“The Capacity of the Relay Channel”: Solution to Cover’s Problem in the Gaussian Case

Consider a memoryless relay channel, where the relay is connected to the destination with an isolated bit pipe of capacity <inline-formula> <tex-math notation="LaTeX">$C_{0}$ </tex-math></inline-formula>. Let <inline-formula> <tex-math notation="LaTeX">$C(C_{0})$ </tex-math></inline-formula> denote the capacity of this channel as a function of <inline-formula> <tex-math notation="LaTeX">$C_{0}$ </tex-math></inline-formula>. What is the critical value of <inline-formula> <tex-math notation="LaTeX">$C_{0}$ </tex-math></inline-formula>, such that <inline-formula> <tex-math notation="LaTeX">$C(C_{0})$ </tex-math></inline-formula> first equals <inline-formula> <tex-math notation="LaTeX">$C(\infty )$ </tex-math></inline-formula>? This is a long-standing open problem posed by Cover and named “The Capacity of the Relay Channel,” in <italic>Open Problems in Communication and Computation</italic>, Springer-Verlag, 1987. In this paper, we answer this question in the Gaussian case and show that <inline-formula> <tex-math notation="LaTeX">$C(C_{0})$ </tex-math></inline-formula> cannot equal to <inline-formula> <tex-math notation="LaTeX">$C(\infty )$ </tex-math></inline-formula> unless <inline-formula> <tex-math notation="LaTeX">$C_{0}=\infty $ </tex-math></inline-formula>, regardless of the SNR of the Gaussian channels. This result follows as a corollary to a new upper bound we develop on the capacity of this channel. Instead of “single-letterizing” expressions involving information measures in a high-dimensional space as is typically done in converse results in information theory, our proof directly quantifies the tension between the pertinent <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-letter forms. This is done by translating the information tension problem to a problem in high-dimensional geometry. As an intermediate result, we develop an extension of the classical isoperimetric inequality on a high-dimensional sphere, which can be of interest in its own right.

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