On the Multiple Access Channel with Asymmetric Noisy State Information at the Encoders

We consider the problem of reliable communication over multiple-access channels (MAC) where the channel is driven by an independent and identically distributed state process and the encoders and the decoder are provided with various degrees of asymmetric noisy channel state information (CSI). For the case where the encoders observe causal, asymmetric noisy CSI and the decoder observes complete CSI, we provide inner and outer bounds to the capacity region, which are tight for the sum-rate capacity. We then observe that, under a Markov assumption, similar capacity results also hold in the case where the receiver observes noisy CSI. Furthermore, we provide a single letter characterization for the capacity region when the CSI at the encoders are asymmetric deterministic functions of the CSI at the decoder and the encoders have non-causal noisy CSI (its causal version is recently solved in \cite{como-yuksel}). When the encoders observe asymmetric noisy CSI with asymmetric delays and the decoder observes complete CSI, we provide a single letter characterization for the capacity region. Finally, we consider a cooperative scenario with common and private messages, with asymmetric noisy CSI at the encoders and complete CSI at the decoder. We provide a single letter expression for the capacity region for such channels. For the cooperative scenario, we also note that as soon as the common message encoder does not have access to CSI, then in any noisy setup, covering the cases where no CSI or noisy CSI at the decoder, it is possible to obtain a single letter characterization for the capacity region. The main component in these results is a generalization of a converse coding approach, recently introduced in [1] for the MAC with asymmetric quantized CSI at the encoders and herein considerably extended and adapted for the noisy CSI setup.