Binary erasure channel

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 $p_e$

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:

$\begin{array}{rl}\mathrm{Pr}[Y=0|X=0]& =1-P_e\\ \mathrm{Pr}[Y=0|X=1]& =0\\ \mathrm{Pr}[Y=1|X=0]& =0\\ \mathrm{Pr}[Y=1|X=1]& =1-P_e\\ \mathrm{Pr}[Y=e|X=0]& =P_e\\ \mathrm{Pr}[Y=e|X=1]& =P_e\end{array}$

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).

Proof

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

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 $1-{\mathrm{H}}_{\text{b}}(P_e)$ (for the binary entropy function ${\mathrm{H}}_{\text{b}}$), 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.

See also

• Erasure code
• Packet erasure channel

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. http://www.inference.phy.cam.ac.uk/mackay/itila/book.html. 
• "A survey of results for deletion channels and related synchronization channels", Probability Surveys 6: 1–33, 2009, doi:10.1214/08-PS141