The geometry of the relay channel
Xiugang Wu, Leighton Pate Barnes, Ayfer Özgür · 2017
Consider a memoryless relay channel, where the channel from the relay to the destination is an isolated bit pipe of capacity C0. Let C(C0) denote the capacity of this channel as a function of C0. What is the critical value of C0such that C(C0) first equals C(∞)? This is a long-standing open problem posed by Cover and named “The Capacity of the Relay Channel,” in Open Problems in Communication and Computation, Springer-Verlag, 1987. In our recent work, we answered this question in the case when the channels from the source to the relay and destination are symmetric, which is the original assumption imposed by Cover, and when these channels are Gaussian. We showed that C(C0) can not equal to C(∞) unless C0= ∞, regardless of the SNR of the Gaussian channels, while the cut-set bound would suggest that C(∞) can be achieved at finite C0. In this paper, we show that our techniques for solving Cover's problem can be naturally extended to the general Gaussian case, where the channels from the source to the relay and destination may be asymmetric, and prove an upper bound on the capacity C(C0) of a general Gaussian relay channel for any C0. This upper bound immediately implies that our previous conclusion, i.e. C(C0) can not equal to C(∞) unless C0= ∞, also holds in the asymmetric case. Our approach is geometric and relies on a strengthening of the isoperimetric inequality on the sphere by using the Riesz rearrangement inequality.