# Computation Over NOMA: Improved Achievable Rate Through Sub-Function Superposition

## TL;DR

This paper introduces a NOMA-based approach for computation over multi-access channels, decomposing functions into sub-functions superposed over resource blocks to enhance spectrum efficiency and achieve higher, non-vanishing computation rates in large networks.

## Contribution

It proposes a novel NOMA-CoMAC scheme that improves achievable computation rates by superposing sub-functions, addressing limitations of traditional CoMAC in large-scale networks.

## Key findings

- NOMA-CoMAC achieves higher computation rates than traditional CoMAC.
- The scheme prevents vanishing rates as the number of nodes increases.
- The system's performance is limited by the worst channel gain among sub-functions.

## Abstract

Massive numbers of nodes will be connected in future wireless networks. This brings great difficulty to collect a large amount of data. Instead of collecting the data individually, computation over multi-access channel (CoMAC) provides an intelligent solution by computing a desired function over the air based on the signal-superposition property of wireless channel. To improve the spectrum efficiency in conventional CoMAC, we propose the use of non-orthogonal multiple access (NOMA) for functions in CoMAC. The desired functions are decomposed into several sub-functions, and multiple sub-functions are selected to be superposed over each resource block (RB). The corresponding achievable rate is derived based on sub-function superposition, which prevents a vanishing computation rate for large numbers of nodes. In order to gain more insights, we further study the limiting case when the number of nodes goes to infinity. An exact expression of the rate is derived which provides a lower bound on the computation rate. Compared with existing CoMAC, the NOMA-based CoMAC (NOMA-CoMAC) not only achieves a higher computation rate, but also provides an improved non-vanishing rate. Furthermore, the diversity order of the computation rate of NOMA-CoMAC is derived, and it shows that the system performance is dominated by the node with the worst channel gain among these sub-functions in each RB.