A new upper bound on the capacity of a class of primitive relay channels

We obtain a new upper bound on the capacity of a class of discrete memoryless relay channels. For this class of relay channels, the relay observes an i.i.d. sequence T, which is independent of the channel input X. The channel is described by a set of probability transition functions p(y|x,t) for all (x,t,y) ∈ X × T × Y. Furthermore, a noiseless link of finite capacity R ₀ exists from the relay to the receiver. Although the capacity for these channels is not known in general, the capacity of a subclass of these channels, namely when T = g(X, Y), for some deterministic function g, was obtained in [1] and it was shown to be equal to the cut-set bound. Another instance where the capacity was obtained was in [2], where the channel output Y can be written as Y = X ∠ Z, where ∠ denotes modulo-m addition, Z is independent of X, |X| = |y| = m, and T is some stochastic function of Z. The compress-and-forward (CAF) achievability scheme [3] was shown to be capacity achieving in both cases. Using our upper bound we recover the capacity results of [1] and [2]. We also obtain the capacity of a class of channels which does not fall into either of the classes studied in [1] and [2]. For this class of channels, CAF scheme is shown to be optimal but capacity is strictly less than the cut-set bound for certain values of R ₀. We further illustrate the usefulness of our bound by evaluating it for a particular relay channel with binary multiplicative states and binary additive noise for which the channel is given as Y = T X + N. We show that our upper bound is strictly better than the cut-set upper bound for certain values of R ₀ but it lies strictly above the rates yielded by the CAF achievability scheme.