SumComp: Coding for Digital Over-the-Air Computation via the Ring of Integers

TL;DR

This study considers a specific category of digital modulations for over-the-air computations, QAM and PAM, for which a novel coding scheme called SumComp is introduced, showing superior performance compared to traditional analog over- the-air computation and the original coding in ChannelComp approaches regarding both MSE and MAE over a noisy multiple access channel.

Abstract

Communication and computation are traditionally treated as separate entities, allowing for individual optimizations. However, many applications focus on local information's functionality rather than the information itself. For such cases, harnessing interference for computation in a multiple access channel through digital over-the-air computation can notably increase the computation, as established by the ChannelComp method. However, the coding scheme originally proposed in ChannelComp may suffer from high computational complexity because it is general and is not optimized for specific modulation categories. Therefore, this study considers a specific category of digital modulations for over-the-air computations, QAM and PAM, for which we introduce a novel coding scheme called SumComp. Furthermore, we derive an MSE analysis for SumComp coding in the computation of the arithmetic mean function and establish an upper bound on the MAE for a set of nomographic functions. Simulation results affirm the superior performance of SumComp coding compared to traditional analog over-the-air computation and the original coding in ChannelComp approaches regarding both MSE and MAE over a noisy multiple access channel. Specifically, SumComp coding shows approximately $10$ dB improvements for computing arithmetic and geometric mean on the normalized MSE for low noise scenarios.

Figures (9)

• Figure 1: Block diagram illustrating the complete transmission process for computing a function over the MAC with $K$ nodes. The process begins with the signals $s_{1}, s_{2}, \ldots, s_{K}$ being passed through an encoder $\mathscr{E}(\cdot)$ and decoder $\mathscr{D}{\rho}$. The modulated signals $x_{1}$, $x_{2}, \ldots x_{K}$ are then transmitted over the MAC, leading to ${r}$ contaminated by the noise ${z}$. The received signal is ${r}$, which undergoes quantization $\mathcal{Q}(\cdot)$. Then, it is passed through an inverse function $\mathcal{G}_{\rho}^{-1}$ to obtain ${g}\in \mathbb{G}$, which is finally decoded by the decoder $\mathcal{D}(\cdot)$ and processed by function $\psi (\cdot)$ to yield the estimated function $\hat{f}$. The dashed boxes denote the Encoding region $\mathscr{E}$ and the Decoding region $\mathscr{D}$. In a standard communication system, $\varphi$, $\mathcal{E}$, and $\mathcal{G}_{\rho}$ correspond to source coding, channel encoding, and modulation mapper blocks, respectively. Similarly, $\psi$, $\mathcal{D}$, $\mathcal{G}_{\rho}^{-1}$ corresponds to source decoding, channel decoding, and demodulation mapper blocks, respectively.
• Figure 2: Constellation diagram showcasing the transmission of Gray-coded PAM-modulated signals by two nodes ($K=2$) with $q=4$ ($2$ bits). Destructive overlaps, depicted in red, occur when transmitting values $\{3,2,3\}$ and $\{2,4,6\}$ concurrently, demonstrating the challenges in uniquely decoding the received signals due to interference in the same channel.
• Figure 3: The complete encoding procedure. The output of preprocessing function $c_k\in \mathbb{Z}_q$ is first encoded to Gaussian integers $\mathbb{G}$ using the encoder $\mathcal{E}(\cdot)$. Then, using $\mathcal{G}_{\rho}$, we map the output value to a general ring of integers $\mathbb{Z}[\rho]$. Finally, node $k$ selects subset $\Lambda_q$ of $\mathbb{Z}[\rho]$ as the final constellation points to transmit over the MAC.
• Figure 4: SumComp coded Hexagonal QAM of order $8$ for two different choices of $(q_1,q_2)$. In Figure \ref{['fig:Hexagonal(a)']}, $(q_1, q_2) = (2, 3)$ shows that the input value $s_k$ can have integer values between $0$ and $8$, except for $4$. Adding the value $4$ requires new constellation points on one of the gray constellation points in the dashed lines, avoiding overlaps. In Figure \ref{['fig:Hexagonal(b)']}, the SumComp coded Hexagonal QAM $8$ with $(q_1, q_2) = (1, 2)$ shows that the input value $s_k$ can have integer values between $0$ and $5$. Here, the numbers $4$ and $1$ are repeated. Replacing the numbers $4$ or $1$ with any other numbers results in overlaps among the constellation points. Note that for the repeated constellation points, every node needs to select one of the constellation points for transmission.
• Figure 5: Gray code vs SumComp code for QAM $16$ modulation with $(q_1,q_2) = (1,4)$.

Theorems & Definitions (25)

• Definition 1
• Theorem 1
• Theorem 2
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Example 1
• Example 2
• Remark 5