signals includes the following operations and algorithms with application examples:^[1]

• Filtering (for example in tone controls and equalizers)
• Smoothing, deblurring (for example in image enhancement)
• Adaptive filtering (for example for echo-cancellation in a conference telephone, or denoising for aircraft identification by radar)
• Spectrum analysis (for example in magnetic resonance imaging, tomographic reconstruction and OFDM modulation)
• Digitization, reconstruction and compression (for example, image compression, sound coding and other source coding)
• Storage (in digital delay lines and reverb)
• Modulation (in modems and radio receivers and transmitters)
• Wavetable synthesis (in modems and music synthesizers)
• Feature extraction (for example speech-to-text conversion and optical character recognition)
• Pattern recognition and correlation analysis (in spread spectrum receivers and computer vision)
• Prediction

• A variety of other operations

In communication systems, signal processing may occur at OSI layer 1, the Physical Layer (modulation, equalization, multiplexing, etc.) in the seven layer OSI model, as well as at OSI layer 6, the Presentation Layer (source coding, including analog-to-digital conversion and data compression).

History[link]

According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the "digitalization" or digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s.^[2]

Mathematical topics embraced by signal processing[link]

• Linear signals and systems, and transform theory
• System identification and classification
• Calculus
• Differential equations
• Vector spaces and Linear algebra
• Functional analysis
• Probability and stochastic processes
• Detection theory
• Estimation theory
• Optimization
• Programming
• Numerical methods
• Iterative methods

Categories of signal processing[link]

Analog signal processing[link]

Analog signal processing is for signals that have not been digitized, as in legacy radio, telephone, radar, and television systems. This involves linear electronic circuits such as passive filters, active filters, additive mixers, integrators and delay lines. It also involves non-linear circuits such as compandors, multiplicators (frequency mixers and voltage-controlled amplifiers), voltage-controlled filters, voltage-controlled oscillators and phase-locked loops.

Discrete time signal processing[link]

Discrete time signal processing is for sampled signals that are considered as defined only at discrete points in time, and as such are quantized in time, but not in magnitude.

Analog discrete-time signal processing is a technology based on electronic devices such as sample and hold circuits, analog time-division multiplexers, analog delay lines and analog feedback shift registers. This technology was a predecessor of digital signal processing (see below), and is still used in advanced processing of gigahertz signals.

The concept of discrete-time signal processing also refers to a theoretical discipline that establishes a mathematical basis for digital signal processing, without taking quantization error into consideration.

Digital signal processing[link]

Digital signal processing is the processing of digitised discrete time sampled signals. Processing is done by general-purpose computers or by digital circuits such as ASICs, field-programmable gate arrays or specialized digital signal processors (DSP chips). Typical arithmetical operations include fixed-point and floating-point, real-valued and complex-valued, multiplication and addition. Other typical operations supported by the hardware are circular buffers and look-up tables. Examples of algorithms are the Fast Fourier transform (FFT), finite impulse response (FIR) filter, Infinite impulse response (IIR) filter, and adaptive filters such as the Wiener and Kalman filters.

Fields of signal processing[link]

• Statistical signal processing – analyzing and extracting information from signals and noise based on their stochastic properties
• Audio signal processing – for electrical signals representing sound, such as speech or music
• Speech signal processing – for processing and interpreting spoken words
• Image processing – in digital cameras, computers, and various imaging systems
• Video processing – for interpreting moving pictures
• Array processing – for processing signals from arrays of sensors
• Time-frequency analysis – for processing non-stationary signals^[3]
• Filtering – used in many fields to process signals
• Seismic signal processing
• Data mining

See also[link]

• Signal subspace

Notes and references[link]

External links[link]

• Signal Processing for Communications — free online textbook by Paolo Prandoni and Martin Vetterli (2008)
• Scientists and Engineers Guide to Digital Signal Processing — free online textbook by Stephen Smith
• Speech Enhancement for Android — Signal processing on speech, an educational platform for Android

Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Since its inception it has broadened to find applications in many other areas, including statistical inference, natural language processing, cryptography generally, networks other than communication networks — as in neurobiology,^[1] the evolution^[2] and function^[3] of molecular codes, model selection^[4] in ecology, thermal physics,^[5] quantum computing, plagiarism detection^[6] and other forms of data analysis.^[7]

A key measure of information is known as entropy, which is usually expressed by the average number of bits needed to store or communicate one symbol in a message. Entropy quantifies the uncertainty involved in predicting the value of a random variable. For example, specifying the outcome of a fair coin flip (two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (six equally likely outcomes).

Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossy data compression (e.g. MP3s and JPGs), and channel coding (e.g. for Digital Subscriber Line (DSL)). The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields. Important sub-fields of information theory are source coding, channel coding, algorithmic complexity theory, algorithmic information theory, information-theoretic security, and measures of information.

Overview[link]

The main concepts of information theory can be grasped by considering the most widespread means of human communication: language. Two important aspects of a concise language are as follows: First, the most common words (e.g., "a", "the", "I") should be shorter than less common words (e.g., "roundabout", "generation", "mediocre",) so that sentences will not be too long. Such a tradeoff in word length is analogous to data compression and is the essential aspect of source coding. Second, if part of a sentence is unheard or misheard due to noise — e.g., a passing car — the listener should still be able to glean the meaning of the underlying message. Such robustness is as essential for an electronic communication system as it is for a language; properly building such robustness into communications is done by channel coding. Source coding and channel coding are the fundamental concerns of information theory.

