The Capacity of Several New Classes of Semi-Deterministic Relay Channels

The relay channel consists of a transmitter input x₁, a relay input x₂, a relay output y₂ , and a receiver output y₃. In this paper, we establish the capacity of three new classes of semi-deterministic relay channels: 1) a class of degraded semi-deterministic relay channels, 2) a class of semi-deterministic orthogonal relay channels, and 3) a class of semi-deterministic relay channels with relay-transmitter feedback. For the first class of relay channels, the output of the relay y₂ depends on a deterministic function of the transmitter´s input x₁, i.e., on s=f₁(x₁), rather than on x₁ directly. In addition, the relay channels satisfy the condition that S → (X₂,Y₂) → Y₃ forms a Markov chain for all input probability distributions p(x₁,x₂). Hence, the first class of relay channels includes, but is strictly not limited to, the class of degraded relay channels previously considered by Cover and El Gamal. The partial decode-and-forward strategy achieves the capacity of the class of degraded semi-deterministic relay channels. Next, we consider the class of semi-deterministic orthogonal relay channels where there are orthogonal channels from the relay to the receiver and from the transmitter to the receiver. In addition, the output of the relay y₂ is a deterministic function of x₁, x₂ and y₃ , i.e., y₂=f₄(x₁,x₂,y₃). The class of semi-deterministic orthogonal relay channels is a generalization of the class of deterministic relay channels considered by Kim. The compress-and-forward strategy achieves the capacity of the class of semi- - -deterministic orthogonal relay channels. For the third class of relay channels, there is a causal and noiseless feedback from the relay to the transmitter. In addition, similar to the second class of relay channels, the output of the relay y₂ is a deterministic function of x₁, x₂, and y₃ . Both the generalized strategy of Gabbai and Bross and the hash-and-forward strategy of Kim achieve the capacity of the class of semi-deterministic relay channels with relay-transmitter feedback.