arXiv subjects

Saeed Razavikia

Publications and source records attributed to Saeed Razavikia.

At least 19 recordsLinked to original sources

arXiv2026

The Computing Channel: How Modulation Programs the Airwaves

Distributed computing and distributed artificial intelligence require frequent exchanges of intermediate results, although many applications need only an aggregate rather than messages from individual devices. Conventional systems recover each message before computing the aggregate, whereas over-the-air computation (OAC) exploits simultaneous transmission to obtain it directly. However, dominant OAC implementations rely on analog signaling, creating a mismatch with finite-precision data and digital communication procedures. This article presents digital function-oriented communication, in which finite-alphabet symbol representations and receiver decisions are jointly designed so that multiple-access superposition encodes the desired function without recovering individual inputs. We introduce its computational-constellation principle, main design approaches, extensions, and implementation challenges. Federated edge learning illustrates how the framework can reduce user-dependent data-bearing resources while operating directly on quantized model updates.

cs.DC↗

arXiv2026

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

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.

cs.IT↗

arXiv2026

Optimal QAM Constellation for Over-the-Air Computation in the Presence of Heavy-Tailed Channel Noise

Over-the-air computation (OAC) enables low-latency aggregation over multiple-access channels (MACs) by exploiting the superposition property of the wireless medium to compute functions efficiently in distributed networks. A critical but often overlooked challenge is that electromagnetic interference in practical radio channels frequently exhibits heavy-tailed behavior, causing strong impulsive noise that severely degrades computation performance. This work studies digital OAC with QAM-based signaling under heavy-tailed interference modeled by a Cauchy distribution (lacking a finite second moment). We seek QAM-like constellations that minimize the mean-squared error (MSE) of sum aggregation subject to an average-power constraint. The problem is formulated as a constrained optimization, whose solution yields unique optimality conditions. Numerical results confirm the effectiveness of the proposed design. Notably, the framework extends naturally to nomographic functions, broader constellation families, and alternative noise models.

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arXiv2026

VecComp: Vector Computing via MIMO Digital Over-the-Air Computation

Recently, the ChannelComp framework has proposed digital over-the-air computation by designing digital modulations that enable the computation of arbitrary functions. Unlike traditional analog over-the-air computation, which is restricted to nomographic functions, ChannelComp enables a broader range of computational tasks while maintaining compatibility with digital communication systems. This framework is intended for applications that favor local information processing over the mere acquisition of data. However, ChannelComp is currently designed for scalar function computation, while numerous data-centric applications necessitate vector-based computations, and it is susceptible to channel fading. In this work, we introduce a generalization of the ChannelComp framework, called VecComp, by integrating ChannelComp with multiple-antenna technology. This generalization not only enables vector function computation but also ensures scalability in the computational complexity, which increases only linearly with the vector dimension. As such, VecComp remains computationally efficient and robust against channel impairments, making it suitable for high-dimensional, data-centric applications. We establish a non-asymptotic upper bound on the mean squared error of VecComp, affirming its computation efficiency under fading channel conditions. Numerical experiments show the effectiveness of VecComp in improving the computation of vector functions and fading compensation over noisy and fading multiple-access channels.

cs.LG↗

arXiv2026

Function Computation Over Multiple Access Channels via Hierarchical Constellations

We study function computation over a Gaussian multiple-access channel (MAC), where multiple transmitters aim at computing a function of their values at a common receiver. To this end, we propose a novel coded-modulation framework for over-the-air computation (OAC) based on hierarchical constellation design, which supports reliable computation of multiple function outputs using a single channel use. Moreover, we characterize the achievable computation rate and show that the proposed hierarchical constellations can compute R output functions with decoding error probability epsilon while the gap to the optimal computation rate scales as O(\log_2(1/ε)/K) for independent source symbols, where K denotes the number of transmitters. Consequently, this gap vanishes as the network size grows, and the optimal rate is asymptotically attained. Furthermore, we introduce a shielding mechanism based on variable-length block coding that mitigates noise-induced error propagation across constellation levels while preserving the superposition structure of the MAC. We show that the shielding technique improves reliability, yielding a gap that scales optimally as O(\log_2\ln{(1/ε)}), regardless of the source distribution. Together, these results identify the regimes in which uncoded or lightly coded OAC is information-theoretically optimal, providing a unified framework for low-latency, channel-agnostic function computation.