Note that these concerns have nothing to do with the importance of messages. For example, a platitude such as "Thank you; come again" takes about as long to say or write as the urgent plea, "Call an ambulance!" while the latter may be more important and more meaningful in many contexts. Information theory, however, does not consider message importance or meaning, as these are matters of the quality of data rather than the quantity and readability of data, the latter of which is determined solely by probabilities.

Information theory is generally considered to have been founded in 1948 by Claude Shannon in his seminal work, "A Mathematical Theory of Communication". The central paradigm of classical information theory is the engineering problem of the transmission of information over a noisy channel. The most fundamental results of this theory are Shannon's source coding theorem, which establishes that, on average, the number of bits needed to represent the result of an uncertain event is given by its entropy; and Shannon's noisy-channel coding theorem, which states that reliable communication is possible over noisy channels provided that the rate of communication is below a certain threshold, called the channel capacity. The channel capacity can be approached in practice by using appropriate encoding and decoding systems.

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 over the past half century or more: adaptive systems, anticipatory systems, artificial intelligence, complex systems, complexity science, cybernetics, informatics, machine learning, along with systems sciences of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory.

Coding theory is concerned with finding explicit methods, called codes, of increasing the efficiency and reducing the net error rate of data communication over a noisy channel to near the limit that Shannon proved is the maximum possible for that channel. These codes can be roughly subdivided into data compression (source coding) and error-correction (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms (both codes and ciphers). Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis. See the article ban (information) for a historical application.

Information theory is also used in information retrieval, intelligence gathering, gambling, statistics, and even in musical composition.

Historical background[link]

The landmark event that established the discipline of information theory, and brought it to immediate worldwide attention, was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948.

Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation Failed to parse (Missing texvc executable; please see math/README to configure.): W = K \log m , where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant. Ralph Hartley's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish one sequence of symbols from any other, thus quantifying information as Failed to parse (Missing texvc executable; please see math/README to configure.): H = \log S^n = n \log S , where S was the number of possible symbols, and n the number of symbols in a transmission. The natural unit of information was therefore the decimal digit, much later renamed the hartley in his honour as a unit or scale or measure of information. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers.

Much of the mathematics behind information theory with events of different probabilities was developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored in Entropy in thermodynamics and information theory.

In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion that

"The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."

With it came the ideas of

• the information entropy and redundancy of a source, and its relevance through the source coding theorem;
• the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem;
• the practical result of the Shannon–Hartley law for the channel capacity of a Gaussian channel; as well as
• the bit—a new way of seeing the most fundamental unit of information.

Quantities of information[link]

Information theory is based on probability theory and statistics. The most important quantities of information are entropy, the information in a random variable, and mutual information, the amount of information in common between two random variables. The former quantity indicates how easily message data can be compressed while the latter can be used to find the communication rate across a channel.

The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. The most 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 hartley, which is based on the common logarithm.

In what follows, an expression of the form Failed to parse (Missing texvc executable; please see math/README to configure.): p \log p \,

is considered by convention to be equal to zero whenever Failed to parse (Missing texvc executable; please see math/README to configure.): p=0.
 This is justified because Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{p \rightarrow 0+} p \log p = 0
for any logarithmic base.

Entropy[link]

Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function, Failed to parse (Missing texvc executable; please see math/README to configure.): H_\mbox{b}(p) . The entropy is maximized at 1 bit per trial when the two possible outcomes are equally probable, as in an unbiased coin toss.

The entropy, Failed to parse (Missing texvc executable; please see math/README to configure.): H , of a discrete random variable Failed to parse (Missing texvc executable; please see math/README to configure.): X

is a measure of the amount of uncertainty associated with the value of Failed to parse (Missing texvc executable; please see math/README to configure.): X

.

Suppose one transmits 1000 bits (0s and 1s). If these bits are known ahead of transmission (to be a certain value with absolute probability), logic dictates that no information has been transmitted. If, however, each is equally and independently likely to be 0 or 1, 1000 bits (in the information theoretic sense) have been transmitted. Between these two extremes, information can be quantified as follows. If Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{X}

is the set of all messages Failed to parse (Missing texvc executable; please see math/README to configure.): \{x_1, ..., x_n\}
that Failed to parse (Missing texvc executable; please see math/README to configure.): X
could be, and Failed to parse (Missing texvc executable; please see math/README to configure.): p(x)
is the probability of Failed to parse (Missing texvc executable; please see math/README to configure.): x
given some Failed to parse (Missing texvc executable; please see math/README to configure.): x \in \mathbb X

, then the entropy of Failed to parse (Missing texvc executable; please see math/README to configure.): X

is defined:[8]

Failed to parse (Missing texvc executable; please see math/README to configure.): H(X) = \mathbb{E}_{X} [I(x)] = -\sum_{x \in \mathbb{X}} p(x) \log p(x).

(Here, Failed to parse (Missing texvc executable; please see math/README to configure.): I(x)

is the self-information, which is the entropy contribution of an individual message, and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E}_{X}
is the expected value.) An important property of entropy is that it is maximized when all the messages in the message space are equiprobable Failed to parse (Missing texvc executable; please see math/README to configure.): p(x)=1/n

,—i.e., most unpredictable—in which case Failed to parse (Missing texvc executable; please see math/README to configure.): H(X)=\log n .

