Foundations of Computation Via Digital Communications

The explosive growth of distributed data generation — spanning data centers, sensor
networks, massive IoT, and edge learning places an unsustainable burden on modern infrastructure, where the energy and latency costs of moving raw data often outstrip those
of processing it. While analog over-the-air computation (OAC) promises a solution by
exploiting the natural superposition of wireless waveforms to aggregate data in-channel,
it remains fragile against noise and fundamentally incompatible with the ubiquitous
digital hardware that powers all modern communication systems.
This thesis introduces a digital-native framework that unifies communication and
computation at the physical layer. Rather than treating channel interference as an
obstacle, we engineer the geometry of digital constellations so that the superposition
of signals directly yields the desired function value. This paradigm shift transforms
the communication link from a passive data pipe into an active computational engine,
applicable to any multiple-access channel—whether wired or wireless—without requiring
the decoding of individual inputs.
We generalize this framework along three axes to ensure scalability and reliability
across diverse network environments. First, we develop noise-aware constellation designs that optimize inter-symbol geometry for non-Gaussian and heavy-tailed interference, ensuring robustness beyond standard Euclidean metrics. Second, we introduce a
sampling-based reduction strategy that leverages the symmetry of aggregation functions
to cut design complexity by orders of magnitude, enabling deployment in massive-scale
networks. Third, we extend the framework to vector-valued computation, utilizing spatial degrees of freedom to perform complex, multi-dimensional aggregations in a single
transmission shot without relying on perfect channel state information.
Finally, to bridge the gap to immediate deployment, we present a closed-form algebraic coding scheme for exact summation. The proposed solution integrates seamlessly
with standard quadrature amplitude modulation, eliminating the need for complex optimization and offering a plug-and-play solution for digital aggregation. We validate
these contributions through the lens of Federated Edge Learning, demonstrating that
computation-by-communication is not only feasible using standard digital protocols but
significantly outperforms traditional orthogonal transmission. Collectively, these works
prove that computation-by-communication is not only feasible on digitally modulated
signals but superior to analog alternatives, paving the way for the next generation
of compute-aware networks, enabling energy-efficient, scalable, and robust intelligence
across any digital infrastructure.