Binary erasure channel
[Image:Binaryerasurechannel.png|thumb|right|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 ]
In coding theory and information theory, a binary erasure channel is a communications channel model. A transmitter sends a bit, and the receiver either receives the bit correctly, or with some probability receives a message that the bit was not received .
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:Capacity
The channel capacity of a BEC is, attained with a uniform distribution for .Observe that, for the binary entropy function,
as is known from y unless, which has probability.
By definition, so
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.