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

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

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

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

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

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

which means that

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

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