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): H_\mbox{b}(p) = - p \log_2 p - (1-p)\log_2 (1-p).\,

Joint entropy[link]

The joint entropy of two discrete random variables Failed to parse (Missing texvc executable; please see math/README to configure.): X

and Failed to parse (Missing texvc executable; please see math/README to configure.): Y
is merely the entropy of their pairing: Failed to parse (Missing texvc executable; please see math/README to configure.): (X, Y)

. This implies that if Failed to parse (Missing texvc executable; please see math/README to configure.): X

and Failed to parse (Missing texvc executable; please see math/README to configure.): Y
are independent, then their joint entropy is the sum of their individual entropies.

For example, if Failed to parse (Missing texvc executable; please see math/README to configure.): (X,Y)

represents the position of a chess piece — Failed to parse (Missing texvc executable; please see math/README to configure.): X
the row and Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

Failed to parse (Missing texvc executable; please see math/README to configure.): H(X, Y) = \mathbb{E}_{X,Y} [-\log p(x,y)] = - \sum_{x, y} p(x, y) \log p(x, y) \,

Despite similar notation, joint entropy should not be confused with cross entropy.

Conditional entropy (equivocation)[link]

The conditional entropy or conditional uncertainty of Failed to parse (Missing texvc executable; please see math/README to configure.): X

given random variable Failed to parse (Missing texvc executable; please see math/README to configure.): Y
(also called the equivocation of Failed to parse (Missing texvc executable; please see math/README to configure.): X
about Failed to parse (Missing texvc executable; please see math/README to configure.): Y

) is the average conditional entropy over Failed to parse (Missing texvc executable; please see math/README to configure.): Y

^[9]

Failed to parse (Missing texvc executable; please see math/README to configure.): H(X|Y) = \mathbb E_Y [H(X|y)] = -\sum_{y \in Y} p(y) \sum_{x \in X} p(x|y) \log p(x|y) = -\sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(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:

Failed to parse (Missing texvc executable; please see math/README to configure.): H(X|Y) = H(X,Y) - H(Y) .\,

Mutual information (transinformation)[link]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): X

relative to Failed to parse (Missing texvc executable; please see math/README to configure.): Y
is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): I(X;Y) = \mathbb{E}_{X,Y} [SI(x,y)] = \sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(x)\, p(y)}

where Failed to parse (Missing texvc executable; please see math/README to configure.): SI

(Specific mutual Information) is the pointwise mutual information.

A basic property of the mutual information is that

Failed to parse (Missing texvc executable; please see math/README to configure.): I(X;Y) = H(X) - H(X|Y).\,

That is, knowing Y, we can save an average of Failed to parse (Missing texvc executable; please see math/README to configure.): I(X; Y)

bits in encoding X compared to not knowing Y.

Mutual information is symmetric:

Failed to parse (Missing texvc executable; please see math/README to configure.): I(X;Y) = I(Y;X) = H(X) + H(Y) - H(X,Y).\,

Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) of the posterior probability distribution of X given the value of Y to the prior distribution on X:

Failed to parse (Missing texvc executable; please see math/README to configure.): I(X;Y) = \mathbb E_{p(y)} [D_{\mathrm{KL}}( p(X|Y=y) \| p(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:

Failed to parse (Missing texvc executable; please see math/README to configure.): I(X; Y) = D_{\mathrm{KL}}(p(X,Y) \| p(X)p(Y)).

Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ² test: mutual information can be considered a statistic for assessing independence between a pair of variables, and has a well-specified asymptotic distribution.

Kullback–Leibler divergence (information gain)[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): D_{\mathrm{KL}}(p(X) \| q(X)) = \sum_{x \in X} -p(x) \log {q(x)} \, - \, \left( -p(x) \log {p(x)}\right) = \sum_{x \in X} p(x) \log \frac{p(x)}{q(x)}.

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).

Kullback–Leibler divergence of a prior from the truth[link]

Another interpretation of KL divergence is this: 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 the 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's expected to make him.

Other quantities[link]

Other important information theoretic quantities include Rényi entropy (a generalization of entropy), differential entropy (a generalization of quantities of information to continuous distributions), and the conditional mutual information.

Coding theory[link]

A picture showing scratches on the readable surface of a CD-R. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using error detection and correction.

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.

• Data compression (source coding): There are two formulations for the compression problem:

1. lossless data compression: the data must be reconstructed exactly;
2. lossy data compression: allocates bits needed to reconstruct the data, within a specified fidelity level measured by a distortion function. This subset of Information theory is called rate–distortion theory.

• Error-correcting codes (channel coding): While data compression removes as much redundancy as possible, an error correcting code adds just the right kind of redundancy (i.e., error correction) needed to transmit the data efficiently and faithfully across a noisy channel.

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.

Source theory[link]

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

Rate[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): r = \lim_{n \to \infty} H(X_n|X_{n-1},X_{n-2},X_{n-3}, \ldots);

that is, the conditional entropy of a symbol given all the previous symbols generated. For the more general case of a process that is not necessarily stationary, the average rate is

Failed to parse (Missing texvc executable; please see math/README to configure.): r = \lim_{n \to \infty} \frac{1}{n} H(X_1, X_2, \dots X_n);

that is, the limit of the joint entropy per symbol. For stationary sources, these two expressions give the same result.^[10]

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.

