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 $$P_e$$ receives a message that the bit was not received ("erased").

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


 * $$\begin{align}

\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{align}$$

Capacity
The channel capacity of a BEC is $$1-P_e$$, attained with a uniform distribution for $$X$$ (i.e. half of the inputs should be 0 and half should be 1).


 * {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"

!Proof
 * By symmetry of the input values, the optimal input distribution is $$X \sim \mathrm{Bernoulli}\left(\frac{1}{2}\right)$$. The channel capacity is:
 * $$\operatorname{I}(X;Y) = \operatorname{H}(X)-\operatorname{H}(X|Y)$$
 * $$\operatorname{I}(X;Y) = \operatorname{H}(X)-\operatorname{H}(X|Y)$$

Observe that, for the binary entropy function $$\operatorname{H}_\text{b}$$ (which has value 1 for input $$\frac{1}{2}$$),
 * $$\operatorname{H}(X|Y)=\sum_y P(y)\operatorname{H}(X|y)=P_e \operatorname{H}_{\text{b}}\left(\frac{1}{2}\right) = P_e$$

as $$X$$ is known from (and equal to) y unless $$y=e$$, which has probability $$P_e$$.

By definition $$\operatorname{H}(X)=\operatorname{H}_{\text{b}}\left(\frac{1}{2}\right)=1$$, so
 * $$\operatorname{I}(X;Y) = 1-P_e$$.


 * }

If the sender is notified when a bit is erased, they can repeatedly transmit each bit until it is correctly received, attaining the capacity $$1-P_e$$. However, by the noisy-channel coding theorem, the capacity of $$1-P_e$$ can be obtained even without such feedback.

Related channels
If bits are flipped rather than erased, the channel is a binary symmetric channel (BSC), which has capacity $$1 - \operatorname H_\text{b}(P_e)$$ (for the binary entropy function $$\operatorname{H}_\text{b}$$), which is less than the capacity of the BEC for $$0<P_e<1/2$$. If bits are erased but the receiver is not notified (i.e. does not receive the output $$e$$) then the channel is a deletion channel, and its capacity is an open problem.

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