Calculate the capacity of the channel

Question

Solve the following problem:

In Problem 1 assume that the same noisy versions of state information are available at both sides; i.e., Z = U = V is available where Z is a binary-valued random variable with

P[Z = 0 | S = 0] = P[Z = 1 | S = 1] = 1

P[Z = 0 | S = 2] = P[Z = 1 | S = 2] = 1/2

Determine the capacity of this channel.

Problem 1:  Consider a BSC in which the channel can be in three states. In state S = 0 the output of the channel is always 0, regardless of the channel input. In state S = 1, the output is always 1, again regardless of the channel input. In state S = 2 the channel in noiseless, i.e., the output is always equal to the input. We assume that P(S = 0) = P(S = 1) = p².

1. Determine the capacity of this channel, assuming no state information is available to the transmitter or the receiver.

2. Determine the capacity of the channel, assuming that channel state information S is available at both sides.