Channel capacity[link]

Communications over a channel—such as an ethernet cable—is the primary motivation of information theory. As anyone who's ever used a telephone (mobile or landline) knows, 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. How much information can one hope to communicate over a noisy (or otherwise imperfect) channel?

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 Failed to parse (Missing texvc executable; please see math/README to configure.): p(y|x)

be the conditional probability distribution function of Y given X. We will consider Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): C = \max_{f} I(X;Y).\!

This capacity has the following property related to communicating at information rate R (where R is usually bits per symbol). For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε; that is, it is always possible to transmit with arbitrarily small block error. In addition, for any rate R > C, it is impossible to transmit with arbitrarily small block error.

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.

Capacity of particular channel models[link]

• A continuous-time analog communications channel subject to Gaussian noise — see Shannon–Hartley theorem.

• A binary symmetric channel (BSC) with crossover probability p is a binary input, binary output channel that flips the input bit with probability p. The BSC has a capacity of Failed to parse (Missing texvc executable; please see math/README to configure.): 1 - H_\mbox{b}(p)

bits per channel use, where Failed to parse (Missing texvc executable; please see math/README to configure.): H_\mbox{b}
is the binary entropy function to the base 2 logarithm:

• A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output channel. The possible channel outputs are 0, 1, and a third symbol 'e' called an erasure. The erasure represents complete loss of information about an input bit. The capacity of the BEC is 1 - p bits per channel use.

Applications to other fields[link]

Intelligence uses and secrecy applications[link]

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 WWII 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. In other words, an eavesdropper would not be able to improve his or her guess of the plaintext by gaining knowledge of the ciphertext but not of the key. However, as in any other cryptographic system, care must be used to correctly apply even information-theoretically secure methods; the Venona project was able to crack the one-time pads of the Soviet Union due to their improper reuse of key material.

Pseudorandom number generation[link]

Pseudorandom number generators are widely available in computer language libraries and application programs. They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software. A class of improved random number generators is termed cryptographically secure pseudorandom number generators, but even they require external to the software random seeds to work as intended. These can be obtained via extractors, if done carefully. The measure of sufficient randomness in extractors is min-entropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluating randomness in cryptographic systems. Although related, the distinctions among these measures mean that a random variable with high Shannon entropy is not necessarily satisfactory for use in an extractor and so for cryptography uses.

Seismic exploration[link]

One early commercial application of information theory was in the field seismic oil exploration. Work in this field made it possible to strip off and separate the unwanted noise from the desired seismic signal. Information theory and digital signal processing offer a major improvement of resolution and image clarity over previous analog methods.^[11]

Miscellaneous applications[link]

Information theory also has applications in gambling and investing, black holes, bioinformatics, and music.

See also[link]

Mathematics portal

• Communication theory
• List of important publications
• Philosophy of information

Applications[link]

• Cryptanalysis
• Cryptography
• Cybernetics
• Entropy in thermodynamics and information theory
• Gambling
• Intelligence (information gathering)
• Seismic exploration

History[link]

• Hartley, R.V.L.
• History of information theory
• Shannon, C.E.
• Timeline of information theory
• Yockey, H.P.

Theory[link]

Concepts[link]

References[link]

The classic work[link]

• Shannon, C.E. (1948), "A Mathematical Theory of Communication", Bell System Technical Journal, 27, pp. 379–423 & 623–656, July & October, 1948. PDF.
  Notes and other formats.
• R.V.L. Hartley, "Transmission of Information", Bell System Technical Journal, July 1928
• Andrey Kolmogorov (1968), "Three approaches to the quantitative definition of information" in International Journal of Computer Mathematics.

Other journal articles[link]

• J. L. Kelly, Jr., Saratoga.ny.us, "A New Interpretation of Information Rate" Bell System Technical Journal, Vol. 35, July 1956, pp. 917–26.
• R. Landauer, IEEE.org, "Information is Physical" Proc. Workshop on Physics and Computation PhysComp'92 (IEEE Comp. Sci.Press, Los Alamitos, 1993) pp. 1–4.
• R. Landauer, IBM.com, "Irreversibility and Heat Generation in the Computing Process" IBM J. Res. Develop. Vol. 5, No. 3, 1961

Textbooks on information theory[link]

• Claude E. Shannon, Warren Weaver. The Mathematical Theory of Communication. Univ of Illinois Press, 1949. ISBN 0-252-72548-4
• Robert Gallager. Information Theory and Reliable Communication. New York: John Wiley and Sons, 1968. ISBN 0-471-29048-3
• Robert B. Ash. Information Theory. New York: Interscience, 1965. ISBN 0-470-03445-9. New York: Dover 1990. ISBN 0-486-66521-6
• Thomas M. Cover, Joy A. Thomas. Elements of information theory, 1st Edition. New York: Wiley-Interscience, 1991. ISBN 0-471-06259-6.

2nd Edition. New York: Wiley-Interscience, 2006. ISBN 0-471-24195-4.