cs.IT↗

arXiv2025

Multi-Symbol Digital AirComp via Modulation Design and Power Adaptation

Recently, over-the-air computation (AirComp) leverages the superposition property of wireless channels to enable efficient function computation over a multiple access channel (MAC). However, existing digital AirComp methods either rely on single-symbol modulation, which limits flexibility and robustness, or on multi-symbol extensions that suffer from high complexity or approximation errors. To overcome these limitations, we propose a new multi-symbol modulation framework, termed sequential modulation for AirComp (SeMAC), which encodes each input into a sequence of symbols with distinct constellation diagrams across multiple time slots. This approach increases design flexibility and robustness against channel noise. Specifically, the modulation design is formulated as a non-convex optimization problem and efficiently solved through a successive convex approximation (SCA) combined with stochastic subgradient descent (SSD). For fixed modulation formats, we further develop SeMAC with power adaptation (SeMAC-PA) to adjusts transmit power and phase while preserving the modulation structure. Notably, numerical results show that SeMAC improves computation accuracy by up to 14 dB compared to the existing methods for computing nonlinear functions such as the product function.

eess.SP↗

arXiv2025

Towards Optimal Constellation Design for Digital Over-the-Air Computation

Over-the-air computation (OAC) has emerged as a key technique for efficient function computation over multiple-access channels (MACs) by exploiting the waveform superposition property of the wireless domain. While conventional OAC methods rely on analog amplitude modulation, their performance is often limited by noise sensitivity and hardware constraints, motivating the use of digital modulation schemes. This paper proposes a novel digital modulation framework optimized for computation over additive white Gaussian noise (AWGN) channels. The design is formulated as an additive mapping problem to determine the optimal constellation that minimizes the mean-squared error (MSE) under a transmit power constraint. We express the optimal constellation design as a system of nonlinear equations and establish the conditions guaranteeing the uniqueness of its solution. In the high signal-to-noise-ratio (SNR) regime, we derive closed-form expressions for the optimal modulation parameters using the generalized Lambert function, providing analytical insight into the system's behavior. Furthermore, we discuss extensions of the framework to higher-dimensional grids corresponding to multiple channel uses, to non-Gaussian noise models, and to computation over real-valued domains via hybrid digital-analog modulation.

cs.IT↗

arXiv2025

Waveforms for Computing Over the Air

Over-the-air computation (AirComp) leverages the signal-superposition characteristic of wireless multiple access channels to perform mathematical computations. Initially introduced to enhance communication reliability in interference channels and wireless sensor networks, AirComp has more recently found applications in task-oriented communications, like wireless distributed learning and in wireless control systems. Its adoption aims to address latency challenges arising from an increased number of edge devices or Internet of Things (IoT) devices accessing the constrained wireless spectrum. This paper is the first one to focus on the physical layer of these systems. We present a unified framework, specifically on the waveform and the signal processing aspects at the transmitter and receiver to meet the challenges that AirComp presents within the different contexts and use cases.

