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Stacked Intelligent Metasurfaces for Efficient Holographic MIMO Communications in 6G

Jiancheng An, Chao Xu, Derrick Wing Kwan Ng, George C. Alexandropoulos, Chongwen Huang, Chau Yuen, Lajos Hanzo

arXiv:2305.08079v1cs.ITeess.SP

TL;DR

The paper tackles practical HMIMO implementation by using stacked intelligent metasurfaces instead of excessive RF-chain processing. It jointly optimizes TX- and RX-SIM phase shifts for diagonal channel fitting, analyzes capacity bounds, and reports performance gains over benchmark systems, including a 150% capacity gain in the stated comparison.

  • Problem

    Existing HMIMO research lacks practical implementations, while single-layer metasurfaces face hardware constraints and RIS links suffer severe two-hop path loss.

  • Method

    The paper integrates SIMs at the transmitter and receiver, optimizes their layer phase shifts to fit a diagonal end-to-end channel, and solves the non-convex problem with gradient descent.

  • Results

    150% capacity gain was attained over the conventional massive MIMO and RIS-assisted counterparts, while simulations also found strong channel fitting with a 7-layer SIM.

  • Takeaways & Limitations

    The multilayer SIM structure can carry out signal processing in the wave domain while providing spatial gains with fewer RF chains.

Abstract

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The revolutionary technology of \emph{Stacked Intelligent Metasurfaces (SIM)} has been recently shown to be capable of carrying out advanced signal processing directly in the native electromagnetic (EM) wave domain. An SIM is fabricated by a sophisticated amalgam of multiple stacked metasurface layers, which may outperform its single-layer metasurface counterparts, such as reconfigurable intelligent surfaces (RISd) and metasurface lenses. We harness this new SIM concept for implementing efficient holographic multiple-input multiple-output (HMIMO) communications that dot require excessive radio-frequency (RF) chains, which constitutes a substantial benefit compared to existing implementations. We first present an HMIMO communication system based on a pair of SIMs at the transmitter (TX) and receiver (RX), respectively. In sharp contrast to the conventional MIMO designs, the considered SIMs are capable of automatically accomplishing transmit precoding and receiver combining, as the EM waves propagate through them. As such, each information data stream can be directly radiated and recovered from the corresponding transmit and receive ports. Secondly, we formulate the problem of minimizing the error between the actual end-to-end SIMs'parametrized channel matrix and the target diagonal one, with the latter representing a flawless interference-free system of parallel subchannels. This is achieved by jointly optimizing the phase shifts associated with all the metasurface layers of both the TX-SIM and RX-SIM. We then design a gradient descent algorithm to solve the resultant non-convex problem. Furthermore, we theoretically analyze the HMIMO channel capacity bound and provide some useful fundamental insights. Extensive simulation results are provided for characterizing our SIM-based HMIMO system, quantifying its substantial performance benefits.

I. INTRODUCTION

The paper addresses practical HMIMO implementation by integrating stacked intelligent metasurfaces with transceivers to perform wave-domain processing and form physical parallel subchannels. It develops a channel-fitting method, capacity analysis, and simulations to characterize the resulting system and compare it with existing schemes.

  • I. INTRODUCTION: 6G networks face increasingly demanding heterogeneous Internet-of-Everything services, motivating communication technologies with higher data rates and wider connectivity.The introduction cites an expected 500 billion connected devices by 2030.
  • C. Contributions: SIMs cascade multiple metasurfaces to implement signal processing in the electromagnetic-wave regime and are integrated at both transmitter and receiver for HMIMO.The proposed framework targets spatial gains while reducing the number of transmit and receive RF chains.
  • B. Motivation: Existing HMIMO research lacks practical implementations, while single-layer metasurfaces remain limited by tunable amplitude and phase constraints.The paper motivates multilayer architectures as a way to increase spatial gain and design degrees of freedom.
  • C. Contributions: The paper formulates end-to-end diagonal-channel fitting through jointly optimized metasurface phase shifts and solves the resulting constrained non-convex problem with gradient descent.The design aims to let spatial streams be radiated and recovered independently through interference-free parallel subchannels.
  • C. Contributions: The study derives HMIMO channel-capacity bounds and scaling laws, then uses numerical experiments to assess fitting, capacity, SIM design, and comparisons with conventional and RIS-aided schemes.The numerical study is organized around channel fitting, capacity, and performance comparisons.

