The Capacity of the Relay Channel

Consider the following seemingly simple discrete memoryless relay channel: Open image in new window Here Y 1, Y 2 are conditionally independent and conditionally identically distributed given X, that is, \(p(y_1,\, y_2\, |\, x) = p(y_1\, |\, x) p(y_2\, |\, x)\). Also, the channel from Y 1 to Y 2 does not interfere with Y 2. A (2nR, n) code for this channel is a map \(x : 2^{nR} \rightarrow X^n\), a relay function \(r :Y_1^{n}\rightarrow 2^{nC_{0}}\), and a decoding function \(g : 2^{nC_{0}} \times Y_2^{n} \rightarrow 2^{nR}\). The probability of error is given by $$ P_e^{(n)} = P \{\,g(r(y_1),y_2) \ne W\}$$ , where W is uniformly distributed over 2nR and $$ p(w, y_1, y_2) = 2^{-nR}\,\, \underset {i=1}{\overset {n}{\Pi}} p(y_{1i}\, |\, x_i(w)) \,\,\underset {i=1}{\overset {n}{\Pi}}\,\, p(y_{2i}\, |\, x_i(w))$$