Digitalizing Over-the-Air Computation via The Novel Complement Coded Modulation

TL;DR

This paper introduces a novel digital coding scheme for over-the-air computation that improves accuracy and robustness, especially at low SNR, by using two's complement encoding, optimized decoding, and power allocation strategies.

Contribution

It proposes a two's complement-based coding scheme for digital AirComp, ensuring asymptotic error-free computation and enhanced performance over existing methods.

Findings

Outperforms existing digital AirComp schemes at low SNR.

Provides a closed-form optimal LMMSE detector.

Demonstrates the effectiveness of uneven power allocation.

Abstract

To overcome inherent limitations of analog signals in over-the-air computation (AirComp), this letter proposes a two's complement-based coding scheme for the AirComp implementation with compatible digital modulations. Specifically, quantized discrete values are encoded into binary sequences using the two's complement and transmitted over multiple subcarriers. At the receiver, we design a decoder that constructs a functional mapping between the superimposed digital modulation signals and the target of computational results, theoretically ensuring asymptotic error free computation with the minimal codeword length. To further mitigate the adverse effects of channel fading, we adopt a truncated inversion strategy for pre-processing. Benefiting from the unified symbol distribution after the proposed encoding, we derive the optimal linear minimum mean squared error (LMMSE) detector in closed form and propose a low complexity algorithm seeking for the optimal truncation selection. Furthermore, the inherent importance differences among the coded outputs motivate an uneven power allocation strategy across subcarriers to improve computational accuracy. Numerical results validate the superiority of the proposed scheme over existing digital AirComp approaches, especially at low signal to-noise ratio (SNR) regimes.