E. Notations

The paper models SIM-based HMIMO transmission with stacked TX/RX metasurfaces that perform wave-domain precoding and combining, creating parallel subchannels in physical space. It defines the metasurface geometry, phase-shift coefficients, notation, practical calibration caveats, and the point-to-point scope.

  • Transmission paradigm: Unlike conventional MIMO, SIM-based HMIMO forms parallel subchannels in the physical space rather than the eigenspace.Conventional MIMO relies on digital precoding and receiver combining, whereas SIM-based HMIMO processes streams through the wave domain.
  • System model: SIM-based HMIMO uses stacked metasurface layers at both transmitter and receiver to perform precoding and combining as EM waves propagate.The TX/RX-SIMs are closed containers with controllable meta-atoms connected to a smart controller.
  • System benefits: The design reduces the number of active RF chains and supports low-precision, power-efficient DACs and ADCs for individually processed data streams.The paper gives 1-bit resolution for BPSK as an example of low-precision conversion.
  • Notations: Each TX and RX metasurface layer is represented by transmission-coefficient vectors and matrices whose phase shifts lie in [0, 2π).The coefficients are indexed by meta-atom and layer, with separate notation for TX and RX layers.
  • Geometry assumptions: The model assumes isomorphic lattice arrangements, uniform planar arrays, square layouts, and uniform spacing between adjacent metasurface layers.For TX-SIM and RX-SIM, adjacent-layer spacings satisfy dt = Dt/L and dr = Dr/K.
  • Scope and practical caveats: Hardware imperfections, fabrication shortcomings, and modeling errors can alter interlayer transmission coefficients, motivating calibration before deployment.The paper considers point-to-point HMIMO; multiuser zero-forcing precoding and combining are outside its scope.

B. Spatially-Correlated HMIMO Channel Model

The HMIMO channel model accounts for spatial correlation among tightly packed metasurface meta-atoms and incorporates path loss and shadow fading. The correlation matrices depend strongly on the surrounding scattering environments, limiting universal applicability.

  • The channel between the TX-SIM and RX-SIM models spatial correlation among tightly packed meta-atoms.
  • The fading channel is represented using an i.i.d. Rayleigh component together with TX- and RX-side spatial correlation matrices.
  • The TX- and RX-side correlation matrices are expressed from the spatial arrangement of meta-atoms under far-field propagation and isotropic scattering.
  • The transmitter–receiver path loss model includes free-space loss at a reference distance, a path-loss exponent, and Gaussian shadow-fading fluctuations.
  • The spatial correlation matrix depends heavily on the scattering environments around both transceivers, so no universal correlated-fading model generally applies.

C. SIM-Aided HMIMO Channel Capacity with Limited Number of Streams

The section characterizes HMIMO transmission and capacity for a limited number of data streams, then contrasts this digital benchmark with SIMs that form parallel physical subchannels through wave-based precoding and combining.

  • For a fixed channel and number of data streams, optimal HMIMO transmission uses a truncated singular value decomposition policy.
  • SVD-based transmit precoding and receive combining select the leading singular modes, producing a diagonal end-to-end channel for the chosen streams.
  • Water-filling allocates transmit power across data streams subject to the total available power constraint and receiver noise power.
  • The HMIMO channel capacity for a finite number of data streams follows from the selected singular modes and their optimized power allocation.
  • SIMs optimize EM-wave-domain precoding and combining to form parallel physical subchannels between corresponding transmit and receive antennas.
  • SIMs perform precoding and combining at the speed of light through multilayer wave-based computing, rather than relying on digital processing.