• Imre Csiszar, Janos Korner. Information Theory: Coding Theorems for Discrete Memoryless Systems Akademiai Kiado: 2nd edition, 1997. ISBN 963-05-7440-3
• Raymond W. Yeung. A First Course in Information Theory Kluwer Academic/Plenum Publishers, 2002. ISBN 0-306-46791-7
• David J. C. MacKay. Information Theory, Inference, and Learning Algorithms Cambridge: Cambridge University Press, 2003. ISBN 0-521-64298-1
• Raymond W. Yeung. Information Theory and Network Coding Springer 2008, 2002. ISBN 978-0-387-79233-0
• Stanford Goldman. Information Theory. New York: Prentice Hall, 1953. New York: Dover 1968 ISBN 0-486-62209-6, 2005 ISBN 0-486-44271-3
• Fazlollah Reza. An Introduction to Information Theory. New York: McGraw-Hill 1961. New York: Dover 1994. ISBN 0-486-68210-2
• Masud Mansuripur. Introduction to Information Theory. New York: Prentice Hall, 1987. ISBN 0-13-484668-0
• Christoph Arndt: Information Measures, Information and its Description in Science and Engineering (Springer Series: Signals and Communication Technology), 2004, ISBN 978-3-540-40855-0

Other books[link]

• Leon Brillouin, Science and Information Theory, Mineola, N.Y.: Dover, [1956, 1962] 2004. ISBN 0-486-43918-6
• James Gleick, The Information: A History, a Theory, a Flood, New York: Pantheon, 2011. ISBN 978-0-375-42372-7
• A. I. Khinchin, Mathematical Foundations of Information Theory, New York: Dover, 1957. ISBN 0-486-60434-9
• H. S. Leff and A. F. Rex, Editors, Maxwell's Demon: Entropy, Information, Computing, Princeton University Press, Princeton, NJ (1990). ISBN 0-691-08727-X
• Tom Siegfried, The Bit and the Pendulum, Wiley, 2000. ISBN 0-471-32174-5
• Charles Seife, Decoding The Universe, Viking, 2006. ISBN 0-670-03441-X
• Jeremy Campbell, Grammatical Man, Touchstone/Simon & Schuster, 1982, ISBN 0-671-44062-4
• Henri Theil, Economics and Information Theory, Rand McNally & Company - Chicago, 1967.

External links[link]

• alum.mit.edu, Eprint, Schneider, T. D., "Information Theory Primer"
• ND.edu, Srinivasa, S. "A Review on Multivariate Mutual Information"
• Chem.wisc.edu, Journal of Chemical Education, Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of Entropy Increase? Nonsense!
• ITsoc.org, IEEE Information Theory Society and ITsoc.org review articles
• Cam.ac.uk, On-line textbook: "Information Theory, Inference, and Learning Algorithms" by David MacKay - giving an entertaining and thorough introduction to Shannon theory, including state-of-the-art methods from coding theory, such as arithmetic coding, low-density parity-check codes, and Turbo codes.
• UMBC.edu, Eprint, Erill, I., "A gentle introduction to information content in transcription factor binding sites"

Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electronic engineer, and cryptographer known as "the father of information theory".^[1][2]

Shannon is famous for having founded information theory with one landmark paper published in 1948. But he is also credited with founding both digital computer and digital circuit design theory in 1937, when, as a 21-year-old master's student at MIT, he wrote a thesis demonstrating that electrical application of Boolean algebra could construct and resolve any logical, numerical relationship. It has been claimed that this was the most important master's thesis of all time.^[3] Shannon contributed to the field of cryptanalysis during World War II and afterwards, including basic work on code breaking.

Biography[link]

Shannon was born in Petoskey, Michigan. His father, Claude Sr (1862–1934), a descendant of early New Jersey settlers, was a self-made businessman and for a while, Judge of Probate. His mother, Mabel Wolf Shannon (1890–1945), daughter of German immigrants, was a language teacher and for a number of years principal of Gaylord High School, Michigan. The first 16 years of Shannon's life were spent in Gaylord, Michigan, where he attended public school, graduating from Gaylord High School in 1932. Shannon showed an inclination towards mechanical things. His best subjects were science and mathematics, and at home he constructed such devices as models of planes, a radio-controlled model boat and a wireless telegraph system to a friend's house half a mile away. While growing up, he worked as a messenger for Western Union. His childhood hero was Thomas Edison, who he later learned was a distant cousin. Both were descendants of John Ogden, a colonial leader and an ancestor of many distinguished people.^[4][5]

Boolean theory[link]

In 1932 he entered the University of Michigan, where he took a course that introduced him to the works of George Boole. He graduated in 1936 with two bachelor's degrees, one in electrical engineering and one in mathematics. Later he began his graduate studies at the Massachusetts Institute of Technology (MIT), where he worked on Vannevar Bush's differential analyzer, an analog computer.^[6]

While studying the complicated ad hoc circuits of the differential analyzer, Shannon saw that Boole's concepts could be used to great utility. A paper drawn from his 1937 master's thesis, A Symbolic Analysis of Relay and Switching Circuits,^[7] was published in the 1938 issue of the Transactions of the American Institute of Electrical Engineers. It also earned Shannon the Alfred Noble American Institute of American Engineers Award in 1940. Howard Gardner, of Harvard University, called Shannon's thesis "possibly the most important, and also the most famous, master's thesis of the century."

Victor Shestakov, at Moscow State University, had proposed a theory of electric switches based on Boolean logic earlier than Shannon, in 1935, but the first publication of Shestakov's result took place in 1941, after the publication of Shannon's thesis.

In this work, Shannon proved that Boolean algebra and binary arithmetic could be used to simplify the arrangement of the electromechanical relays then used in telephone routing switches, then expanded the concept and also proved that it should be possible to use arrangements of relays to solve Boolean algebra problems. Exploiting this property of electrical switches to do logic is the basic concept that underlies all electronic digital computers. Shannon's work became the foundation of practical digital circuit design when it became widely known among the electrical engineering community during and after World War II. The theoretical rigor of Shannon's work completely replaced the ad hoc methods that had previously prevailed.

