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Space-Time Stacked Intelligent Metasurfaces for Doppler-Resilient LEO Broadcasting

Chandan Kumar Sheemar, Giovanni Iacovelli, Sourabh Solanki, Symeon Chatzinotas

arXiv:2610.34479v1eess.SP

TL;DR

A common Doppler correction leaves broadcast terminals with residuals up to half the Doppler span, and existing SIM designs do not deterministically assign separate frequency shifts to spatial modes. The paper develops space-time SIMs and their synthesis methods; simulations show a four-layer design removes the static-SIM Doppler floor above about 45 km and approaches ideal digital precompensation within 0.5 dB for resolvable terminals.

  • Problem

    A common translation for a broadcast beam serving terminals with different Doppler shifts leaves residuals up to half the Doppler span, while existing SIMs do not deterministically pair spatial modes with distinct frequency translations.

    Source: §I · p. 1

  • Method

    The paper derives an ST-SIM harmonic-domain model, solves minimum-energy space–frequency synthesis, and develops a phase-only algorithm with quantization and control-rate laws.

    Source: B. Contributions · p. 2

  • Results

    A four-layer ST-SIM removed the Doppler floor of a static SIM above about 45 km and matched ideal digital per-terminal precompensation within 0.5 dB at equal radiated peak energy for resolvable terminals.

    Source: §VI · p. 12

  • Takeaways & Limitations

    For registered terminals, the ST-SIM delivers a carrier-Doppler-free signal, reducing frequency search to a single hypothesis during reacquisition and tracking.

    Source: E. Grouped Compensation and Robustness · p. 12

    The architecture’s modal-energy cost grows with terminal count and shrinks with aperture size; at equal feed power, each normalized-cascade layer incurs about 4 dB of diffraction loss.

    Source: §VI · p. 12

Abstract

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Stacked intelligent metasurfaces (SIMs) cascade programmable layers in front of a feed. In the SIM designs reported so far, including those proposed for low-Earth-orbit (LEO) satellites, the layer coefficients are space-only controls or are varied randomly in time to support opportunistic scheduling, so that every radiated beam carries the frequency of the feed. This paper introduces space-time SIMs (ST-SIMs), whose layers are modulated in time and use the temporal dimension of the stack deterministically: each spatial mode of the aperture is paired with its own prescribed frequency translation. The application is LEO broadcasting, where a common waveform reaches terminals with different Doppler shifts, and any common Doppler correction leaves residuals of up to half the Doppler span. We derive the harmonic-domain model of the stack, solve the minimum-energy space-frequency synthesis in closed form, and prove that full column rank of the steering matrix is necessary and sufficient for decoupling K terminal-specific translations, a result that extends to groups of terminals. We further propose a phase-only synthesis algorithm with guaranteed stationarity, together with quantization and control-rate laws. Simulations at 20 GHz over 50 random drops show that a four-layer ST-SIM with a single radio-frequency chain removes the Doppler floor of a static SIM for footprint radii above about 45 km and, under the ideal phase-only model and at equal radiated peak energy, approaches ideal digital per-terminal precompensation within 0.5 dB for resolvable terminals, whereas, in the evaluated geometries, a single time-modulated layer cannot synthesize the required modes.

I. INTRODUCTION

A common Doppler correction leaves broadcast terminals with residuals up to half the Doppler span, while existing SIMs do not deterministically pair spatial modes with separate frequency shifts. The paper proposes ST-SIMs to perform that space–frequency processing in the satellite aperture and develops its theory, design, and system model.

I. INTRODUCTION

  • A common broadcast translation leaves a worst-terminal residual equal to half the Doppler span, which exceeds 300 kHz for a 200-km footprint at 20 GHz.The residual is independent of transmit power, noise level, and digital processing of the common waveform.Source: §I · p. 1

A. State-of-the-Art

  • Prior SIMs use space-only controls or random temporal variation, whereas deterministic pairing of distinct spatial modes with Doppler-matched translations had not been investigated.Related time-modulated surfaces generally use harmonics as channels or offset one link, rather than decoupling users through a multilayer stack.Source: §I · p. 1, A. State-of-the-Art · p. 2

