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One-Shot Beam Tracking for Stacked Intelligent Metasurfaces-Assisted LEO Broadcasting

Chandan Kumar Sheemar, Giovanni Iacovelli, Sourabh Solanki, Wali Ullah Khan, Symeon Chatzinotas

arXiv:2610.05520v1eess.SPcs.ITmath.IT

TL;DR

Rapid LEO motion changes the geometry and beam needed to maintain ground illumination, while repeated SIM reoptimization can offset wave-domain processing’s complexity advantage. The paper predicts geometry and uses a one-shot, regularized Gauss–Newton correction to preserve the normalized SNR profile. At t=60 s, it retains approximately 94% and 92% of full-reoptimization rate for the two reported configurations.

  • Problem

    LEO motion rapidly changes satellite–terminal geometry and the required beam, while adapting the many coupled SIM phase variables through repeated reoptimization can offset wave-domain processing’s complexity advantage.

    Source: §I · p. 1

  • Method

    The method propagates predicted orbital geometry into finite-interval SNR variation and uses a multilayer phase Jacobian with one regularized Gauss–Newton correction to preserve the normalized SNR profile.

    Source: §I · p. 1

  • Results

    At t=60 s, the tracker retains approximately 94% and 92% of full-reoptimization rate for (L, N) = (2, 100) and (4, 225), respectively.

    Source: §IV · p. 5

  • Takeaways & Limitations

    The one-shot tracker maintains near-full-reoptimization rates while avoiding iterative optimization.

    Source: §IV · p. 5

    The remaining rate gap is attributed to drift of the fixed reference profile from the max–min optimum, the first-order phase update, clipping, and quantization.

    Source: §IV · p. 5

Abstract

from arXiv · show

This letter develops a physics-informed one-shot beam tracking method for stacked intelligent metasurface (SIM)-enabled low-Earth-orbit (LEO) links with $K$ users and an $L$-layer SIM with $N$ meta-atoms per layer. The predictable orbital motion is propagated through the line-of-sight (LoS) geometry to forecast the finite-interval variation of the user signal-to-noise ratio (SNR) profile, and an analytical multilayer Jacobian relates this variation to the $LN$ SIM phase coefficients. Beam tracking is formulated as normalized SNR-profile preservation and solved through a single regularized Gauss--Newton correction that requires one Jacobian evaluation and a $K\times K$ linear solve, thereby avoiding iterative reoptimization. A tracking-error bound separates the regularization residual from the nonlinear Taylor remainder and establishes a second-order local error under a full-row-rank Jacobian. Numerical results show that the proposed method retains approximately $94\%$ and $92\%$ of the iterative-reoptimization rate for $L=2$ and $L=4$, respectively, while running about $350\times$ and $260\times$ faster at $N=225$, and confirm the predicted second-order residual error behavior.

I. INTRODUCTION

LEO motion changes the geometry and beam needed to maintain ground illumination, while repeatedly reoptimizing a many-phase SIM can erase its computational advantage. The paper uses orbital prediction and a joint multilayer correction to track the normalized SNR profile without iterative reoptimization.

I. INTRODUCTION

  • Rapidly changing satellite–terminal geometry makes beam tracking a signal-processing challenge, while repeated SIM reoptimization can offset the complexity advantage of wave-domain processing.The SIM transformation depends on coupled phase variables across its layers.Source: §I · p. 1
  • Prior work does not provide a multilayer SIM sensitivity model for predictable geometry changes or a noniterative rule for jointly updating all layers.Existing studies focus mainly on static or slowly varying channels, or use conventional array-domain predictive beamforming.Source: §I · p. 1
  • The proposed tracker predicts future geometry and compensates the resulting signal variation in the SIM control wave-domain instead of repeatedly solving a high-dimensional beamforming problem.It replaces iterative SIM reoptimization with a single joint multilayer correction at each tracking instant.Source: §I · p. 1
  • Under a circular Keplerian orbit, an exact two-step position recurrence is propagated through range, direction, Earth rotation, and LVLH attitude to predict finite-interval SNR evolution without linearizing satellite displacement.The recurrence coefficient is fixed by the orbital rate.Source: §I · p. 1
  • Tracking preserves the normalized SNR profile using a closed-form K × LN multilayer phase Jacobian and one regularized Gauss–Newton step that updates all layers jointly.The update needs one Jacobian evaluation and only a K × K solve.Source: §I · p. 1
  • For a full-row-rank Jacobian, the unregularized update cancels linearized SNR mismatch and leaves a second-order Taylor residual.The error bound separates regularization residual from nonlinear phase-domain error.Source: §I · p. 1

A. Multilayer SIM-Enabled Broadcasting

  • The broadcasting model uses one RF feed and an L-layer SIM to send a common unit-power symbol to K single-antenna ground terminals.The feed coupling and inter-layer propagation matrices are fixed by the SIM geometry; terminals remain visible during the tracking interval and synchronization compensates Doppler and delay evolution.Source: A. Multilayer SIM-Enabled Broadcasting · p. 2

B. Geometry-Based LoS Channel

The geometry-based LoS channel is specified from satellite and terminal positions in an Earth-centered inertial frame, together with satellite attitude and the SIM response.