Vannevar Bush suggested that Shannon, flush with this success, work on his dissertation at Cold Spring Harbor Laboratory, funded by the Carnegie Institution headed by Bush, to develop similar mathematical relationships for Mendelian genetics, which resulted in Shannon's 1940 PhD thesis at MIT, An Algebra for Theoretical Genetics.^[8]

In 1940, Shannon became a National Research Fellow at the Institute for Advanced Study in Princeton, New Jersey. At Princeton, Shannon had the opportunity to discuss his ideas with influential scientists and mathematicians such as Hermann Weyl and John von Neumann, and even had the occasional encounter with Albert Einstein. Shannon worked freely across disciplines, and began to shape the ideas that would become information theory.^[9]

Wartime research[link]

Shannon then joined Bell Labs to work on fire-control systems and cryptography during World War II, under a contract with section D-2 (Control Systems section) of the National Defense Research Committee (NDRC).

He met his wife Betty when she was a numerical analyst at Bell Labs. They married in 1949.^[10]

For two months early in 1943, Shannon came into contact with the leading British cryptanalyst and mathematician Alan Turing. Turing had been posted to Washington to share with the US Navy's cryptanalytic service the methods used by the British Government Code and Cypher School at Bletchley Park to break the ciphers used by the German U-boats in the North Atlantic.^[11] He was also interested in the encipherment of speech and to this end spent time at Bell Labs. Shannon and Turing met at teatime in the cafeteria.^[11] Turing showed Shannon his seminal 1936 paper that defined what is now known as the "Universal Turing machine"^[12][13] which impressed him, as many of its ideas were complementary to his own.

In 1945, as the war was coming to an end, the NDRC was issuing a summary of technical reports as a last step prior to its eventual closing down. Inside the volume on fire control a special essay titled Data Smoothing and Prediction in Fire-Control Systems, coauthored by Shannon, Ralph Beebe Blackman, and Hendrik Wade Bode, formally treated the problem of smoothing the data in fire-control by analogy with "the problem of separating a signal from interfering noise in communications systems."^[14] In other words it modeled the problem in terms of data and signal processing and thus heralded the coming of the information age.

His work on cryptography was even more closely related to his later publications on communication theory.^[15] At the close of the war, he prepared a classified memorandum for Bell Telephone Labs entitled "A Mathematical Theory of Cryptography," dated September, 1945. A declassified version of this paper was subsequently published in 1949 as "Communication Theory of Secrecy Systems" in the Bell System Technical Journal. This paper incorporated many of the concepts and mathematical formulations that also appeared in his A Mathematical Theory of Communication. Shannon said that his wartime insights into communication theory and cryptography developed simultaneously and "they were so close together you couldn’t separate them".^[16] In a footnote near the beginning of the classified report, Shannon announced his intention to "develop these results ... in a forthcoming memorandum on the transmission of information."^[17]

While at Bell Labs, he proved that the one-time pad is unbreakable in his World War II research that was later published in October 1949. He also proved that any unbreakable system must have essentially the same characteristics as the one-time pad: the key must be truly random, as large as the plaintext, never reused in whole or part, and kept secret.^[18]

Postwar contributions[link]

In 1948 the promised memorandum appeared as "A Mathematical Theory of Communication", an article in two parts in the July and October issues of the Bell System Technical Journal. This work focuses on the problem of how best to encode the information a sender wants to transmit. In this fundamental work he used tools in probability theory, developed by Norbert Wiener, which were in their nascent stages of being applied to communication theory at that time. Shannon developed information entropy as a measure for the uncertainty in a message while essentially inventing the field of information theory.

The book, co-authored with Warren Weaver, The Mathematical Theory of Communication, reprints Shannon's 1948 article and Weaver's popularization of it, which is accessible to the non-specialist. Shannon's concepts were also popularized, subject to his own proofreading, in John Robinson Pierce's Symbols, Signals, and Noise.

Information theory's fundamental contribution to natural language processing and computational linguistics was further established in 1951, in his article "Prediction and Entropy of Printed English", showing upper and lower bounds of entropy on the statistics of English - giving a statistical foundation to language analysis. In addition, he proved that treating whitespace as the 27th letter of the alphabet actually lowers uncertainty in written language, providing a clear quantifiable link between cultural practice and probabilistic cognition.

Another notable paper published in 1949 is "Communication Theory of Secrecy Systems", a declassified version of his wartime work on the mathematical theory of cryptography, in which he proved that all theoretically unbreakable ciphers must have the same requirements as the one-time pad. He is also credited with the introduction of sampling theory, which is concerned with representing a continuous-time signal from a (uniform) discrete set of samples. This theory was essential in enabling telecommunications to move from analog to digital transmissions systems in the 1960s and later.

He returned to MIT to hold an endowed chair in 1956.

