Binary erasure channel
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A binary erasure channel (or BEC) is a common communications channel model used in coding theory and information theory. In this model, a transmitter wishes to send 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.
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[edit] Description
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 2 symbols, possibly even an infinite number of choices) The channel is not perfect however, and sometimes the bit gets "erased"; that is, the bit gets scrambled and the receiver has no idea what the bit was.
Note that 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.
[edit] Definition
A binary erasure channel with erasure probability p is a channel with binary input, ternary output, and probability of erasure p. 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
- Pr( Y = 0 | X = 0) = 1-p
- Pr( Y = e | X = 0) = p
- Pr( Y = 1 | X = 0) = 0
- Pr( Y = 0 | X = 1) = 0
- Pr( Y = e | X = 1) = p
- Pr( Y = 1 | X = 1) = 1-p
[edit] Capacity of the BEC
The capacity of a BEC is 1 - p.
The converse can be explained intuitively. Suppose there is an omniscient "genie" that tells the source whenever its bit gets erased. There is nothing the source can do to avoid erasure, but it can fix them when they happen. For example, the source could repeatedly transmit a bit until it gets through. There is no need for X to code, as Y will simply ignore erasures, knowing that the next successfully received bit is the one that X intended to send. Therefore, having a genie allows us to achieve a rate of 1 - p on average. We cannot hope to do any better than this.
[edit] References
- David J. C. MacKay. Information Theory, Inference, and Learning Algorithms Cambridge: Cambridge University Press, 2003. ISBN 0-521-64298-1
- Thomas M. Cover, Joy A. Thomas. Elements of information theory, 1st Edition. New York: Wiley-Interscience, 1991. ISBN 0-471-06259-6.