B. Geometry-Based LoS Channel

  • The satellite and terminal positions are expressed in an Earth-centered inertial frame, with RBI transforming coordinates into the satellite body frame.The final-layer meta-atom positions are measured relative to the layer’s phase center.Source: B. Geometry-Based LoS Channel · p. 2
  • Under the adopted LoS model, the channel is deterministically specified by satellite state, attitude, and terminal positions.The directional transmit response is produced by G(θ)f and is not included in the fixed feed/element gain.Source: B. Geometry-Based LoS Channel · p. 2

C. Physics-Constrained Orbital Recurrence

Under a circular two-body orbit, sampled satellite positions follow an exact recurrence whose coefficient is set by the orbital rate. The recurrence and the stated Earth-rotation and LVLH relations determine the next line-of-sight geometry state.

C. Physics-Constrained Orbital Recurrence

  • The sampled satellite position obeys an exact two-step recurrence derived from the circular-orbit position model.The proof uses sine and cosine sum identities.Source: C. Physics-Constrained Orbital… · p. 2
  • The recurrence coefficient is dictated by orbital dynamics, while unmodeled perturbations and ephemeris errors are represented as model mismatch.The orbital rate is specified using the Earth gravitational parameter and orbital radius.Source: C. Physics-Constrained Orbital… · p. 2
  • The recurrence, Earth-rotation relation, and LVLH attitude model completely determine the next line-of-sight geometry state under the adopted assumptions.Source: C. Physics-Constrained Orbital… · pp. 2–3

III. PROPOSED ONE-SHOT BEAM TRACKING

The method predicts the finite-interval SNR-profile variation from the forecast geometry and seeks to restore a normalized reference footprint. Each update also compensates for error accumulated in preceding intervals.

III. PROPOSED ONE-SHOT BEAM TRACKING

  • The SIM is fully optimized at initialization, after which tracking targets preservation of the normalized SNR profile relative to its reference.The normalization gives equal weight to each terminal's relative SNR variation.Source: A. Finite-Interval Observation… · p. 3
  • The predicted finite-interval SNR variation follows from the forecast geometry without linearizing the orbital displacement under the adopted models.The geometry is propagated using the stated orbital and attitude relations, and the SNR mapping is deterministic.Source: A. Finite-Interval Observation… · p. 3
  • Each update compensates for both the newly predicted variation and residual error accumulated over preceding intervals.Source: A. Finite-Interval Observation… · p. 3

B. Multilayer Phase Sensitivity

The multilayer transfer matrix is factored around each layer to express phase sensitivity. Collecting these derivatives yields a normalized Jacobian that captures the whole footprint’s sensitivity to all coupled SIM phases.

B. Multilayer Phase Sensitivity

  • Factoring the SIM transfer matrix around layer l expresses it as G = V_lΦ_lU_l, with the surrounding-layer products specified for the first, interior, and final layers.Source: B. Multilayer Phase Sensitivity · p. 3
  • The derivative of the complete multilayer transfer function is collected across users, layers, and meta-atoms to form the phase Jacobian.Source: B. Multilayer Phase Sensitivity · p. 3
  • The resulting normalized phase Jacobian is a K×LN matrix capturing sensitivity of the whole normalized footprint to all LN coupled phases.Source: B. Multilayer Phase Sensitivity · p. 3

C. One-Shot Nonlinear Correction

The nonlinear tracking correction is posed as regularized least squares and approximated by one damped Gauss–Newton step. It jointly updates all SIM layers using a K×K solve, avoiding iterative reoptimization at each tracking instant.

