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Stacked Intelligent Metasurfaces for Holographic MIMO Aided Cell-Free Networks

Qingchao Li, Mohammed El-Hajjar, Chao Xu, Jiancheng An, Chau Yuen, Lajos Hanzo

arXiv:2405.09753v1cs.ITeess.SP

TL;DR

The paper addresses practical limitations of large-scale MIMO and cellular HMIMO by proposing a distributed SIM-based HMIMO architecture for uplink cell-free networks. It combines locally optimized SIM beamforming and receiver combining with CPU-level MMSE fusion; simulations show improved achievable rate, while RF-chain hardware impairments limit high-SNR performance.

  • Problem

    Large-scale MIMO and existing HMIMO architectures face high hardware and power costs, imperfect RF hardware, and cell-edge degradation from path loss and inter-cell interference.

  • Method

    The proposed architecture uses layer-by-layer iterative optimization for SIM coefficients and AP receiver combiners, followed by CPU fusion of local detections using MMSE weights that account for hardware impairments.

  • Results

    The SIM-based hybrid beamformer outperforms the conventional full-digital beamformer in low-SNR conditions with imperfect hardware and has higher average achievable rate than full-digital beamforming with perfect hardware.

  • Takeaways & Limitations

    Increasing SIM layers and elements improves achievable rate, but RF-chain impairments at APs and UEs cause achievable-rate saturation in the high-SNR region.

Abstract

from arXiv · show

Large-scale multiple-input and multiple-output (MIMO) systems are capable of achieving high date rate. However, given the high hardware cost and excessive power consumption of massive MIMO systems, as a remedy, intelligent metasurfaces have been designed for efficient holographic MIMO (HMIMO) systems. In this paper, we propose a HMIMO architecture based on stacked intelligent metasurfaces (SIM) for the uplink of cell-free systems, where the SIM is employed at the access points (APs) for improving the spectral- and energy-efficiency. Specifically, we conceive distributed beamforming for SIM-assisted cell-free networks, where both the SIM coefficients and the local receiver combiner vectors of each AP are optimized based on the local channel state information (CSI) for the local detection of each user equipment (UE) information. Afterward, the central processing unit (CPU) fuses the local detections gleaned from all APs to detect the aggregate multi-user signal. Specifically, to design the SIM coefficients and the combining vectors of the APs, a low-complexity layer-by-layer iterative optimization algorithm is proposed for maximizing the equivalent gain of the channel spanning from the UEs to the APs. At the CPU, the weight vector used for combining the local detections from all APs is designed based on the minimum mean square error (MMSE) criterion, where the hardware impairments (HWIs) are also taken into consideration based on their statistics. The simulation results show that the SIM-based HMIMO outperforms the conventional single-layer HMIMO in terms of the achievable rate. We demonstrate that both the HWI of the radio frequency (RF) chains at the APs and the UEs limit the achievable rate in the high signal-to-noise-ratio (SNR) region.

I. INTRODUCTION

HMIMO and related metasurface architectures address the hardware cost and energy consumption of large-scale MIMO, while cell-free networking and distributed processing target coverage and CSI-sharing challenges. This paper proposes SIM-aided cell-free HMIMO with iterative AP beamforming and MMSE CPU fusion under RF hardware impairments.

  • Motivation: Large-scale MIMO increases throughput but requires many RF chains, raising hardware cost and energy consumption.HMIMO uses near-continuous apertures and holographic radios to improve hardware and energy efficiency.
  • Motivation: Cell-free networking deploys distributed APs to reduce path loss and inter-cell interference without cell boundaries.The architecture also reduces CSI-sharing overhead through distributed processing.
  • Proposed architecture: The paper proposes SIM-based uplink HMIMO in which APs locally detect UE data and the CPU fuses local detections.SIM coefficients and receiver combining vectors are optimized locally, while CPU weights account statistically for RF hardware impairments.
  • Beamforming design: A low-complexity layer-by-layer iterative algorithm alternates SIM-coefficient and receiver-combiner optimization until convergence.The SIM coefficients are optimized layer by layer, while receiver vectors use the MRC criterion when SIM coefficients are fixed.
  • Results: The SIM-based architecture outperforms conventional single-layer intelligent-surface HMIMO in average achievable rate, while RF impairments cause high-SNR saturation.The conclusion also reports rate improvements with more SIM layers and elements per layer.

