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Binary Symmetric Erasure Channel

Definition (Binary symmetric erasure channel)

With alphabet X={0,1}\mathcal{X}=\{0,1\} and Y={0,E,1}\mathcal{Y}=\{0,E,1\} with PY∣X(b∣a)={1−α−ϵ\mboxifa=b,b∈{0,1}ϵ\mboxifa≠b,a,b∈{0,1}α\mboxifb=E,a∈{0,1}P_{Y|X}(b|a)=\begin{cases}1-\alpha-\epsilon&\mbox{if }a=b,b\in\{0,1\}\\\epsilon&\mbox{if }a\not=b,a,b\in\{0,1\}\\\alpha&\mbox{if }b=E,a\in\{0,1\}\end{cases}where ϵ\epsilon and α\alpha are the crossover and erasure probabilities. The transition matrix is Q=[PXY]=[PY∣X(0∣0)PY∣X(E∣0)PY∣X(1∣0)PY∣X(0∣1)PY∣X(E∣1)PY∣X(1∣1)]=[1−α−ϵαϵϵα1−α−ϵ]Q=[P_{XY}]=\begin{bmatrix}P_{Y|X}(0|0)&P_{Y|X}(E|0)&P_{Y|X}(1|0)\\P_{Y|X}(0|1)&P_{Y|X}(E|1)&P_{Y|X}(1|1)\end{bmatrix}=\begin{bmatrix}1-\alpha-\epsilon&\alpha&\epsilon\\\epsilon&\alpha&1-\alpha-\epsilon\end{bmatrix}

Remark (Explicit Representation of BSEC)

The BSEC(ϵ,α)(\epsilon,\alpha) can be explicitly modeled via a binary-input channel with an iid noise-erasure process {Zi}i=1∞\{Z_{i}\}_{i=1}^{\infty} with alphabet Z={0,E,1}\mathcal{Z}=\{0,E,1\} and pZ(1)=ϵp_{Z}(1)=\epsilon and pZ(E)=αp_{Z}(E)=\alpha.

Special Cases

For BSEC(ϵ,α)(\epsilon,\alpha)

  • α=0  ⟹  \alpha=0\implies BSC(ϵ)(\epsilon)
  • ϵ=0  ⟹  \epsilon=0\implies BEC(α)(\alpha)

Information Capacity

The information capacity of the BSEC(ϵ,α)(\epsilon,\alpha) can be found using Information Capacity of Quasi-Symmetric Channels where we find that it evaluates to C=(1−α)[1−hb(ϵ1−α)]C=(1-\alpha)\left[1-h_{b}\left(\frac{\epsilon}{1-\alpha}\right)\right]