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

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

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]

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

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

See also

Notes

References

  • 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
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