C. One-Shot Nonlinear Correction

  • The nonlinear footprint-restoration problem couples LN phase variables, so solving it iteratively at every tracking instant would erode the advantage of predicting orbital evolution.Source: C. One-Shot Nonlinear Correction · p. 3
  • Neglecting second- and higher-order phase terms yields one damped Gauss–Newton, equivalently Levenberg–Marquardt, correction to the regularized least-squares problem.Source: C. One-Shot Nonlinear Correction · pp. 3–4
  • For η > 0, the quadratic one-step problem is strictly convex and has a unique solution.Source: C. One-Shot Nonlinear Correction · p. 4
  • The joint phase update requires only a K×K matrix inversion and one Jacobian evaluation, rather than an LN×LN inversion or layer-by-layer optimization.The regularization parameter η bounds phase excursion; decreasing it prioritizes footprint restoration, while increasing it yields smoother, smaller excursions.Source: C. One-Shot Nonlinear Correction · p. 4
  • Two initial ephemeris positions suffice for open-loop operation because subsequent positions are earlier predictions.The trajectory-to-SIM relation maps those positions through predicted geometry and SNR variation to a phase correction and the next SIM configuration.Source: C. One-Shot Nonlinear Correction · p. 4

D. Tracking Accuracy

The tracking-error bound separates regularization residual from nonlinear phase error, with the latter second order in the update size. Under a full-row-rank Jacobian, the minimum-norm update cancels the linearized normalized SNR mismatch; the stated analysis assumes continuous phases.

D. Tracking Accuracy

  • The two terms in the tracking-error bound quantify the regularization residual and a nonlinear phase error that is second order in the update size.The bound assumes the normalized Jacobian is Lipschitz along the segment between the current and updated phases.Source: D. Tracking Accuracy · p. 4
  • With a full-row-rank Jacobian and vanishing regularization, the minimum-norm update cancels the linearized normalized SNR mismatch, leaving a second-order phase-domain residual.The full-row-rank condition requires LN ≥ K.Source: D. Tracking Accuracy · p. 4
  • The analysis applies to unconstrained continuous phases; clipping and finite-resolution quantization add hardware-projection errors.The full-row-rank regime is described as particularly relevant because SIMs typically satisfy LN ≫ K.Source: D. Tracking Accuracy · p. 4

E. Implementation and Complexity

The implementation reuses forward and backward fields to compute multilayer SIM sensitivities, with per-update costs expressed in terms of K, L, and N.

E. Implementation and Complexity

  • Forward and backward field recursions provide all KLN sensitivities from shared computations.The forward fields are propagated layer by layer, and the backward-field expression supplies the corresponding user sensitivities.Source: E. Implementation and Complexity · p. 4
  • For dense inter-layer matrices, the forward/backward passes require O((K + 1)LN^2) operations, while the K × K inversion costs O(K^3).Forming the matrix used in the small solve is reported to require O(K^2LN) operations.Source: E. Implementation and Complexity · p. 4
  • Fig. 2 covers both computational complexity and local approximation accuracy of the one-shot tracker.The provided figure caption does not give numerical values for either comparison.Source: Figure 2 · p. 5

IV. NUMERICAL RESULTS

The numerical results compare fixed SIM, one-shot tracking, and full reoptimization across LEO passes and SIM configurations. The one-shot tracker retains most of the fully reoptimized common rate, while its local Taylor residual follows the predicted second-order behavior.

IV. NUMERICAL RESULTS

  • Experiments average over 50 terminal/channel realizations and include accumulated orbital model mismatch from open-loop propagation initialized with two ephemeris positions.The predictor is not evaluated on the ideal trajectory from which it is derived; the compared strategies use the same phase resolution and phase-increment constraints whenever relevant.Source: §IV · p. 5
  • For both SIM configurations, the measured Taylor residual closely follows a quadratic in perturbation magnitude over the considered phase-perturbation range, confirming predicted second-order local behavior from the second term of (40).Source: §IV · p. 5
  • At t = 60 s, one-shot tracking achieves 2.4 and 3.7 bit/s/Hz versus 2.55 and 4.0 bit/s/Hz under full reoptimization for (L, N) = (2, 100) and (4, 225), retaining approximately 94% and 92% of the rate.The fixed SIM rapidly loses alignment, while the one-shot tracker maintains an almost constant common rate throughout the pass.Source: §IV · p. 5, Figure 3 · p. 5

V. CONCLUSION

The conclusion combines orbital prediction, an analytical multilayer Jacobian, and a regularized Gauss–Newton update into a one-shot tracker, with second-order local error and lower complexity than full reoptimization.

V. CONCLUSION

  • The proposed physics-informed tracker jointly adjusts all LN SIM phases using one Jacobian evaluation and a K × K solve, with second-order local residual and substantially lower complexity than full SIM reoptimization.Its update combines orbital prediction and an analytical multilayer Jacobian.Source: §V · p. 5
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