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.

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.

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, quadrature amplitude modulation (QAM) and pulse-amplitude modulation (PAM), for which we introduce a novel coding scheme called SumComp. Furthermore, we derive a mean squared error (MSE) analysis for SumComp coding in the computation of the arithmetic mean function and establish an upper bound on the mean absolute error (MAE) for a set of nomographic functions. Simulation results are presented to affirm the superior performance of SumComp coding compared to traditional analog over-the-air computation and the original coding in ChannelComp approaches in terms of both MSE and MAE over a noisy multiple access channel. Specifically, SumComp coding shows at least 10 dB improvements for computing arithmetic and geometric mean on the normalized MSE for low noise scenarios.

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 Repetition for Multiple Access Computing (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 4.5 dB compared to standard ChannelComp, particularly for the max function.

The Internet of Things interconnects billions of devices and forms 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 independent and identically distributed 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 lifting-type optimization frameworks. Numerical experiments confirm the efficacy of the proposed method in accurately estimating closely-spaced delay parameters and recovering messages.

In this paper, we consider the ChannelComp frame-work, which facilitates the computation of desired functions by multiple transmitters over a common receiver using digital mod-ulations 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 ReCh-CompCode can reduce the computation error by approximately up to 30 dB compared to standard ChannelComp, particularly for product functions.

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.

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%.

We consider the problem of gridless blind deconvolution and demixing (GB2D) in scenarios where multiple users communicate messages through multiple unknown channels, and a single base station (BS) collects their contributions. This scenario arises in various communication fields, including wireless communications, the Internet of Things, over-the-air computation, and integrated sensing and communications. In this setup, each user's message is convolved with a multi-path channel formed by several scaled and delayed copies of Dirac spikes. The BS receives a linear combination of the convolved signals, and the goal is to recover the unknown amplitudes, continuous-indexed delays, and transmitted waveforms from a compressed vector of measurements at the BS. However, without prior knowledge of the transmitted messages and channels, GB2D is highly challenging and intractable in general. To address this issue, we assume that each user's message follows a distinct modulation scheme living in a known low-dimensional subspace. By exploiting these subspace assumptions and the sparsity of the multipath channels for different users, we transform the nonlinear GB2D problem into a matrix tuple recovery problem from a few linear measurements. To achieve this, we propose a semidefinite programming optimization that exploits the specific low-dimensional structure of the matrix tuple to recover the messages and continuous delays of different communication paths from a single received signal at the BS. Finally, our numerical experiments show that our proposed method effectively recovers all transmitted messages and the continuous delay parameters of the channels with sufficient samples.

Over-the-air computation (AirComp) is a known technique in which wireless devices transmit values by analog amplitude modulation so that a function of these values is computed over the communication channel at a common receiver. The physical reason is the superposition properties of the electromagnetic waves, which naturally return sums of analog values. Consequently, the applications of AirComp are almost entirely restricted to analog communication systems. However, the use of digital communications for over-the-air computations would have several benefits, such as error correction, synchronization, acquisition of channel state information, and easier adoption by current digital communication systems. Nevertheless, a common belief is that digital modulations are generally unfeasible for computation tasks because the overlapping of digitally modulated signals returns signals that seem to be meaningless for these tasks. This paper breaks through such a belief and proposes a fundamentally new computing method, named ChannelComp, for performing over-the-air computations by any digital modulation. In particular, we propose digital modulation formats that allow us to compute a wider class of functions than AirComp can compute, and we propose a feasibility optimization problem that ascertains the optimal digital modulation for computing functions over-the-air. The simulation results verify the superior performance of ChannelComp in comparison to AirComp, particularly for the product functions, with around 10 dB improvement of the computation error.

Resource allocation and multiple access schemes are instrumental for the success of communication networks, which facilitate seamless wireless connectivity among a growing population of uncoordinated and non-synchronized users. In this paper, we present a novel random access scheme that addresses one of the most severe barriers of current strategies to achieve massive connectivity and ultra reliable and low latency communications for 6G. The proposed scheme utilizes wireless channels' angular continuous group-sparsity feature to provide low latency, high reliability, and massive access features in the face of limited time-bandwidth resources, asynchronous transmissions, and preamble errors. Specifically, a reconstruction-free goal oriented optimization problem is proposed which preserves the angular information of active devices and is then complemented by a clustering algorithm to assign active users to specific groups. This allows to identify active stationary devices according to their line of sight angles. Additionally, for mobile devices, an alternating minimization algorithm is proposed to recover their preamble, data, and channel gains simultaneously, enabling the identification of active mobile users. Simulation results show that the proposed algorithm provides excellent performance and supports a massive number of devices. Moreover, the performance of the proposed scheme is independent of the total number of devices, distinguishing it from other random access schemes. The proposed method provides a unified solution to meet the requirements of machine-type communications and ultra reliable and low latency communications, making it an important contribution to the emerging 6G networks.

In this paper, we study the problem of estimating the direction of arrival (DOA) using a sparsely sampled uniform linear array (ULA). Based on an initial incomplete ULA measurements, our strategy is to choose a sparse subset of array elements for measuring the next snapshot. Then, we use a Hankel-structured matrix completion to interpolate for the missing ULA measurements. Finally, the source DOAs are estimated using a subspace method such as Prony on the fully recovered ULA. We theoretically provide a sufficient bound for the number of required samples (array elements) for perfect recovery. The numerical comparisons of the proposed method with existing techniques such as atomic-norm minimization and off-the-grid approaches confirm the superiority of the proposed method.

In this letter, we consider the problem of blind source separation under certain nonlinear mixing conditions using a deep learning approach. Conventionally, the separation of sources within linear mixtures is achieved by applying the independence property of the sources. In the nonlinear regime, however, this property is no longer sufficient. In this letter, we consider nonlinear mixing operators where the non-linearity could be fairly approximated using a Taylor series. Next, for solving the nonlinear BSS problem, we design an end-to-end recurrent neural network (RNN) that learns the inverse of the system, and ultimately separates the sources. For training the RNN, we employ a set of multi-variate polynomial functions to simulate the Taylor expansion of the nonlinear mixture. Numerical experiments show that the proposed method successfully separates the sources with a performance superior to the state of the art approaches.

With the dominance of digital imaging systems, we are often dealing with discrete-domain samples of an analog image. Due to physical limitations, all imaging devices apply a blurring kernel on the input image before taking samples to form the output pixels. In this paper, we focus on the reconstruction of binary shape images from few blurred samples. This problem has applications in medical imaging, shape processing, and image segmentation. Our method relies on representing the analog shape image in a discrete grid much finer than the sampling grid. We formulate the problem as the recovery of a rank r matrix that is formed by a Hankel structure on the pixels. We further propose efficient ADMM-based algorithms to recover the low-rank matrix in both noiseless and noisy settings. We also analytically investigate the number of required samples for successful recovery in the noiseless case. For this purpose, we study the problem in the random sampling framework, and show that with O(r log4(n1n2)) random samples (where the size of the image is assumed to be n1 x n2) we can guarantee the perfect reconstruction with high probability under mild conditions. We further prove the robustness of the proposed recovery in the noisy setting by showing that the reconstruction error in the noisy case is bounded when the input noise is bounded. Simulation results confirm that our proposed method outperform the conventional total variation minimization in the noiseless settings.