Hobbies and inventions[link]

Outside of his academic pursuits, Shannon was interested in juggling, unicycling, and chess. He also invented many devices, including rocket-powered flying discs, a motorized pogo stick, and a flame-throwing trumpet for a science exhibition^{[citation needed]}. One of his more humorous devices was a box kept on his desk called the "Ultimate Machine", based on an idea by Marvin Minsky. Otherwise featureless, the box possessed a single switch on its side. When the switch was flipped, the lid of the box opened and a mechanical hand reached out, flipped off the switch, then retracted back inside the box. Renewed interest in the "Ultimate Machine" has emerged on YouTube and Thingiverse. In addition he built a device that could solve the Rubik's Cube puzzle.^[4]

He is also considered the co-inventor of the first wearable computer along with Edward O. Thorp.^[19] The device was used to improve the odds when playing roulette.

Legacy and tributes[link]

Shannon came to MIT in 1956 to join its faculty and to conduct work in the Research Laboratory of Electronics (RLE). He continued to serve on the MIT faculty until 1978. To commemorate his achievements, there were celebrations of his work in 2001, and there are currently six statues of Shannon sculpted by Eugene L. Daub: one at the University of Michigan; one at MIT in the Laboratory for Information and Decision Systems; one in Gaylord, Michigan; one at the University of California, San Diego; one at Bell Labs; and another at AT&T Shannon Labs.^[20] After the breakup of the Bell system, the part of Bell Labs that remained with AT&T was named Shannon Labs in his honor.

According to Neil Sloane, an AT&T Fellow who co-edited Shannon's large collection of papers in 1993, the perspective introduced by Shannon's communication theory (now called information theory) is the foundation of the digital revolution, and every device containing a microprocessor or microcontroller is a conceptual descendant of Shannon's 1948 publication:^[21] "He's one of the great men of the century. Without him, none of the things we know today would exist. The whole digital revolution started with him."^[22]

Shannon developed Alzheimer's disease, and spent his last few years in a Massachusetts nursing home. He was survived by his wife, Mary Elizabeth Moore Shannon; a son, Andrew Moore Shannon; a daughter, Margarita Shannon; a sister, Catherine S. Kay; and two granddaughters.^[10][23]

Shannon was oblivious to the marvels of the digital revolution because his mind was ravaged by Alzheimer's disease. His wife mentioned in his obituary that had it not been for Alzheimer's "he would have been bemused" by it all.^[22]

Other work[link]

Shannon's mouse[link]

Theseus, created in 1950, was a magnetic mouse controlled by a relay circuit that enabled it to move around a maze of 25 squares. Its dimensions were the same as an average mouse.^[2] The maze configuration was flexible and it could be modified at will.^[2] The mouse was designed to search through the corridors until it found the target. Having travelled through the maze, the mouse would then be placed anywhere it had been before and because of its prior experience it could go directly to the target. If placed in unfamiliar territory, it was programmed to search until it reached a known location and then it would proceed to the target, adding the new knowledge to its memory thus learning.^[2] Shannon's mouse appears to have been the first artificial learning device of its kind.^[2]

Shannon's computer chess program[link]

In 1950 Shannon published a paper on computer chess entitled Programming a Computer for Playing Chess. It describes how a machine or computer could be made to play a reasonable game of chess. His process for having the computer decide on which move to make is a minimax procedure, based on an evaluation function of a given chess position. Shannon gave a rough example of an evaluation function in which the value of the black position was subtracted from that of the white position. Material was counted according to the usual relative chess piece relative value (1 point for a pawn, 3 points for a knight or bishop, 5 points for a rook, and 9 points for a queen).^[24] He considered some positional factors, subtracting ½ point for each doubled pawns, backward pawn, and isolated pawn. Another positional factor in the evaluation function was mobility, adding 0.1 point for each legal move available. Finally, he considered checkmate to be the capture of the king, and gave the king the artificial value of 200 points. Quoting from the paper:

The coefficients .5 and .1 are merely the writer's rough estimate. Furthermore, there are many other terms that should be included. The formula is given only for illustrative purposes. Checkmate has been artificially included here by giving the king the large value 200 (anything greater than the maximum of all other terms would do).

The evaluation function is clearly for illustrative purposes, as Shannon stated. For example, according to the function, pawns that are doubled as well as isolated would have no value at all, which is clearly unrealistic.

The Las Vegas connection: information theory and its applications to game theory[link]

Shannon and his wife Betty also used to go on weekends to Las Vegas with M.I.T. mathematician Ed Thorp,^[25] and made very successful forays in blackjack using game theory type methods co-developed with fellow Bell Labs associate, physicist John L. Kelly Jr. based on principles of information theory.^[26] They made a fortune, as detailed in the book Fortune's Formula by William Poundstone and corroborated by the writings of Elwyn Berlekamp,^[27] Kelly's research assistant in 1960 and 1962.^[3] Shannon and Thorp also applied the same theory, later known as the Kelly criterion, to the stock market with even better results.^[28] Over the decades, Kelly's scientific formula has become a part of mainstream investment theory^[29] and the most prominent users, well-known and successful billionaire investors Warren Buffett,^[30][31] Bill Gross^[32], and Jim Simons use Kelly methods. Warren Buffett met Thorp the first time in 1968. It's said that Buffett uses a form of the Kelly criterion in deciding how much money to put into various holdings. Also Elwyn Berlekamp had applied the same logical algorithm for Axcom Trading Advisors, an alternative investment management company, that he had founded. Berlekamp's company was acquired by Jim Simons and his Renaissance Technologies Corp hedge fund in 1992, whereafter its investment instruments were either subsumed into (or essentially renamed as) Renaissance's flagship Medallion Fund. But as Kelly's original paper demonstrates, the criterion is only valid when the investment or "game" is played many times over, with the same probability of winning or losing each time, and the same payout ratio.^[33]

The theory was also exploited by the famous MIT Blackjack Team, which was a group of students and ex-students from the Massachusetts Institute of Technology, Harvard Business School, Harvard University, and other leading colleges who used card-counting techniques and other sophisticated strategies to beat casinos at blackjack worldwide. The team and its successors operated successfully from 1979 through the beginning of the 21st century. Many other blackjack teams have been formed around the world with the goal of beating the casinos.

