Low‐Complexity separable beamformers for massive antenna array systems

Abstract

Future cellular systems will likely employ massive bi-dimensional arrays to improve performance by large array gain and more accurate spatial filtering, motivating the design of low-complexity signal processing methods. We propose optimising a Kroneckerseparable beamforming filter that takes advantage of the bi-dimensional array geometry to reduce computational costs. The Kronecker factors are obtained using two strategies: alternating optimisation, and sub-array minimum mean square error beamforming with Tikhonov regularization. According to the simulation results, the proposed methods are computationally efficient but come with source recovery degradation, which becomes negligible when the sources are sufficiently separated in space.

Separable Least-Mean Squares Beamforming

“…To address the computational complexity issue and to exploit the URA separability, we have put forward separable extensions of the classical Wiener filter in [6]. Specifically, we proposed optimizing a separable beamforming filter w = w v ⊗ w h , with w h ∈ C N h and w v ∈ C Nv by calculating deterministic gradients of (4).…”

“…The first solution, referred to as TLMS, was first introduced in [7], [12] by Rupp and Schwarz to tackle the slow convergence issue of the NLMS algorithm. The second solution, hereafter called ATLMS consists of a different implementation of the TLMS algorithm inspired by the alternating minimization strategy of [4], [6], [8]. In the remainder of this section, we present the TLMS and ATLMS algorithms and briefly comment on their stability, convergence and computational complexity properties.…”

“…This present work is a follow-up of our contributions [4], [6] and should be regarded as a proof-of-concept paper, in which we sketch the potential of separable adaptive filters to the large-scale array beamforming problem. In [4], [6], we have employed deterministic gradients in the calculation of the separable beamformers. We now adopt the stochastic gradient technique, which leads to the LMS solution.…”

“…Simulation results show that the proposed filter, which is based on an alternating minimization strategy, exhibits modest computational complexity reduction compared to the classical Wiener beamformer [5]. In the follow-up work [6], we obtain analytical expressions for the separable filter of [4], which is based on sample estimates, and conduct an asymptotic performance analysis. In [7], the authors propose a separable least mean squares (LMS) algorithm to identify a second-order separable system, i.e., a linear and time-invariant system whose impulse response h can be well approximated as a Kronecker product h ≈ a ⊗ b.…”

Low-Rank Tensor MMSE Equalization

“…In previous works [3], [4], we consider rank-1 separable tensor filters, i.e., W is written as an outer product of vectors. However, such a structure is too strict for some applications.…”