Two-dimensional Fourier Compressed Sensing Under a Fixed Readout Budget Per Channel

Jul 16, 2026

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Recovering sparse signals from subsampled Fourier data is central to communications, radar, and imaging. This letter studies reconstruction of sparse 2D signals (matrices) when each channel is limited to a fixed number of Fourier samples, such as sampling only K entries per row or column. A lower bound is derived for the mutual coherence of the resulting compressed sensing matrix, shown to exceed the classical Welch bound under restricted readout. Deterministic subsampling patterns meeting the bound are constructed and validated against random sampling via simulations.

LanguageEnglish

CategoryResearch & Report

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Pages5

Reading Time13 min

Introduction

2D-Fourier CS under a Readout Budget

Two-dimensional Fourier compressed sensing under a fixed readout budget per channel Nitin Jonathan Myers, Senior Member, IEEE Delft Center for Systems and Control, Delft University of Technology arXiv :2607 .03611v1 [ ee ss . SP] 3 Jul 2026 Abstract—Recovering sparse signals from their subsampled Fourier representation is an important problem in communications, radar, and imaging. In this letter, we focus on reconstructing sparse 2D signals (matrices) under the constraint that only a fixed number of entries can be sampled from each channel, e.g., a row or a column in the Fourier domain. For a specified perchannel readout budget, we derive a lower bound on the mutual coherence of the corresponding compressed sensing matrix. We show that our bound is larger than the classical Welch bound, due to a limited readout budget. We also construct deterministic subsampling patterns that attain this bound for a class of matrix dimensions and readout budgets, and benchmark them against random subsampling through simulations. Index Terms—Multi-dimensional sampling, cyclic difference sets, Zadoff-Chu sequences, point spread function, circular shifts I. INTRODUCTION Sparse signals can be recovered from fewer samples in their Fourier representation using compressed sensing (CS) [1] . In some 2D-CS settings, hardware constraints impose a readout limit per channel. For example, analog correlator-based radars acquire code-domain measurements using correlator banks [2], [3] . In a MIMO analog correlator-based radar, chip-area constraints can limit the number of physical analog correlators per receiver and hence the number of measurements acquired per receive channel [4] . Similar per-channel readout limits arise in stepped-frequency radar and imaging systems [5], [6] . The measurement pattern, called subsampling pattern in CS, directly influences the reconstruction quality. For instance, the subsampling pattern determines the mutual coherence of the CS matrix; lower coherence reduces the upper bound on the reconstruction error [7] with the orthogonal matching pursuit (OMP) [8] . Therefore, prior work on Fourier CS in [9],[10] has designed subsampling patterns to minimize mutual coherence, whose fundamental limit is given by the Welch bound [11] . For CS of 1D signals (vectors), special partial Fourier sensing matrices based on cyclic difference sets [12] and almost difference sets [13], [14] are known to achieve or approach the Welch bound; these matrices exist only for certain combinations of the vector’s dimension and the total measurement budget. When such constructions do not exist, numerical methods to minimize coherence-related objectives were developed in [15],[16] . Recent work has also considered 2D subsampling pattern design using difference sets [17] and data-driven methods [18]–[21] . These works, however, do not address the per-channel readout constraint considered in this paper. The closest related work is [22], which designs 2Dsubsampling patterns under a fixed readout constraint, but only for the special case of one readout per channel. This paper generalizes the construction in [22] to the case where multiple measurements can be acquired from each channel. Specifically, we consider the recovery of a P × Q matrix, whose 2D-discrete Fourier transform (2D-DFT) is sparse, when only K distinct entries can be sampled from each row. Our contributions are listed below. • We derive a tight lower bound on the mutual coherence of the CS matrix under a per-channel readout budget and show that this bound is strictly larger than the classical Welch bound for a comparable total measurement budget. • We construct deterministic subsampling patterns that attain the proposed mutual-coherence bound for a specified per-channel readout budget. Our construction applies when P = Q, P is prime, and Q − 1 divides K (K − 1), as it relies on cyclic difference sets [23] and prime-length Zadoff-Chu sequences [24] . • We show, using simu

FAQs

How are the proposed subsampling patterns evaluated?

Simulations compare the proposed deterministic patterns with comparable random patterns under the same per-channel readout budget, demonstrating lower signal reconstruction error for the constructed patterns.

Jul 16, 2026

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