Binary erasure channel

A binary erasure channel (or BEC) is a common communications channel model used in coding theory and information theory. In this model, a transmitter sends a bit (a zero or a one), and the receiver either receives the bit or it receives a message that the bit was not received ("erased"). This channel is used frequently in information theory because it is one of the simplest channels to analyze. The BEC was introduced by Peter Elias of MIT in 1955 as a toy example.

Closely related to the binary erasure channel is the packet erasure channel which shares many similar theoretical results with the binary erasure channel.

The BEC is a binary channel; that is, it can transmit only one of two symbols (usually called 0 and 1). (A non-binary channel would be capable of transmitting more than two symbols, possibly even an infinite number of choices.) The channel is not perfect and sometimes the bit gets "erased"; that is, the bit gets scrambled so the receiver has no idea what the bit was.

The BEC is, in a sense, error-free. Unlike the binary symmetric channel, when the receiver gets a bit, it is 100% certain that the bit is correct. The only confusion arises when the bit is erased.

This channel is often used by theorists because it is one of the simplest noisy channels to analyze. Many problems in communication theory can be reduced to a BEC.

A binary erasure channel with erasure probability Pe is a channel with binary input, ternary output, and probability of erasure Pe. That is, let X be the transmitted random variable with alphabet {0, 1}. Let Y be the received variable with alphabet {0, 1, e}, where e is the erasure symbol. Then, the channel is characterized by the conditional probabilities

The capacity of a BEC is 1 - Pe.

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