Simultaneous Over-the-Air Computation and Digital Communication over a Gaussian Multiple-Access Channel

Core Concepts

There is a meaningful tradeoff between the rates of digital communication and the number of analog over-the-air computations that can be performed simultaneously over a Gaussian multiple-access channel.

Abstract

The paper studies a communication system over a Gaussian multiple-access channel (MAC) where there are two types of transmitters:

Digital transmitters that hold messages from discrete sets and need to communicate them to the receiver with vanishing error probability.
Analog transmitters that hold sequences of analog values, and some functions of these distributed values (but not the values themselves) need to be conveyed to the receiver, subject to a fidelity criterion such as mean squared error (MSE) or a certain maximum error with given confidence.

For the case where the computed function for the analog transmitters is a sum of values in [-1, 1], the authors derive inner and outer bounds for the tradeoff between the digital and analog rates of communication under peak and average power constraints for digital transmitters and a peak power constraint for analog transmitters.
The authors then extend the achievability result to a class of functions that includes all linear and some non-linear functions. The practicality of the proposed communication scheme is shown through channel simulations that use low-density parity-check (LDPC) coding and evaluate the system performance for different block lengths and Gaussian as well as non-Gaussian noise distributions.
The key insights are:

There is a meaningful tradeoff between digital communication rates and the number of analog over-the-air computations that can be performed simultaneously.
The proposed hybrid digital-analog communication scheme can achieve this tradeoff by combining digital coding techniques and analog transmissions in the same system.
The scheme is compatible with practical coding techniques like LDPC and can handle both Gaussian and non-Gaussian noise distributions.

Stats

The total power output of the analog pre-processors is at most L * A_a^2/β' ≤ A_a^2n, satisfying the average power constraint.
The total power of the output of digital transmitter k is at most ⌊n(1 - β')⌋P_k/(1 - β') ≤ nP_k, satisfying the average power constraint.
The analog function estimates have MSE β'σ^2/A_a^2.

Quotes

"We propose a novel hybrid communication scheme in which digital communications are encoded in such a way that they do not disturb and are not disturbed by analog OTA-C that is executed concurrently through the same channel."
"The hybrid communication schemes we propose for the achievability part of our main result are derived from codes for standard MAC communication by the use of computationally inexpensive additional processing steps at the transmitters and the receiver."

Key Insights Distilled From

Simultaneous Computation and Communication over MAC

by Matt... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2401.16751.pdf

Deeper Inquiries

How can the proposed scheme be extended to handle more general function computations beyond sums of analog values

The proposed scheme can be extended to handle more general function computations beyond sums of analog values by considering a broader class of functions that can be computed over the shared channel. In the context of the research presented, the functions to be computed are represented by sums of values in the interval [-1, 1]. To extend this to more general functions, one approach would be to consider a wider range of functions that can be expressed as linear combinations or transformations of the analog values held by the transmitters. By allowing for more complex operations on the analog values, such as multiplication, division, or non-linear transformations, a larger class of functions can be computed through the simultaneous computation and communication scheme. This extension would involve designing appropriate encoding and decoding procedures that can handle the computation of these more general functions while ensuring efficient communication over the shared channel.

What are the fundamental limits on the tradeoff between digital communication rates and analog computation rates, and how tight are the bounds derived in this work

The fundamental limits on the tradeoff between digital communication rates and analog computation rates in the proposed scheme are determined by the achievable rate region defined in the research. The tradeoff region, denoted as $H_{\beta,V}$, represents the set of achievable digital and analog rates for simultaneous computation and communication over the Gaussian multiple-access channel. The bounds derived in the work provide insights into the achievable tradeoff between digital and analog rates under peak and average power constraints for the transmitters. The inner and outer bounds presented in the research offer a characterization of the achievable rate region, showing the potential tradeoffs between digital communication efficiency and analog computation accuracy. The tightness of these bounds depends on the specific system parameters and constraints, and further research may aim to refine these bounds to provide a more precise understanding of the tradeoff limits.

Can the proposed hybrid digital-analog communication scheme be applied to other communication scenarios beyond the Gaussian multiple-access channel, such as relay networks or interference channels

The proposed hybrid digital-analog communication scheme can potentially be applied to other communication scenarios beyond the Gaussian multiple-access channel, such as relay networks or interference channels. The key idea of combining digital communication with analog computation in a shared channel can be adapted to different network configurations where multiple transmitters need to convey information to a common receiver. By modifying the encoding and decoding procedures to suit the specific channel characteristics and requirements of relay networks or interference channels, the hybrid scheme can be extended to facilitate simultaneous computation and communication in these scenarios. The principles of leveraging analog computation for function approximation and digital communication for message transmission can be applied in a variety of network settings to improve efficiency and reliability in data exchange. Further research and experimentation would be needed to tailor the scheme to the unique challenges and opportunities presented by relay networks and interference channels.