Simultaneous Computation and Communication over MAC

TL;DR

This work investigates simultaneous computation and communication over a Gaussian MAC with both analog and digital transmitters, proposing a hybrid digital/analog coding scheme that enables computing sums (and broader monomial functions via nomographic representations) of distributed analog values while delivering digital messages under amplitude and power constraints. The authors derive inner and outer bounds on the digital rate–analog computation tradeoffs for sums in $[-1,1]$, and extend the achievability to a broader function class and to fading channels with full CSIT; they also provide practical validation through LDPC-based simulations across finite block lengths and various noise environments. A key contribution is a constructive achievability framework that separates digital and analog processing via a zerosum transformation, enabling concurrent operation without foul interference and offering concrete bounds that reveal a meaningful digital–analog tradeoff. The results have practical implications for distributed sensing, control, and learning (e.g., federated learning) where networked devices must both convey information and compute aggregate statistics over shared wireless resources.

Abstract

We study communication over a Gaussian multiple-access channel (MAC) with two types of transmitters: Digital transmitters hold a message from a discrete set that needs to be communicated to the receiver with vanishing error probability. Analog transmitters hold sequences of analog values. 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 in which the computed function for the analog transmitters is a sum of values in [-1,1], we derive inner and outer bounds for the tradeoff of digital and analog rates of communication under peak and average power constraints for digital transmitters and a peak power constraint for analog transmitters. We then extend the achievability result to a class of functions that includes all linear and some non-linear functions. This extended scheme works over fading channels as long as full channel state information is available at the transmitter. The practicality of our proposed communication scheme is shown in channel simulations that use a version of the scheme based on low density parity check (LDPC) coding. We evaluate the system performance for different block lengths and Gaussian as well as non-Gaussian noise distributions.

Figures (6)

• Figure 1: System model for socc.
• Figure 2: Theorem \ref{['theorem:capacity']} sum rate bounds for $\sigma^2 = \qty{0}{\dB}, \mathcal{A}_a = \qty{2.5}{\dB}, \mathcal{P}_{K_a+1} = \dots = \mathcal{P}_{K} = \qty{8}{\dB}, \mathcal{A}_{K_a+1} = \cdots = \mathcal{A}_K = 2\sqrt{2\mathcal{P}_k}$. The analog approximation is with $V := \beta' \sigma^2/\mathcal{A}_a^2$. Trivial converse is the known sum rate bound for power-constrained Gaussian without analog transmitters or , converse is the second inclusion in Theorem \ref{['theorem:capacity']}, and achievable is the first inclusion in Theorem \ref{['theorem:capacity']}, where $\hat{C}_{\mathcal{W}_\mathrm{AWGN}}$ defined in Section \ref{['sec:gaussian-mac']} is used as a bound for $\bar{C}_{\mathcal{W}_\mathrm{AWGN}}$. There need to be at least four analog transmitters in the system for all converse bounds shown in these figures to hold.
• Figure 3: Graphical overview of the proof idea of Lemma \ref{['lemma:zerosum']}. The notation $U_{n_1} \oplus \dots \oplus U_{n_L}$ means that the operations $U_{n_1}, \dots, U_{n_L}$ are applied in parallel to consecutive, nonoverlapping blocks of lengths $n_1 - 1, \dots, n_L - 1$.
• Figure 4: Example of the amplitude-constrained rate region for $K=2, \mathcal{P}_1 = \qty{1}{\dB}, \mathcal{P}_2 = \qty{4}{\dB}, \sigma^2 = 1, \mathcal{A}_1 = 2\sqrt{\mathcal{P}_1}, \mathcal{A}_2 = 2\sqrt{\mathcal{P}_2}$. The solid lines outline $\bar{C}_{\mathcal{W}_\mathrm{AWGN}}(\mathcal{P}_1,\mathcal{P}_2,\mathcal{A}_1,\mathcal{A}_2)$, where in the region of the sum rate constraint, the boundary of the capacity region is somewhere between the two solid lines.
• Figure 5: Digital and analog error performance for the simulated scenario. Every data point shown is averaged over the transmission of approximately $2 \cdot 10^9$ digital bits.

Theorems & Definitions (28)

• Definition 1
• Lemma 1
• proof
• Definition 2
• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Lemma 2