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.

翻译：毫米波载波频率下波传播的高方向性导致用户设备与基站之间仅存在少量显著传输路径。这种波传播的稀疏特性在波束空间域中得以揭示，传统上通过在基站处的均匀线性天线阵列上进行空间离散傅里叶变换获得，其中每个DFT输出与一个特定波束相关联。近年来，波束空间处理已成为一种有前景的技术，用于降低工作在毫米波频率的全数字大规模多用户多输入多输出系统中的基带复杂度和功耗。然而，对于有限维天线阵列，DFT是否是最优的稀疏化变换仍不明确。本文中，我们将翟等人通过$\ell^4$范数最大化进行完备字典学习的框架扩展至复数情形，以学习新的稀疏化变换。我们为$\ell^4$范数最大化提供了理论基础，并提出了两种适用的学习算法。随后，我们利用这些算法（i）从理论上并通过仿真评估DFT对信道向量进行稀疏化的最优性，以及（ii）为真实世界和合成生成的信道向量学习改进的稀疏化变换。