eess.SP↗

arXiv2025

DOA Estimation via Optimal Weighted Low-Rank Matrix Completion

This paper presents a novel method for estimating the direction of arrival (DOA) for a non-uniform and sparse linear sensor array using the weighted lifted structure low-rank matrix completion. The proposed method uses a single snapshot sample in which a single array of data is observed. The method is rooted in a weighted lifted-structured low-rank matrix recovery framework. The method involves four key steps: (i) lifting the antenna samples to form a low-rank stature, then (ii) designing left and right weight matrices to reflect the sample informativeness, (iii) estimating a noise-free uniform array output through completion of the weighted lifted samples, and (iv) obtaining the DOAs from the restored uniform linear array samples. We study the complexity of steps (i) to (iii) above, where we analyze the required sample for the array interpolation of step (iii) for DOA estimation. We demonstrate that the proposed choice of weight matrices achieves a near-optimal sample complexity. This complexity aligns with the problem's degree of freedom, equivalent to the number of DOAs adjusted for logarithmic factors. Numerical evaluations show the proposed method's superiority against the non-weighted counterpart and atomic norm minimization-based methods. Notably, our proposed method significantly improves, with approximately a 10 dB reduction in normalized mean-squared error over the non-weighted method at low-noise conditions.

eess.SP↗

arXiv2025

On Designing Modulation for Over-the-Air Computation -- Part I: Noise-Aware Design

Over-the-air computation (OAC) leverages the physical superposition property of wireless multiple access channels (MACs) to compute functions while communication occurs, enabling scalable and low-latency processing in distributed networks. While analog OAC methods suffer from noise sensitivity and hardware constraints, existing digital approaches are often limited in design complexity, which may hinder scalability and fail to exploit spectral efficiency fully. This two-part paper revisits and extends the ChannelComp framework, a general methodology for computing arbitrary finite-valued functions using digital modulation. In Part I, we develop a novel constellation design approach that is aware of the noise distribution and formulates the encoder design as a max-min optimization problem using noise-tailored distance metrics. Our design supports noise models, including Gaussian, Laplace, and heavy-tailed distributions. We further demonstrate that, for heavy-tailed noise, the optimal ChannelComp setup coincides with the solution to the corresponding max-min criterion for the channel noise with heavy-tailed distributions. Numerical experiments confirm that our noise-aware design achieves a substantially lower mean-square error than leading digital OAC methods over noisy MACs. In Part II, we consider a constellation design with a quantization-based sampling scheme to enhance modulation scalability and computational accuracy for large-scale digital OAC.

eess.SP↗

arXiv2025

On Designing Modulation for Over-the-Air Computation -- Part II: Pyramid Sampling

Over-the-air computation (OAC) harnesses the natural superposition of wireless signals to compute aggregate functions during transmission, thereby collapsing communication and computation into a single step and significantly reducing latency and resource usage. In Part I, digital OAC was formulated as a noise-aware constellation design problem by casting encoder design as a max-min optimization that aligns minimum Euclidean distances between superimposed constellation points with squared differences of their corresponding function outputs. In this paper, Part II, we address the prohibitive complexity and quantization challenges inherent in digital OAC constellation design for large-scale edge networks. More precisely, we introduce a pyramid sampling strategy that judiciously selects a subset of superimposed constellation points to reduce the encoder design complexity from $\mathcal{O}(q^K)$ to $\mathcal{O}(q^{K-p+1})$, where $p\in\{1,\dots, K\}$ denotes the sampling order, $q$ levels of modulation, and $K$ denotes the number nodes in the network. Under the assumption of symmetric aggregation, this approach enables a controlled trade-off between computational complexity and function computation accuracy. As a special case, we propose majority-based sampling ($p=K$), which confines aggregation to only $q$ consensus points, inherently avoiding destructive overlaps and permitting the use of standard digital modulations (e.g., QAM, PSK, ASK) without bespoke constellation designs. We also show via several simulations, across various aggregation functions, modulation levels, and noise levels, that moderate sampling orders attain acceptable performance with orders-of-magnitude fewer constraints than exhaustive designs.

eess.SP↗

arXiv2025

Timely and Painless Breakups: Off-the-grid Blind Message Recovery and Users' Demixing

