On the Capacity and State Estimation Error of “Beam-Pointing” Channels: The Binary Case

Abstract

Motivated by beamformed millimeter-wave (mmWave) communication, we consider the optimal tradeoff between reliable communication rate and state estimation error for a new state-dependent channel model with in-block memory referred to as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">binary beam-pointing</i> (BBP) channel. The multiantenna base station (BS) uses a finite beamforming codebook (i.e., discrete beam directions) and associates the target user to the most favorable beam, i.e., the beam directed along the strongest propagation path from BS to the user in 5G and IEEE 802.11ad mmWave communication systems. We model the target user’s Angle-of-Departure (AoD) as the state of a state-dependent channel. Since the AoD remains almost constant over several consecutive time slots, the channel has memory. In addition, we assume the BS receives implicit causal feedback (e.g., modeling a backscatter signal as in radar) and consider a joint communication and sensing (JCAS) problem where the BS is also interested in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">explicitly</i> estimate the channel state (quantized AoD). We derive a closed-form solution to the capacity of the BBP channel model under the peak input cost constraint. Under the average constraint, we present an implicit capacity result, where the capacity is given as the solution of an optimization problem, and we provide an algorithm for its numerical computation. Finally, for the JCAS problem at hand, we provide the minimum distortion under the peak input constraint and show that this coincides with that obtained by the capacity-achieving strategy. This <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">completely characterizes the capacity-distortion tradeoff</i> for the BBP channel under peak input constraint. For the average constraint case, numerical results show that our capacity-achieving strategy yields a state estimation error very close to the theoretical minimum, showing the near-optimality of the proposed strategy for the JCAS problem under the average cost constraint.