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

Hon-Fah Chong,Mehul Motani

doi:10.1109/tit.2011.2165131

Abstract

The relay channel consists of a transmitter input x 1 , a relay input x 2 , a relay output y 2 , and a receiver output y 3 . 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 y 2 depends on a deterministic function of the transmitter's input x 1 , i.e., on s = f 1 ( x 1 ), rather than on x 1 directly. In addition, the relay channels satisfy the condition that S → ( X 2 , Y 2 ) → Y 3 forms a Markov chain for all input probability distributions p ( x 1 , x 2 ). 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 y 2 is a deterministic function of x 1 , x 2 and y 3 , i.e., y 2 = f 4 ( x 1 , x 2 , y 3 ). 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 y 2 is a deterministic function of x 1 , x 2 , and y 3 . 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.

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