Binary erasure channel

A binary erasure channel with erasure probability ${\displaystyle P_{e}}$  is a channel with binary input, ternary output, and probability of erasure ${\displaystyle P_{e}}$ . That is, let ${\displaystyle X}$  be the transmitted random variable with alphabet ${\displaystyle \{0,1\}}$ . Let ${\displaystyle Y}$  be the received variable with alphabet ${\displaystyle \{0,1,{\text{e}}\}}$ , where ${\displaystyle {\text{e}}}$  is the erasure symbol. Then, the channel is characterized by the conditional probabilities:^[1]

${\displaystyle {\begin{aligned}\operatorname {Pr} [Y=0|X=0]&=1-P_{e}\\\operatorname {Pr} [Y=0|X=1]&=0\\\operatorname {Pr} [Y=1|X=0]&=0\\\operatorname {Pr} [Y=1|X=1]&=1-P_{e}\\\operatorname {Pr} [Y=e|X=0]&=P_{e}\\\operatorname {Pr} [Y=e|X=1]&=P_{e}\end{aligned}}}$ 

The channel capacity of a BEC is ${\displaystyle 1-P_{e}}$ , attained with a uniform distribution for ${\displaystyle X}$  (i.e. half of the inputs should be 0 and half should be 1).^[2]

Proof[2]

By symmetry of the input values, the optimal input distribution is ${\displaystyle X\sim \mathrm {Bernoulli} \left({\frac {1}{2}}\right)}$ . The channel capacity is: ${\displaystyle \operatorname {I} (X;Y)=\operatorname {H} (X)-\operatorname {H} (X|Y)}$  Observe that, for the binary entropy function ${\displaystyle \operatorname {H} _{\text{b}}}$  (which has value 1 for input ${\displaystyle {\frac {1}{2}}}$ ), ${\displaystyle \operatorname {H} (X|Y)=\sum _{y}P(y)\operatorname {H} (X|y)=P_{e}\operatorname {H} _{\text{b}}\left({\frac {1}{2}}\right)=P_{e}}$  as ${\displaystyle X}$  is known from (and equal to) y unless ${\displaystyle y=e}$ , which has probability ${\displaystyle P_{e}}$ . By definition ${\displaystyle \operatorname {H} (X)=\operatorname {H} _{\text{b}}\left({\frac {1}{2}}\right)=1}$ , so ${\displaystyle \operatorname {I} (X;Y)=1-P_{e}}$ .

If the sender is notified when a bit is erased, they can repeatedly transmit each bit until it is correctly received, attaining the capacity ${\displaystyle 1-P_{e}}$ . However, by the noisy-channel coding theorem, the capacity of ${\displaystyle 1-P_{e}}$  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 ${\displaystyle 1-\operatorname {H} _{\text{b}}(P_{e})}$  (for the binary entropy function ${\displaystyle \operatorname {H} _{\text{b}}}$ ), which is less than the capacity of the BEC for ${\displaystyle 0<P_{e}<1/2}$ .^[4][5] If bits are erased but the receiver is not notified (i.e. does not receive the output ${\displaystyle e}$ ) then the channel is a deletion channel, and its capacity is an open problem.^[6]

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

• Erasure code
• Packet erasure channel

• 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