The Blackwell channel is a

deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...

broadcast Broadcasting is the distribution of audio or video content to a dispersed audience via any electronic mass communications medium, but typically one using the electromagnetic spectrum ( radio waves), in a one-to-many model. Broadcasting began ...

channel Channel, channels, channeling, etc., may refer to: Geography * Channel (geography), in physical geography, a landform consisting of the outline (banks) of the path of a narrow body of water. Australia * Channel Country, region of outback Austral ...

model used in

coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are stud ...

and

information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...

. It was first proposed by mathematician

David Blackwell David Harold Blackwell (April 24, 1919 – July 8, 2010) was an American statistician and mathematician who made significant contributions to game theory, probability theory, information theory, and statistics. He is one of the eponyms of the ...

. In this model, a transmitter transmits one of three symbols to two receivers. For two of the symbols, both receivers receive exactly what was sent; the third symbol, however, is received differently at each of the receivers. This is one of the simplest examples of a non-trivial capacity result for a non-

stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...

channel.

Definition

The Blackwell channel is composed of one input (transmitter) and two outputs (receivers). The channel input is ternary (three symbols) and is selected from . This symbol is

to the receivers; that is, the transmitter sends one symbol simultaneously to both receivers. Each of the channel outputs is

binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...

(two symbols), labeled . Whenever a 0 is sent, both outputs receive a 0. Whenever a 1 is sent, both outputs receive a 1. When a 2 is sent, however, the first output is 0 and the second output is 1. Therefore, the symbol 2 is confused by each of the receivers in a different way. The operation of the channel is

memoryless In probability and statistics, memorylessness is a property of certain probability distributions. It usually refers to the cases when the distribution of a "waiting time" until a certain event does not depend on how much time has elapsed already ...

and completely

Capacity of the Blackwell channel

The capacity of the channel was found by S. I. Gel'fand. It is defined by the region: : 1. ''R''₁ = 1, 0 ≤ ''R''₂ ≤ : 2. ''R''₁ = H(''a''), ''R''₂ = 1 − ''a'', for ≤ ''a'' ≤ : 3. ''R''₁ + ''R''₂ = log₂ 3, log₂ 3 - ≤ ''R''₁ ≤ : 4. ''R''₁ = 1 − ''a'', ''R''₂ = H(''a''), for ≤ ''a'' ≤ : 5. 0 ≤ ''R''₁ ≤ , ''R''₂ = 1 A solution was also found by Pinkser et al. (1995).

References

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IEEE Transactions on Information Theory ''IEEE Transactions on Information Theory'' is a monthly peer-reviewed scientific journal published by the IEEE Information Theory Society. It covers information theory and the mathematics of communications. It was established in 1953 as ''IRE Tran ...

, publisher=

Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operation ...

, location=

New York City New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the L ...

New York New York most commonly refers to: * New York City, the most populous city in the United States, located in the state of New York * New York (state), a state in the northeastern United States New York may also refer to: Film and television * '' ...

, United States , volume=41 , number=6 , pages=1877–1888 , date=November 1995 , doi=10.1109/18.476313 , author1-link=Mark Semenovich Pinsker {{cite journal , title=Proof of shannon's transmission theorem for finite-state indecomposable channels , author1=L Breiman , author2-link=David Blackwell , author2=D Blackwell , author3=A J Thomasian , journal=

The Annals of Mathematical Statistics The ''Annals of Mathematical Statistics'' was a peer-reviewed statistics journal published by the Institute of Mathematical Statistics from 1930 to 1972. It was superseded by the ''Annals of Statistics'' and the ''Annals of Probability''. In 1938, ...

, publisher=

Institute of Mathematical Statistics The Institute of Mathematical Statistics is an international professional and scholarly society devoted to the development, dissemination, and application of statistics and probability. The Institute currently has about 4,000 members in all parts o ...

, location=United States , volume=29 , issue=4 , pages=1209–2220 , year=1958 , doi=10.1214/aoms/1177706452 , author1-link=Leo Breiman , doi-access=free {{cite journal , title=Capacity of one broadcast channel , author=S I Gel'fand , journal=Problemy Peredachi Informatsii , publisher=

Russian Academy of Sciences The Russian Academy of Sciences (RAS; russian: Росси́йская акаде́мия нау́к (РАН) ''Rossíyskaya akadémiya naúk'') consists of the national academy of Russia; a network of scientific research institutes from across t ...

, Branch of Informatics, Computer Equipment and Automatization , location=

Moscow Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 million ...

, Russia , volume=13 , number=3 , pages=106–108 , year=1977 {{cite journal , title=A survey of multi-way channels in information theory: 1961-1976 , author=E van der Meulen , journal=

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, United States , volume=23 , issue=1 , pages=1–37 , year=1977 , doi=10.1109/tit.1977.1055652 Coding theory