A. Problem Formulation

The formulation fits the SIM-parametrized end-to-end channel to a desired diagonal channel by jointly optimizing TX- and RX-SIM phase shifts and a compensating gain. Because the problem is non-convex and strongly coupled, an iterative gradient-descent procedure updates phase shifts, gain, and learning rate while preserving constant-modulus constraints.

  • A. Problem Formulation: The optimization minimizes the Frobenius-norm error between the end-to-end channel H = QGP and the desired diagonal channel Λ1:S,1:S.
  • A. Problem Formulation: The formulation includes a SIM-compensated scaling factor α to support the channel-fitting objective.
  • A. Problem Formulation: The constant-modulus constraints and coupled variables make the channel-fitting problem non-convex and difficult to solve optimally.
  • B. The Proposed Gradient Descent Algorithm: The proposed gradient-descent algorithm differentiates the loss with respect to TX- and RX-SIM phase shifts, preserving constant-modulus constraints throughout iterations.
  • B. The Proposed Gradient Descent Algorithm: The partial derivatives use entries of the actual and target channels together with cascaded channels through selected TX- and RX-SIM meta-atoms.
  • B. The Proposed Gradient Descent Algorithm: Derivative normalization mitigates gradient explosion and vanishing problems, while phase shifts are updated using a learning rate η that sets each step size.
  • B. The Proposed Gradient Descent Algorithm: Given the phase shifts, the scaling factor is updated by least squares using vectorized actual and target channel matrices.
  • B. The Proposed Gradient Descent Algorithm: The learning rate decays exponentially to avoid overshooting, and multiple initial phase-shift sets are tested to reduce entrapment in local optima.

A. HMIMO Channel Capacity Analysis

The paper bounds HMIMO channel capacity and analyzes how it scales with data streams and meta-atoms. Capacity saturates as active streams increase, while channel gain can scale quadratically with meta-atoms in a special case.

  • Capacity bounds: The HMIMO channel capacity is bounded using best- and worst-subchannel assumptions because the exact capacity expression lacks a readily available closed form.The upper and lower bounds use numerical approximations of the first and S-th eigenvalues.
  • Scaling with data streams: Increasing the number of active data streams eventually yields diminishing capacity gains because spatial multiplexing is intrinsically limited.Further increasing active components may severely degrade energy efficiency.
  • Scaling with meta-atoms: For S = 1 and L = K = 1 under i.i.d. Rayleigh fading, Proposition 2 characterizes the asymptotic ergodic-capacity scaling as M, N → ∞.The analysis uses normalized channel coefficients for links through the optimal scatterer.
  • Scaling with meta-atoms: The channel gain follows a quadratic scaling law with the number of meta-atoms, with both TX-SIM and RX-SIM contributing spatial gains.In an ideal setup, doubling meta-atoms at both ends can provide about 4 bps/Hz of capacity improvement.
  • Scope of analysis: Rigorous capacity analysis for arbitrary numbers of metasurface layers remains complex because forward propagation involves many matrix multiplications.Numerical results nevertheless indicate that moderate-layer SIMs can fit the end-to-end channel with high accuracy.

B. Computational Complexity Analysis

The proposed gradient descent solver has an explicitly derived polynomial-time computational complexity. Its cost combines forward propagation, derivative computation, update steps, and the number of iterations.

  • Per-iteration cost: The complexity of Step 1 includes forward propagation and calculation of all partial derivatives, with costs expressed in terms of S, M, N, L, and K.The forward-propagation term is O1−1 = 4SM (ML −M + L) + 4SN (NK −N + K) + 4MSN + 2 (M + N) S2, while derivative computation is O1−2 = 2 (ML + NK) S2.
  • Per-iteration cost: Steps 2 through 5 contribute O2 = 2 (ML + NK) to the computational complexity.The total complexity aggregates these update costs with the Step 1 terms and additional terms O3, O4, and O5.
  • Total complexity: The total algorithmic complexity is obtained by multiplying the aggregate per-iteration cost by the iteration count I.The expression includes the forward-propagation, derivative, and update-step costs.
  • Iteration count: Under the empirical setup, the number of iterations is shown to be less than 20.This iteration count is used to characterize the practical computational burden of the solver.
  • Complexity class: The proposed gradient descent algorithm has polynomial-time complexity when solving Problem (22).The paper introduces numerical experiments to characterize the performance of the SIM-aided HMIMO system.

