On the Capacity and State Estimation Error of Binary "Beam-Pointing" Channels with Block Memory and Feedback

To counter the large isotropic pathloss in Millimeter-wave (mmWave) communications, high beamforming gain by means of large antenna arrays is required. Joint Communication and Sensing (JCAS) is predicted to be a major feature of future communication systems, but suitable channel models and performance metrics are still in development. This paper investigates the information-theoretic limits of JCAS using a channel model proposed in [1], which consists of a binary state-dependent channel with unit-delayed feedback and an in-block memory (iBM) [2] of the fixed block length. This model is sufficiently simple to treat, yet captures some fundamental aspects of beam acquisition (BA) where the feedback models the backscatter signal as in a radar system, thus fitting the paradigm of JCAS. When the transmission cost is bounded on average, we simplify the capacity computation as an optimization problem by exploring a small set of optimal average input costs at each channel use based on feedback. The converse proof inspires a capacity-achieving strategy that uniformly explores a given number of beam directions under the cost constraint. The sensing performance is characterized in terms of the state estimation error. We present the minimum distortion without communication constraint as another optimization problem, which can be solved by the successive convex approximation (SCA) method similar to the capacity computation problem. Numerical examples are provided to illustrate the performance of the proposed strategies and show a very small gap between the estimation error achieved by the capacity-achieving strategy and the minimum possible error. Hence, for the model at hand sensing comes essentially "for free", i.e., there exists a JCAS strategy that pays very little in terms of sensing performance while being optimal with respect to the communication rate.