Over-the-Air Computation with Imperfect Channel State Information

TL;DR

This paper explores how imperfect channel information impacts over-the-air computation, proposing joint transmit and receive designs to minimize mean squared error under estimation errors, with solutions for SISO and SIMO systems.

Contribution

It introduces optimal and high-quality solutions for joint transmit and receive design in AirComp systems considering channel estimation errors, including closed-form and iterative methods.

Findings

Non-zero MSE due to estimation errors with finite antennas

Massive antennas enable vanishing MSE with independent channels

Proposed designs outperform baseline methods

Abstract

This paper investigates the effect of imperfect channel state information (CSI) on the over-the-air computation (AirComp) system, in which multiple wireless devices (WDs) send individual messages to one access point (AP) for distributed functional computation. By particularly considering the channel estimation errors, we jointly design the transmit coefficients at the WDs and the receive strategy at the AP, for minimizing the computation mean squared error (MSE). First, we consider the single-input single-output (SISO) case with each WD and AP equipped with one single antenna, in which the globally optimal solution to the computation MSE minimization problem is obtained in closed form. Next, we consider the single-input multiple-output (SIMO) case with multiple receive antennas at the AP, in which a high-quality solution is obtained based on alternating optimization and convex optimization. For both cases, the optimized power control solution at the WDs follows a threshold-based regularized channel inversion structure; while for the SIMO case, the receive beamforming at the AP follows a sum-minimum MSE (MMSE) structure. It is shown that with finite receive antennas, a non-zero computation MSE is inevitable due to the channel estimation errors even when the WDs' transmit powers become infinity; while with massive receive antennas, a vanishing MSE is achievable when the channel vectors are independent and identically distributed. Finally, numerical results are provided to demonstrate the effectiveness of the proposed designs.