In the near future, the Internet of Things will interconnect billions of devices, forming a vast network where users sporadically transmit short messages through multi-path wireless channels. These channels are characterized by the superposition of a small number of scaled and delayed copies of Dirac spikes. At the receiver, the observed signal is a sum of these convolved signals, and the task is to find the amplitudes, continuous-indexed delays, and transmitted messages from a single signal. This task is inherently ill-posed without additional assumptions on the channel or messages. In this work, we assume the channel exhibits sparsity in the delay domain and that i.i.d. random linear encoding is applied to the messages at the devices. Leveraging these assumptions, we propose a semidefinite programming optimization capable of simultaneously recovering both messages and the delay parameters of the channels from only a single received signal. Our theoretical analysis establishes that the required number of samples at the receiver scales proportionally to the sum-product of sparsity and message length of all users, aligning with the degrees of freedom in the proposed convex optimization framework. Numerical experiments confirm the efficacy of the proposed method in accurately estimating closely-spaced delay parameters and recovering messages.

cs.IT↗

arXiv2025

ReMAC:Digital Multiple Access Computing by Repeated Transmission

In this paper, we consider the ChannelComp framework, where multiple transmitters aim to compute a function of their values at a common receiver while using digital modulations over a multiple access channel. ChannelComp provides a general framework for computation by designing digital constellations for over-the-air computation. Currently, ChannelComp uses a symbol-level encoding. However, encoding repeated transmissions of the same symbol and performing the function computation using the corresponding received sequence may significantly improve the computation performance and reduce the encoding complexity. In this paper, we propose a new scheme where each transmitter repeats the transmission of the same symbol over multiple time slots while encoding such repetitions and designing constellation diagrams to minimize computational errors. We formally model such a scheme by an optimization problem, whose solution jointly identifies the constellation diagram and the repetition code. We call the proposed scheme ReMAC. To manage the computational complexity of the optimization, we divide it into two tractable subproblems. We verify the performance of ReMAC by numerical experiments. The simulation results reveal that ReMAC can reduce the computation error in noisy and fading channels by approximately up to 7.5$dB compared to standard ChannelComp, particularly for product functions.

eess.SP↗

arXiv2024

A Novel Channel Coding Scheme for Digital Multiple Access Computing

In this paper, we consider the ChannelComp framework, which facilitates the computation of desired functions by multiple transmitters over a common receiver using digital modulations across a multiple access channel. While ChannelComp currently offers a broad framework for computation by designing digital constellations for over-the-air computation and employing symbol-level encoding, encoding the repeated transmissions of the same symbol and using the corresponding received sequence may significantly improve the computation performance and reduce the encoding complexity. In this paper, we propose an enhancement involving the encoding of the repetitive transmission of the same symbol at each transmitter over multiple time slots and the design of constellation diagrams, with the aim of minimizing computational errors. We frame this enhancement as an optimization problem, which jointly identifies the constellation diagram and the channel code for repetition, which we call ReChCompCode. To manage the computational complexity of the optimization, we divide it into two tractable subproblems. Through numerical experiments, we evaluate the performance of ReChCompCode. The simulation results reveal that ReChCompCode can reduce the computation error by approximately up to 30 dB compared to standard ChannelComp, particularly for product functions.

eess.SP↗

arXiv2024

Blind Federated Learning via Over-the-Air q-QAM

In this work, we investigate federated edge learning over a fading multiple access channel. To alleviate the communication burden between the edge devices and the access point, we introduce a pioneering digital over-the-air computation strategy employing q-ary quadrature amplitude modulation, culminating in a low latency communication scheme. Indeed, we propose a new federated edge learning framework in which edge devices use digital modulation for over-the-air uplink transmission to the edge server while they have no access to the channel state information. Furthermore, we incorporate multiple antennas at the edge server to overcome the fading inherent in wireless communication. We analyze the number of antennas required to mitigate the fading impact effectively. We prove a non-asymptotic upper bound for the mean squared error for the proposed federated learning with digital over-the-air uplink transmissions under both noisy and fading conditions. Leveraging the derived upper bound, we characterize the convergence rate of the learning process of a non-convex loss function in terms of the mean square error of gradients due to the fading channel. Furthermore, we substantiate the theoretical assurances through numerical experiments concerning mean square error and the convergence efficacy of the digital federated edge learning framework. Notably, the results demonstrate that augmenting the number of antennas at the edge server and adopting higher-order modulations improve the model accuracy up to 60\%.

