Multi-Symbol Digital AirComp via Modulation Design and Power Adaptation

Recently, over-the-air computation (AirComp) leverages the superposition property of wireless channels to enable efficient function computation over a multiple access channel (MAC). However, existing digital AirComp methods either rely on single-symbol modulation, which limits flexibility and robustness, or on multi-symbol extensions that suffer from high complexity or approximation errors. To overcome these limitations, we propose a new multi-symbol modulation framework, termed sequential modulation for AirComp (SeMAC), which encodes each input into a sequence of symbols with distinct constellation diagrams across multiple time slots. This approach increases design flexibility and robustness against channel noise. Specifically, the modulation design is formulated as a non-convex optimization problem and efficiently solved through a successive convex approximation (SCA) combined with stochastic subgradient descent (SSD). For fixed modulation formats, we further develop SeMAC with power adaptation (SeMAC-PA) to adjusts transmit power and phase while preserving the modulation structure. Notably, numerical results show that SeMAC improves computation accuracy by up to 14 dB compared to the existing methods for computing nonlinear functions such as the product function.