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

TL;DR

The paper tackles off-the-grid blind deconvolution and demixing for IoT-style short-packet transmissions over sparse multi-path channels. It recasts the nonlinear recovery problem as a linear lifting via atomic-norm minimization, formulating a semidefinite program to jointly recover continuous delays and transmitted messages from a single snapshot. A rigorous sample-complexity bound ties the required measurements to the sum of per-user delay-path counts and message lengths, and dual polynomial certificates are used to identify delays. Numerical experiments validate accurate delay estimation and message recovery under varying sparsity, teamwork, and noise, underscoring the method's potential for scalable ISAC and OAC deployments in massive IoT deployments.

Abstract

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.

Figures (14)

• Figure 1: An illustration of the mathematical model of OBDD problem. User $i$ transmits a signal $x_i(t)$ to the receiver through the channel $h_i(t)$. The signal transmitted by the $i$-th user comprises a message vector, represented by $\mathbf{f}_i \in \mathbb{C}^{k_i \times 1}$, where the elements are derived from a constellation (e.g., amplitude-shift keying modulation (ASK)). This is followed by a redundant encoding matrix $\mathbf{B}_i \in \mathbb{C}^{N \times k_i}$. The channel corresponding to the $i$-th user denoted by $h_i(t)$ is characterized by a sparse number of scaled and delayed Dirac spikes. The receiver observes the contributions of all users in the signal $y(t)$ and takes $N$ samples of its Fourier transform at a rate of $\frac{1}{{\rm B}_{\max}}$ where ${\rm B}_{\max}$ denotes the shared bandwidth utilized by all users. These samples are then collected into a vector denoted by $\mathbf{y} \in \mathbb{C}^{N \times 1}$.
• Figure 2: (a). A typical application of the OBDD model in ISAC where two users transmit signals $x_i(t), i=1,2$ towards a single-antenna receiver via multi-path channels $h_i(t), i=1,2$. The receiver is capable of both sensing its surroundings and decoding the messages of uplink users simultaneously. The signals from each user, reflected by different paths, are depicted with the same color. (b). User $i$ transmits data $\mathbf{f}_i$ through a waveform signal $x_i(t)$ over the MAC for $i\in [r]$. Due to the superposition property of electromagnetic waveforms, the receiver obtains the weighted sum of waveforms from all transmitting users. The receiver employs the OBDD method to decode individual data streams and subsequently computes the function $g_l(f_{1,l}, \ldots, f_{r,l})$ for each $l$ within the set $\{1, \ldots, k\}$, represented as $\mathbf{g}(\mathbf{f}_1, \ldots, \mathbf{f}_r)$.
• Figure 3: (a): Geometric intuition of the proposed optimization problem \ref{['eq.primalprob']}. The blue hyperplane represents the linear constraint \ref{['eq:linear_measure']} which is the feasible set, while the orange-colored object depicts the convex hull of the atoms corresponding to all users. The solution $\widehat{\bm{\mathcal{H}}}$ provided in \ref{['eq.primalprob']} lies at the minimal contour of the convex hull, where it intersects the hyperplane and contains the least number of atoms. (b): An example demonstrating that each matrix $\mathbf{H}_i\in\mathbb{C}^{k_i\times N}$ comprises $s_i=2$ matrix-valued atoms, denoted as $\mathbf{f}_i\mathbf{a}(\tau^i)^{\mathsf{T}}$, formed by the outer product of the message vector $\mathbf{f}_i\in\mathbb{C}^{k_i\times 1}$ and the steering vector $\mathbf{a}(\tau^i)\in\mathbb{C}^{N\times 1}$.
• Figure 4: This figure shows the $\ell_2$ norm of the dual polynomial functions $\mathbf{q}_i(\tau), i\in[r]$ for two users i.e., $r=2$ and the parameters $N=64$, $s_1=2,s_2=1$ and $k_1=k_2 =5$. The peaks of $\|\mathbf{q}_i(\tau)\|_2$ specify the delay parameters corresponding to user $i$.
• Figure 5: A schematic comparison of DoF in the OBDD problem. The Atomic DoF represents the DoF required to jointly recover the channel and messages by reconstructing their outer product, whereas the Independent Factor DoF treats the channel and message signals independently, using an alternating recovery approach

Theorems & Definitions (32)

• Proposition 1
• Remark 1
• Theorem 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Lemma 1
• proof