Symmetric Channel

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Symmetric Channel

Definition (Symmetric Channel)

A DMC $(\mathcal{X},\mathcal{Y},Q=[P_{XY}])$ is called symmetric if the rows of $Q$ are permutations of each other and the columns of $Q$ are permutations of each other

Definition (Weakly Symmetric Channel)

A DMC $(\mathcal{X},\mathcal{Y},Q=[P_{XY}])$ is called weakly symmetric if the rows of $Q$ are permutations of each other and the column sums in $Q$ are equal i.e. $\sum\limits_{a\in\mathcal{X}}P_{Y|X}(b|a)=c \ \mbox{(constant), }\forall b\in \mathcal{Y}$

Lemma (Information Capacity of Weakly Symmetric Channels)

For a $(\mathcal{X},\mathcal{Y},Q=[P_{XY}])$ , its information capacity, $C$ , is achieved by a uniform input distribution (i.e., $p_{X}(a)= \frac{1}{|\mathcal{X}|}, \ \forall a\in\mathcal{X}$ ) and is given by $C=\log_{2}|\mathcal{Y}|-H(q_{1},\cdots,q_{|\mathcal{Y}|}) \ \mbox{(in bits)}$ where $(q_{1},\cdots,q_{|\mathcal{Y}|})$ is any row from $Q$ .

Definition (Quasi-symmetric Channel)

A DMC $(\mathcal{X},\mathcal{Y},Q=[P_{XY}])$ , called quasi-symmetric if $Q$ can be partitioned along its columns into $m$ weakly symmetric sub-matrices $Q_{1},\cdots, Q_{m}$ for some integer $m\ge1$ , where each sub-matrix $Q_{i}$ has size $|\mathcal{X}|\times|\mathcal{Y}_{i}|$ , $i=1,\cdots,m$ , with $\mathcal{Y}_{1}\cup\cdots\cup\mathcal{Y}_{m}=\mathcal{Y}$ and $\mathcal{Y}_{i}\cap\mathcal{Y}_{j}=\emptyset \ \forall i\not=j$

Lemma (Information Capacity of Quasi-Symmetric Channels)

For a DMC $(\mathcal{X},\mathcal{Y},Q=[P_{XY}])$ , its information capacity, $C$ , is achieved by a uniform input distribution and is given by $C=\sum\limits_{i=1}^{m}a_{i}C_{i}$ where $a_{i}=\sum\limits_{y\in\mathcal{Y}_{i}}p_{XY}=\mbox{sum of any row in }Q_{i}$ and $C_{i}=\log_{2}|\mathcal{Y}_{i}|-H(\mbox{any row in matrix } \frac{1}{a_{i}}Q_{i}), \ i=1,\cdots,m$

Linked from

Binary Erasure Channel

Binary Symmetric Erasure Channel

Symmetric Channel