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Discrete Memoryless Channel

Definition (DMC)

A discrete memoryless channel (DMC) is a discrete channel whose sequence of nn-dimensional transition distributions {PYn∣Xn}\{P_{Y^{n}|X^{n}}\} satisfies PYn∣Xn(bn∣an)=Πi=1∞PY∣X(bi∣ai)(**)\tag{**}P_{Y^{n}|X^{n}}(b^{n}|a^{n})=\Pi_{i=1}^{\infty}P_{Y|X}(b_{i}|a_{i})∀n≥1, an∈Xn, bn∈Yn\forall n\ge1, \ a^{n}\in\mathcal{X}^{n}, \ b^{n}\in \mathcal{Y}^{n} where PY∣XP_{Y|X} is a fixed (time-invariant) conditional distribution on X×Y\mathcal{X}\times\mathcal{Y}. In other words, a DMC is fully described by the triplet (X,Y,Q=[PXY])(\mathcal{X},\mathcal{Y},Q=[P_{XY}]) where QQ is called the channel’s transition matrix and PXY:=PY∣X(y∣x), x∈X, y∈YP_{XY}:=P_{Y|X}(y|x), \ x\in\mathcal{X}, \ y\in\mathcal{Y}.

Remark

  • Matrix is row-stochastic (i.e. rows sum to one)
  • It can be verified that a DMC satisfies the consistency property

Lemma

The DMC Property (∗∗)(**) is equivalent to these two conditions:

  1. Output Memoryless Feature: PYn∣Xn,Yn−1(bn∣an,bn−1)=PY∣X(bn∣an)P_{Y_{n}|X^{n},Y^{n-1}}(b_{n}|a^{n},b^{n-1})=P_{Y|X}(b_{n}|a_{n}) ∀n≥1, an∈Xn, bn∈Yn\forall n\ge1, \ a^{n}\in\mathcal{X}^{n}, \ b^{n}\in \mathcal{Y}^{n}. i.e. The current output is conditionally independent of past output given the current input.
  2. Non-Anticipatory Feature: PYn−1∣Xn(bn−1∣an)=PYn−1∣Xn−1(bn−1∣an−1)P_{Y^{n-1}|X^{n}}(b^{n-1}|a^{n})=P_{Y^{n-1}|X^{n-1}}(b^{n-1}|a^{n-1}) ∀n≥2, an∈Xn, bn−1∈Yn−1\forall n\ge2, \ a^{n}\in\mathcal{X}^{n}, \ b^{n-1}\in \mathcal{Y}^{n-1}. i.e. The current output is conditionally independent of future input given the current and past inputs.

Linked from

Binary Erasure Channel

Binary Symmetric Channel

Information Capacity

Lossless Joint Source-Channel Coding Theorem

Operational Channel Capacity

Shannon's Channel Coding Theorem for the DMC

Symmetric Channel

Lossy Source-Channel Coding Theorem