Binary erasure channel

In coding theory and information theory, a binary erasure channel (BEC) is a communications channel model. A transmitter sends a bit (a zero or a one), and the receiver either receives the bit correctly, or with some probability receives a message that the bit was not received ("erased") .

The channel model for the binary erasure channel showing a mapping from channel input X to channel output Y (with known erasure symbol ?). The probability of erasure is

Definition

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A binary erasure channel with erasure probability   is a channel with binary input, ternary output, and probability of erasure  . That is, let   be the transmitted random variable with alphabet  . Let   be the received variable with alphabet  , where   is the erasure symbol. Then, the channel is characterized by the conditional probabilities:[1]

 

Capacity

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The channel capacity of a BEC is  , attained with a uniform distribution for   (i.e. half of the inputs should be 0 and half should be 1).[2]

If the sender is notified when a bit is erased, they can repeatedly transmit each bit until it is correctly received, attaining the capacity  . However, by the noisy-channel coding theorem, the capacity of   can be obtained even without such feedback.[3]

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If bits are flipped rather than erased, the channel is a binary symmetric channel (BSC), which has capacity   (for the binary entropy function  ), which is less than the capacity of the BEC for  .[4][5] If bits are erased but the receiver is not notified (i.e. does not receive the output  ) then the channel is a deletion channel, and its capacity is an open problem.[6]

History

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The BEC was introduced by Peter Elias of MIT in 1955 as a toy example.[citation needed]

See also

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Notes

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  1. ^ MacKay (2003), p. 148.
  2. ^ a b MacKay (2003), p. 158.
  3. ^ Cover & Thomas (1991), p. 189.
  4. ^ Cover & Thomas (1991), p. 187.
  5. ^ MacKay (2003), p. 15.
  6. ^ Mitzenmacher (2009), p. 2.

References

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  • Cover, Thomas M.; Thomas, Joy A. (1991). Elements of Information Theory. Hoboken, New Jersey: Wiley. ISBN 978-0-471-24195-9.
  • MacKay, David J.C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press. ISBN 0-521-64298-1.
  • Mitzenmacher, Michael (2009), "A survey of results for deletion channels and related synchronization channels", Probability Surveys, 6: 1–33, doi:10.1214/08-PS141, MR 2525669