A. SIM-Aided Holographic MIMO Architecture

The cell-free HMIMO AP uses a stacked metasurface beamformer before RF-chain conversion and antenna reception. Its passive layers contain densely spaced reconfigurable elements whose phase shifts are configured to obtain beamforming gain, with near-field channel responses modeled between components.

  • Architecture: The hybrid digital-analog architecture uses few RF chains and a phase-shift array to reduce RF-chain power consumption.The section contrasts this arrangement with fully digital massive MIMO, where RF-chain count matches antenna count.
  • Architecture: Each AP receives signals through a SIM-based beamformer, whose output is captured by antennas and converted to baseband through RF chains.The SIM is a closed structure containing multiple stacked reconfigurable metasurface layers.
  • Metasurface configuration: Each SIM layer contains densely spaced passive elements whose software-controlled phase shifts can produce beamforming gain.The AP antenna array and SIM layers are modeled as uniform rectangular planar arrays with M = M_xM_y and N = N_xN_y elements.
  • Channel structure: The model includes channels between UE-facing SIM elements, inter-layer SIM elements, and AP antennas.Near-field responses are used for the antenna-to-SIM and inter-layer links.

2) Channel links between UEs and APs:

The channel model represents UE-to-AP propagation through large- and small-scale fading, multipath geometry, additive noise, and hardware-induced distortion. Hardware quality factors characterize the severity of impairments at UEs and AP RF chains.

  • UE-to-AP channel: The UE-to-SIM channel includes large-scale fading and small-scale fading between each UE and each AP.The propagation environment is characterized using an mmWave multipath channel model.
  • Multipath model: The mmWave model represents propagation as clustered paths with elevation and azimuth angles of departure for each path.Cluster-specific angular means and spreads describe the path directions.
  • Received signal: The received AP signal includes the desired UE information, transmit power, and additive noise.The equivalent channel spans each UE to the antennas at the AP.
  • Hardware impairments: UE and AP RF hardware impairments introduce information-symbol and RF-chain distortion governed by hardware quality factors.A quality factor of 1 denotes ideal hardware, whereas 0 denotes completely inadequate hardware.

III. BEAMFORMING DESIGN

The beamforming design distributes local detection across APs while jointly optimizing SIM and receiver-combiner parameters. SIM beamforming focuses AP resources toward nearby UEs, and digital combining supports local multi-user detection under hardware distortion.

  • Distributed detection: Each AP performs local UE detection using SIM coefficient matrices and receiver-combiner vectors optimized from local CSI.The CPU later combines local detections from all APs for final recovery.
  • Joint optimization: The design jointly optimizes active receiver-combiner vectors and passive SIM beamformers to maximize each AP’s target-UE SINR.The formulation includes constraints on SIM matrices and receiver-vector norms.
  • Beamforming objective: For a target UE, SIM beamforming focuses on the nearest-user channel gain, while digital beamforming is optimized for all UEs’ local information estimates.The target AP set is defined using UE-to-AP distance.
  • Iterative solution: The non-convex design is decoupled into sub-problems that are optimized iteratively.The resulting local detection produces an estimated symbol and corresponding SINR, including HWI-induced distortion terms.

1) Design of RC vectors at APs:

Given the SIM-based beamformer, each AP estimates the equivalent channels from all UEs and designs its active receiver combiner for local information recovery using MRC.

  • The SIM-based beamformer enables estimation of the equivalent channels from all K UEs to AP-l.
  • The active beamformer b(l)k′ for recovering UE-k′ information at AP-l is optimized using the MRC criterion.

2) Design of SIM coefficient matrices at APs:

The AP SIM coefficients are designed through a layer-by-layer iterative procedure that alternates active-combiner updates with passive SIM-layer optimization to improve the equivalent channel gain.

  • The SIM-coefficient subproblem optimizes the channel gain ||G(l)h(l)k|| after fixing the active beamformer.
  • Because the passive-layer subproblem is non-convex, the algorithm optimizes one SIM layer at a time while fixing the remaining layers.
  • The layer-by-layer iterative optimization procedure is presented as the hybrid-beamformer design algorithm at AP-l.
  • Algorithm 1 initializes passive SIM matrices randomly, alternates channel and active-beamformer updates, and returns optimized active vectors and passive matrices.

