Learning Sparsifying Transforms for mmWave Communication via ℓ4-Norm Maximization

2026-01-08

arXiv.org

AI Summary

It remains unclear whether the conventional discrete Fourier transform (DFT) constitutes an optimal sparsifying transform for finite-dimensional antenna arrays in millimeter-wave (mmWave) communications. This work proposes a complex-domain ℓ⁴-norm maximization framework that extends real-valued dictionary learning techniques to the complex field, enabling the learning of sparsifying transforms tailored to mmWave MIMO channels. Two efficient algorithms are developed and evaluated on both real-world and synthetic channel data, consistently yielding transform bases that outperform DFT in enhancing beamspace sparsity. The results empirically demonstrate the suboptimality of DFT and provide new theoretical insights and algorithmic tools for mmWave channel modeling and beamforming design.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

The high directionality of wave propagation at millimeter-wave (mmWave) carrier frequencies results in only a small number of significant transmission paths between user equipments and the basestation (BS). This sparse nature of wave propagation is revealed in the beamspace domain, which is traditionally obtained by taking the spatial discrete Fourier transform (DFT) across a uniform linear antenna array at the BS, where each DFT output is associated with a distinct beam. In recent years, beamspace processing has emerged as a promising technique to reduce baseband complexity and power consumption in all-digital massive multiuser (MU) multiple-input multiple-output (MIMO) systems operating at mmWave frequencies. However, it remains unclear whether the DFT is the optimal sparsifying transform for finite-dimensional antenna arrays. In this paper, we extend the framework of Zhai et al. for complete dictionary learning via $\ell^4$-norm maximization to the complex case in order to learn new sparsifying transforms. We provide a theoretical foundation for $\ell^4$-norm maximization and propose two suitable learning algorithms. We then utilize these algorithms (i) to assess the optimality of the DFT for sparsifying channel vectors theoretically and via simulations and (ii) to learn improved sparsifying transforms for real-world and synthetically generated channel vectors.

Problem

• mmWave communication
• sparsifying transform
• DFT optimality
• beamspace processing
• channel sparsity

Innovation

• sparsifying transform
• ℓ⁴-norm maximization
• mmWave communication
• beamspace processing
• dictionary learning