Claude Shannon's card count techniques were explained in Bringing Down the House, the best-selling book published in 2003 about the MIT Blackjack Team by Ben Mezrich. In 2008, the book was adapted into a drama film titled 21.

Shannon's maxim[link]

Shannon formulated a version of Kerckhoffs' principle as "the enemy knows the system". In this form it is known as "Shannon's maxim".

Awards and honors list[link]

See also[link]

Shannon–Fano coding Shannon–Hartley theorem Nyquist–Shannon sampling theorem Noisy channel coding theorem Rate distortion theory Information theory Channel Capacity Confusion and diffusion

Systems science portal One-time pad Shannon switching game Shannon number Claude E. Shannon Award Shannon index Shannon's source coding theorem Information entropy Shannon's expansion

References[link]

Further reading[link]

• Claude E. Shannon: A Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, 1948. [1] [2]
• Claude E. Shannon and Warren Weaver: The Mathematical Theory of Communication. The University of Illinois Press, Urbana, Illinois, 1949. ISBN 0-252-72548-4
• Rethnakaran Pulikkoonattu — Eric W. Weisstein: Mathworld biography of Shannon, Claude Elwood (1916–2001) [3]
• Claude E. Shannon: Programming a Computer for Playing Chess, Philosophical Magazine, Ser.7, Vol. 41, No. 314, March 1950. (Available online under External links below)
• David Levy: Computer Gamesmanship: Elements of Intelligent Game Design, Simon & Schuster, 1983. ISBN 0-671-49532-1
• Mindell, David A., "Automation's Finest Hour: Bell Labs and Automatic Control in World War II", IEEE Control Systems, December 1995, pp. 72–80.
• David Mindell, Jérôme Segal, Slava Gerovitch, "From Communications Engineering to Communications Science: Cybernetics and Information Theory in the United States, France, and the Soviet Union" in Walker, Mark (Ed.), Science and Ideology: A Comparative History, Routledge, London, 2003, pp. 66–95.
• Poundstone, William, Fortune's Formula, Hill & Wang, 2005, ISBN 978-0-8090-4599-0
• Gleick, James, The Information: A History, A Theory, A Flood, Pantheon, 2011, ISBN 978-0-375-42372-7

Shannon videos[link]

• Shannon's video machines
• Shannon - father of the information age
• AT&T Tech Channel's Tech Icons - Claude Shannon

External links[link]

Wikiquote has a collection of quotations related to: Claude Elwood Shannon

• C. E. Shannon, An algebra for theoretical genetics, Massachusetts Institute of Technology, Ph.D. Thesis, MIT-THESES//1940–3 (1940) Online text at MIT
• Shannon's math genealogy
• Shannon's NNDB profile
• Works by or about Claude Shannon in libraries (WorldCat catalog)
• A Mathematical Theory of Communication
• Communication Theory of Secrecy Systems
• Communication in the Presence of Noise
• Summary of Shannon's life and career
• Biographical summary from Shannon's collected papers
• Video documentary: "Claude Shannon - Father of the Information Age"
• Mathematical Theory of Claude Shannon In-depth MIT class paper on the development of Shannon's work to 1948.
• Retrospective at the University of Michigan
• Shannon's University of Michigan profile
• Notes on Computer-Generated Text
• Shannon's Juggling Theorem and Juggling Robots
• Color photos of Shannon
• Shannon's paper on computer chess, text
• Shannon's paper on computer chessPDF (175 KiB)
• Shannon's paper on computer chess, text, alternate source
• A Bibliography of His Collected Papers
• A Register of His Papers in the Library of Congress
• The Technium: The (Unspeakable) Ultimate Machine
• The Most Beautiful Machine. (aka the "Ultimate Machine") It's a communication based on the functions ON and OFF.
• Guizzo, "The Essential Message: Claude Shannon and the Making of Information Theory"
• Claude Shannon, Edward O. Thorp, Fortune's Formula
• Claude Shannon : Founding Father of Electronic Communication age,Dream 2047, December,2006, Shivaprasad Khened

Jorma J. Rissanen (born October 20, 1932 in Pielisjärvi (now in Lieksa), Finland) is an information theorist, known for inventing the arithmetic coding technique of lossless data compression, and the minimum description length principle.

An IBM researcher since 1960, Rissanen received his Ph.D. from the Helsinki University of Technology in 1965. Retired from IBM, he is now professor emeritus of Tampere University of Technology, and a fellow of Helsinki Institute for Information Technology. He was awarded the IEEE Richard W. Hamming Medal in 1993,^[1] a Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society in 1998,^[2] the Kolmogorov Medal in 2006, and the IEEE Claude E. Shannon Award in 2009. A Festschrift collection was published to honor his 75th birthday.^[3]

References[link]

External links[link]

• Dr. Rissanen at RHUL
• Rissanen's page at MDL-Research

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