Preliminaries

$\bf{\text{1. Symmetric Capacity:}}$

The symmetric capacity is the highest possible rate that can be achieved when all of the input symbols to the channel are equiprobable.

The symmetric capacity is calculated as follows

\begin{equation} I(W)=\sum_{y \in \mathbb{Y}} \sum_{x \in \mathbb{X}} \ \frac{1}{2} \cdot W(y|x) \cdot \log_2 \ \frac{W(y|x)}{\frac{1}{2} \cdot W(y|0)+\frac{1}{2} \cdot W(y|1)}. \end{equation}

The symmetric capacity is equal to the Shannon capacity when the channel $W$ is a symmetric channel.

$\bf{\text{2. Bhattacharyya Parameter:}}$

The Bhattacharyya parameter $Z(W)$ is the upper bound on the probability of an ML decision error when transmitting $0$ or $1$ over the channel $W$. Thus the Bhattacharyya parameter $Z(W)$ is a channel reliability measure.

The Bhattacharyya parameter can be calculated as follows

\begin{equation} Z(W)= \sum_{y \in \mathbb{Y}} \sqrt{W(y|0) \cdot W(y|1)}. \end{equation}

The relationship between $I(W)$ and $Z(W)$ for any B-DMC $W$ is

\begin{equation} I(W) \ge \log \ \frac{2}{1+Z(W)}, \end{equation} \begin{equation} I(W) \le \sqrt{1-Z(W)^2}. \end{equation}

which means that

a. $I(W)=1$ if and only if $Z(W)=0$.

b. $I(W)=0$ if and only if $Z(W)=1$.