User-Heterogeneous Cell-Free Massive MIMO Downlink and Uplink Beamforming via Tensor Decomposition

Abstract

We consider a cell-free massive MIMO (CF-mMIMO) system in which multiple access points (APs), connected to a common central processing unit (CPU) through unbounded fronthaul, collaboratively serve multiple users in a heterogeneous scenario in which each user equipment (UE) has a different number of antennas, and therefore is capable of communicating via distinct numbers of digital streams. For such a user-heterogeneous system, new joint transmit (TX)/receive (RX) beamforming (BF) algorithms are then proposed, both for downlink and uplink modes and integrated with two alternative transmit (TX) power and spatial resource allocation strategies, which enable interference-free communications. To that end, a novel tensor decomposition scheme is presented, based on an orthogonality-enforcing modification of the recently-proposed multilinear generalized singular value decomposition (ML-GSVD). Simulation results show both that the new orthogonality-enforcing ML-GSVD (OEML-GSVD) achieves greater accuracy than the previous multilinear generalized singular value decomposition (ML-GSVD) without sacrificing convergence speed, and that the corresponding OEML-GSVD-based proposed beamformers outperform state-of-the-art (SotA) techniques, as well as an equivalent beamformer based on the previous ML-GSVD alternative.INDEX TERMS Cell-free massive MIMO (CF-mMIMO), beamforming design, tensor factorization, multilinear generalized singular value decomposition (ML-GSVD).

Robust Tensor Decomposition for Heterogeneous Beamforming Under Imperfect Channel State Information

“…For future convenience, the indices of all APs and UEs will be collected in the sets L ≜ {1, • • • , L} and K ≜ {1, • • • , K}, respectively. Following preceding work on CF-mMIMO [1]- [5] and relying on channel reciprocity, it is assumed that the network operates in time division duplex (TDD) fashion, with UL training phase and subsequent DL and UL data communication phases. It is furthermore assumed, however, that the CSI acquired during the UL training phase is subject to channel estimation errors, with the CSI imperfection modeled as additive noise [17], [18].…”

“…, LN }, and 0 < ε ≪ 1. Furthermore, in order to ensure that the resources allocated to different users are orthogonal [5], all but M k elements of the k-th row of C are zero-forced, i.e., c…”

“…To be truly useful, however, CF-mMIMO systems must be able to handle users with arbitrary and distinct numbers of antennas [8], [9]. To that end, a joint transmit (TX) and receive (RX) BF scheme for user-heterogeneous CF-mMIMO systems based on a tensor decomposition dubbed the orthogonality-enforcing multilinear generalized singular value decomposition (OEML-GSVD) was proposed in [5]. In that design, the instantaneous channel tensor was decomposed by the proposed OEML-GSVD, enabling the available orthogonalized subspaces to be exclusively allocated to each UE.…”

“…Moreover, since BF optimization based on singular value decomposition (SVD)like methods strongly relies on knowledge of instantaneous channel structure, practically achievable SE can be significantly degraded in the presence of CSI imperfection [13]. Indeed, also the OEML-GSVD-based BF design of [5] is heavily affected by CSI imperfection because it is fundamentally a generalization of the SVD-BF of [14], from the earlier point-to-point multiple-input multiple-output (MIMO) setting to more recent heterogeneous multi-user CF-mMIMO setting. It follows from the latter that improving the robustness against CSI imperfection is crucial for practical effectiveness of BF design in CF-mMIMO systems.…”

“…In turn, tensor decomposition algorithms such as OEML-GSVD and the earlier multilinear generalized singular value decomposition (ML-GSVD) [15], [16] have inherent decomposition errors even if perfect CSI is available, such that accuracy of the tensor-based BF approach is exposed to both sources of degradation. This article aims to mitigate this issue, by adding to the OEML-GSVD-based BF design method of [5] robustness to both CSI imperfection and to the tensor decomposition error, yielding a new robust OEML-GSVD (ROEML-GSVD) scheme and corresponding TX and RX BF for both downlink (DL) and uplink (UL) transmissions. The main contributions of this article can be summarized as follows:…”