Capacity theorems for the relay channel

A relay channel consists of an input x₋, a relay output y₁, a channel output y, and a relay sender x₂ (whose transmission is allowed to depend on the past symbols y₁. The dependence of the received symbols upon the inputs is given by p (y, y₁|x₁, x₂). The channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1) If y is a degraded form of y₁, then C \: = \: \!₏ (ₗ_₁, x₂) \, I (X₁, X₂;Y), I (X₁; Y₁|X₂). 2) If y₁ is a degraded form of y, then C \: = \: \!₏ (ₗ_₁) ₗ_₂ I (X₁;Y|x₂). 3) If p (y, y₁|x₁, x₂) is an arbitrary relay channel with feedback from (y, y₁) to both x₁ x₂, then C\: = \: ₏ (ₗ_₁, x₂) \, I (X₁, X₂;Y), I \, (X₁;Y, Y₁|X₂). 4) For a general relay channel, C \: \: ₏ (ₗ_₁, x₂) \, {I \, (X₁, X₂;Y), I (X₁;Y, Y₁|X₂). Superposition block Markov encoding is used to show achievability of C, 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.