Multi-Symbol Digital AirComp via Modulation Design and Power Adaptation

Abstract: In this paper, we consider digital over-the-air computation (AirComp) and introduce a new multi-symbol modulation framework called sequential modulation for AirComp (SeMAC). Building upon ChannelComp, a general framework for designing modulation schemes to support arbitrary function computation over a multiple access channel (MAC), SeMAC maps each input value to a sequence of modulated symbols using distinct constellation diagrams across multiple time slots. This extension generalizes ChannelComp by enabling flexible modulation design across multiple transmissions, thereby enhancing reliability against channel noise. We formulate the modulation design as a non-convex optimization problem, apply matrix lifting to relax it into a semidefinite programming (SDP), and recover a feasible modulation solution by solving a low rank approximation. For scenarios where the modulation formats cannot be changed, we further develop a power adaptation scheme that adjusts amplitude and phase of the modulated symbols while preserving the modulation structure. Numerical results show that SeMAC can achieve a reliable computation by reducing the computation error up to 18 dB compared to other existing methods, particularly for the product function.