Capacity theorems for the relay channel

Abstract

A relay channel consists of an inputx_{l}, a relay outputy_{1}, a channel outputy, and a relay senderx_{2}(whose transmission is allowed to depend on the past symbolsy_{1}. The dependence of the received symbols upon the inputs is given byp(y,y_{1}|x_{1},x_{2}). The channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1)Ifyis a degraded form ofy_{1}, thenC \: = \: \max \!_{p(x_{1},x_{2})} \min \,{I(X_{1},X_{2};Y), I(X_{1}; Y_{1}|X_{2})}. 2)Ify_{1}is a degraded form ofy, thenC \: = \: \max \!_{p(x_{1})} \max_{x_{2}} I(X_{1};Y|x_{2}). 3)Ifp(y,y_{1}|x_{1},x_{2})is an arbitrary relay channel with feedback from(y,y_{1})to bothx_{1} \and x_{2}, thenC\: = \: \max_{p(x_{1},x_{2})} \min \,{I(X_{1},X_{2};Y),I \,(X_{1};Y,Y_{1}|X_{2})}. 4)For a general relay channel,C \: \leq \: \max_{p(x_{1},x_{2})} \min \,{I \,(X_{1}, X_{2};Y),I(X_{1};Y,Y_{1}|X_{2}). Superposition block Markov encoding is used to show achievability ofC, and converses are established. The capacities of the Gaussian relay channel and certain discrete relay channels are evaluated. Finally, an achievable lower bound to the capacity of the general relay channel is established.