A. Simulation Setups

The simulations evaluate how SIM architecture and system parameters affect channel fitting and capacity. Results identify favorable layer counts, element spacing, stream counts, and metasurface sizes.

  • Simulation settings: The evaluated configuration uses 28 GHz operation, 0.05 m TX/RX-SIM thicknesses, Pt = 20 dBm, and receiver sensitivity σ2 = −110 dBm.The gradient-descent simulations use 10 random initializations and up to 100 iterations unless otherwise specified.
  • Metasurface layers: At least five metasurface layers reduce channel-fitting NMSE before performance eventually bottoms out.The NMSE and capacity approach optimal values at L = 7; increasing layers further does not improve either metric.
  • Meta-atoms: Doubling the number of meta-atoms in both SIMs yields about 4 bps/Hz of additional capacity.
  • Element spacing: Channel-fitting NMSE is minimized at half-wavelength spacing, while larger or smaller spacing increases channel correlations.The corresponding capacity improves when channel fitting is better because parallel subchannels experience less interference.
  • Data streams: Increasing data streams degrades fitting accuracy, creating a tradeoff that makes capacity peak at a setup-dependent stream count.The reported capacity maxima occur at S = 2, 3, 4, and 6 across the four considered setups.

C. Validation of the Proposed Algorithm

The proposed gradient-descent design is evaluated through convergence, analytical-capacity validation, and comparisons with conventional, massive-MIMO, RIS-aided, and single-layer systems.

  • Algorithm validation: The gradient-fitting NMSE eventually decreases under all tested setups, although a small decay parameter can converge to a local minimum.The convergence study varies the initial learning rate η0 and decay parameter β.
  • Analytical validation: An 8 bps/Hz capacity increase occurs when quadrupling each metasurface from M = N = 25 to M = N = 100 for S = 4.The observed increase is reported as consistent with the paper’s analytical scaling result.
  • Comparison with existing schemes: SIM-assisted HMIMO outperforms massive MIMO and RIS-aided MIMO in all considered setups, including a 150% capacity gain near RIS.At the cell edge, the reported gain may increase to 200%; massive MIMO remains at least 3 bps/Hz below HMIMO.
  • Error performance: Using L = K = 7 metasurface layers effectively mitigates inter-stream interference at a 4 bpcu transmission rate.The error comparison considers four BPSK data streams over a 200 m link.
  • Multilayer comparison: A multilayer SIM fits the target channel better than a single-layer SIM with the same total number of meta-atoms.The single-layer design provides only marginal capacity improvements as meta-atoms increase.

VI. CONCLUSIONS

The paper introduces SIM-based HMIMO that performs precoding and combining in the EM wave domain while reducing RF-chain requirements. Its simulations and analysis identify effective designs and capacity advantages over benchmark schemes.

  • Contributions: The framework jointly optimizes TX-SIM and RX-SIM phase shifts to fit a capacity-optimal diagonal channel matrix.A gradient-descent algorithm solves the resulting fitting problem, alongside a numerical HMIMO capacity approximation and scaling analysis.
  • Design findings: A 7-layer SIM with half-wavelength element spacing achieves excellent channel fitting approaching MIMO capacity.
  • Scaling findings: Doubling the number of meta-atoms produces a quadratic channel gain according to the theoretical analysis and simulations.
  • Performance comparison: The proposed HMIMO attains a reported 150% capacity gain over conventional massive MIMO and RIS-assisted counterparts.The conclusion also identifies multilayer SIMs as wave-domain signal processors with potential implementation-oriented benefits.
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