Information Bottleneck for an Oblivious Relay with Channel State Information: the Scalar Case

We consider an extension of the information bottleneck problem where underlying Markov Chain is $X-0-(Y,\ S)-0-Z$, and where $P_{X,S,Y}=P_{X}P_{S}P_{Y|X,S}$ is the joint distribution of a source X, a channel state S independent of the source, and the channel output Y of a state-dependent channel. For the case $Y=SX+N$ with $X, S$ and N Gaussian circularly symmetric, we provide an upper bound and two achievable lower bounds on the information bottleneck rate. We relate this problem to the case of an oblivious relay with channel state information. Our results show that simple symbol-by-symbol relay processing, possibly followed by “entropy coding” (data compression) yields a very effective method, virtually achieving the upper bound on a wide range of relevant system parameters.