(a) Define the information capacity of a discrete memoryless channel (DMC).

(b) Consider a DMC where there are two input symbols, $A$ and $B$, and three output symbols, $A, B$ and $\star$. Suppose each input symbol is left intact with probability $1 / 2$, and transformed into a $\star$ with probability $1 / 2$.

(i) Write down the channel matrix, and calculate the information capacity.

(ii) Now suppose the output is further processed by someone who cannot distinguish between $A$ and $\star$, so that the channel matrix becomes

$$\left(\begin{array}{cc} 1 & 0 \\ 1 / 2 & 1 / 2 \end{array}\right)$$

Calculate the new information capacity.