B. Weight Vector Design at the CPU

The CPU combines AP-local estimates with a unit-norm weight vector and chooses that vector using a generalized Rayleigh-quotient maximization of the resulting SINR.

  • The CPU forms each UE estimate as a linear combination of the local estimates from the APs.
  • The CPU weight vector ηk assigns weights to local estimates and satisfies ||ηk||2 = 1.
  • The SINR γk of the CPU estimate is maximized using the generalized Rayleigh quotient.

C. Computational Complexity

The complexity analysis counts floating-point multiplications required by the iterative hybrid-beamformer algorithm and establishes polynomial-time complexity under stated dimension conditions.

  • The complexity depends on the alternating-maximization iteration count τ and the operations required in each iteration.
  • The analysis counts floating-point multiplications while neglecting additions because additions are considered readily implementable in hardware.
  • The total per-iteration multiplication count is assembled from the costs of the algorithm's active and passive beamforming loops.
  • For N > M and N > K, the proposed optimization has polynomial-time complexity with respect to APs, SIM layers, and reconfigurable elements per layer.

2) Computational complexity of the CPU processing:

CPU information recovery computes the matrix R_k, the linear combining vector η_k, and the recovered information ŝ_k. The resulting processing has polynomial time-complexity with respect to the numbers of UEs, APs, and RF chains per AP.

  • CPU recovery of s_k requires calculating R_k, the linear combining vector η_k, and the recovered information ŝ_k.These quantities are defined in equations (32), (38), and (29), respectively.
  • R_k requires L2 + 2L + 3M floating-point multiplication operations.The count follows from equations (32)–(35).
  • The linear combining vector η_k is calculated using Cholesky decomposition of the Hermitian positive-definite matrix R_k^-1.
  • The total multiplication count for recovering s_k combines the costs of calculating R_k, η_k, and ŝ_k.The total is obtained according to equations (46)–(48).
  • Recovering s_1, s_2, · · ·, s_K at the CPU has polynomial time-complexity in the number of UEs, APs, and RF chains at each AP.

D. Convergence Analysis

The layer-by-layer alternating optimization produces a bounded, non-decreasing SINR sequence and is therefore guaranteed to converge. Simulations show how AP count, SIM layers, hardware quality, antenna count, spacing, iterations, and phase quantization affect achievable rate.

  • Convergence Analysis: The received SINR is non-decreasing after separately optimizing the digital combining vectors and holographic SIM beamformer.The two alternating subproblems optimize the digital beamformer with fixed SIM coefficients, then the SIM coefficients with fixed digital beamformer.
  • Convergence Analysis: Because the objective sequence is monotonic and bounded, the proposed layer-by-layer iterative optimization algorithm is guaranteed to converge.
  • Simulation Results: With imperfect hardware, achievable rate saturates as transmit power increases, and increasing SIM layers can degrade high-SNR performance when layer radiation attenuates signals.The SIM-based hybrid beamformer outperforms full-digital beamforming in the low-SNR region, but hardware quality limits high-SNR performance.
  • Simulation Results: Increasing the number of SIM layers improves average achievable rate when the total number of SIM elements is held constant.The comparison uses 1, 2, and 4 layers with 16 × 16, 8 × 8, and 4 × 4 elements per layer, respectively.
  • Simulation Results: Optimal inter-layer distance increases with more surface elements, while increasing AP antennas improves rate but requires higher energy consumption.For small surfaces, inter-layer path loss favors shorter spacing; with more elements, larger spacing improves signal radiation between adjacent layers.
  • Simulation Results: The algorithm converges within 10 iterations, while four-bit phase-shift quantization approaches the rate of infinite phase-shift resolution.Convergence slows as the number of SIM layers increases.

V. CONCLUSIONS

The paper presents a distributed uplink SIM-based cell-free HMIMO architecture with local AP detection and CPU-level fusion. Its achievable rate improves with SIM scaling but saturates at high SNR because of RF-chain hardware impairments.

  • Each AP performs local UE detection using jointly optimized SIM coefficients and receiver-combining vectors, while the CPU fuses detections from all APs.The AP optimization uses a low-complexity layer-by-layer iterative algorithm, and CPU combining follows the MMSE criterion while accounting for RF-chain hardware impairments.
  • The achievable rate improves as the number of SIM layers and elements per layer increases.
  • At high SNR, hardware impairments at both APs and UEs cause the achievable rate to saturate.
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