eess.SP↗

arXiv2023

Harmonic Retrieval Using Weighted Lifted-Structure Low-Rank Matrix Completion

In this paper, we investigate the problem of recovering the frequency components of a mixture of $K$ complex sinusoids from a random subset of $N$ equally-spaced time-domain samples. Because of the random subset, the samples are effectively non-uniform. Besides, the frequency values of each of the $K$ complex sinusoids are assumed to vary continuously within a given range. For this problem, we propose a two-step strategy: (i) we first lift the incomplete set of uniform samples (unavailable samples are treated as missing data) into a structured matrix with missing entries, which is potentially low-rank; then (ii) we complete the matrix using a weighted nuclear minimization problem. We call the method a \emph{ weighted lifted-structured (WLi) low-rank matrix recovery}. Our approach can be applied to a range of matrix structures such as Hankel and double-Hankel, among others, and provides improvement over the unweighted existing schemes such as EMaC and DEMaC. We provide theoretical guarantees for the proposed method, as well as numerical simulations in both noiseless and noisy settings. Both the theoretical and the numerical results confirm the superiority of the proposed approach.

eess.SP↗

arXiv2023

ChannelComp: A General Method for Computation by Communications

Over-the-air computation (AirComp) is a well-known technique by which several wireless devices transmit by analog amplitude modulation to achieve a sum of their transmit signals at a common receiver. The underlying physical principle is the superposition property of the radio waves. Since such superposition is analog and in amplitude, it is natural that AirComp uses analog amplitude modulations. Unfortunately, this is impractical because most wireless devices today use digital modulations. It would be highly desirable to use digital communications because of their numerous benefits, such as error correction, synchronization, acquisition of channel state information, and widespread use. However, when we use digital modulations for AirComp, a general belief is that the superposition property of the radio waves returns a meaningless overlapping of the digital signals. In this paper, we break through such beliefs and propose an entirely new digital channel computing method named ChannelComp, which can use digital as well as analog modulations. We propose a feasibility optimization problem that ascertains the optimal modulation for computing arbitrary functions over-the-air. Additionally, we propose pre-coders to adapt existing digital modulation schemes for computing the function over the multiple access channel. The simulation results verify the superior performance of ChannelComp compared to AirComp, particularly for the product functions, with more than 10 dB improvement of the computation error.

eess.SP↗

arXiv2023

Optimal Receive Filter Design for Misaligned Over-the-Air Computation

Over-the-air computation (OAC) is a promising wireless communication method for aggregating data from many devices in dense wireless networks. The fundamental idea of OAC is to exploit signal superposition to compute functions of multiple simultaneously transmitted signals. However, the time- and phase-alignment of these superimposed signals have a significant effect on the quality of function computation. In this study, we analyze the OAC problem for a system with unknown random time delays and phase shifts. We show that the classical matched filter does not produce optimal results, and generates bias in the function estimates. To counteract this, we propose a new filter design and show that, under a bound on the maximum time delay, it is possible to achieve unbiased function computation. Additionally, we propose a Tikhonov regularization problem that produces an optimal filter given a tradeoff between the bias and noise-induced variance of the function estimates. When the time delays are long compared to the length of the transmitted pulses, our filter vastly outperforms the matched filter both in terms of bias and mean-squared error (MSE). For shorter time delays, our proposal yields similar MSE as the matched filter, while reducing the bias.

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