The Capacity of the Relay Channel

Abstract

Consider the following seemingly simple discrete memoryless relay channel:

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 (2^nR, 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 2^nR and

$$ p(w, y_1, y_2) = 2^{-nR}\,\, \underset {i=1}{\overset {n}{\Pi}} p(y_{1i}\, |\, x_i(w)) \,\,\underset {i=1}{\overset {n}{\Pi}}\,\, p(y_{2i}\, |\, x_i(w))$$