Learning Sparsifying Transforms for mmWave Communication via $\ell^4$-Norm Maximization

TL;DR

This paper investigates whether the discrete Fourier transform (DFT) is the best sparsifying transform for mmWave MIMO systems and proposes a method to learn improved transforms using $ ext{ell}^4$-norm maximization, supported by theory and simulations.

Contribution

It extends dictionary learning via $ ext{ell}^4$-norm maximization to complex channels and develops algorithms to evaluate and improve sparsifying transforms for mmWave channels.

Findings

DFT may not be optimal for channel sparsification.

Proposed algorithms can learn transforms that better sparsify channel vectors.

The learned transforms outperform DFT in simulations.

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.