B. Contributions

  • The paper introduces ST-SIMs for deterministic space–frequency processing, where each frequency-translated mode must also be spatially nulled at unintended terminals.The finite-depth, finite-resolution, and finitely fast phase-only stack must synthesize the resulting field.Source: B. Contributions · p. 2
  • The harmonic-domain model shows that stacking convolves the temporal spectra of layers while mixing their spatial fields, with desired modes and unwanted sidebands sharing a passive energy budget.The minimum-energy synthesis is solved in closed form, and exact decoupling of K terminal-specific translations is possible if and only if the steering matrix has full column rank; grouping relaxes this condition.Source: B. Contributions · p. 2
  • A phase-only synthesis algorithm provides monotone objective decrease to stationary points, quantization-aware refinement, and explicit quantization and control-rate laws.The study evaluates the architecture at 20 GHz over 50 random drops against single-layer, static-SIM, and digital-precompensation alternatives.Source: B. Contributions · p. 2

II. SYSTEM MODEL

  • The system model considers one LEO beam delivering a common data stream to K registered terminals whose positions are known to the network.A single RF feed illuminates an L-layer stack with N meta-atoms per layer.Source: §II · p. 2

A. LEO Propagation and Doppler

  • The propagation model derives received baseband signals from transmit and receive times linked by the light-time delay, with carrier Doppler arising from the delay-induced carrier phase.It models the LoS path; multipath Doppler, phase noise, and oscillator offsets remain for terminal synchronization.Source: A. LEO Propagation and Doppler · p. 3

B. Multilayer ST-SIM Transfer Function

The multilayer ST-SIM cascades time-varying phase layers and fixed propagation operators to shape the aperture field. In the harmonic-domain view, layer spectra combine while propagation mixes their spatial fields, enabling prescribed frequency translations but sharing a finite energy budget.

B. Multilayer ST-SIM Transfer Function

  • The aperture field is a cascade of time-varying phase matrices and fixed inter-layer propagation operators acting on the feed field.The model assumes frequency-flat propagation, negligible modulation during transit, passive propagation, and ideal phase-only layers without multiple reflections or hardware parasitics.Source: B. Multilayer ST-SIM Transfer Function · p. 3

C. Harmonic-Domain Representation

  • Stacking convolves the temporal spectra of the layers, while inter-layer propagation mixes the associated spatial fields.The aperture harmonics are obtained by combining layer harmonics in the cascade and applying the fixed propagation operators.Source: C. Harmonic-Domain Representation · p. 4
  • Each prescribed tone is an aperture mode; synthesis concentrates harmonic content on the selected indices, with unwanted harmonics forming the residual field.The number of modes equals the number of beams: K for one beam per terminal or G for one multicast beam per group. Held control limits synthesized tones to |ν| < fu/2 and finite reconfiguration accuracy diverts energy into unwanted harmonics.Source: C. Harmonic-Domain Representation · p. 4
  • Temporal modulation redistributes rather than adds aperture energy, so desired modes and unwanted sidebands share the same energy budget.A lossless stack has constant instantaneous energy, whereas a multi-tone field has time-varying energy; the difference is carried by unwanted harmonics or absorbed by loss.Source: C. Harmonic-Domain Representation · p. 4

A. Minimum-Energy Multiuser Synthesis

The synthesis problem assigns one aperture mode to each terminal’s Doppler translation while enforcing the required steering responses. Full column rank makes the problem feasible, and the minimum-norm solution exposes how terminal geometry affects energy.

A. Minimum-Energy Multiuser Synthesis

  • The minimum-energy synthesis assigns each terminal a mode whose steering response provides its prescribed aperture gain while canceling the other terminals.The solution is constructed from the steering matrix’s Gram matrix and the desired diagonal gains.Source: A. Minimum-Energy Multiuser Synthesis · p. 5
  • For distinct mode frequencies, the minimum synthesis energy is also the long-time average energy of the modal field.The temporal exponentials are orthogonal in the time average.Source: A. Minimum-Energy Multiuser Synthesis · p. 5
  • When N ≥ K and the steering matrix has full column rank, the decoupling constraint is feasible and has a unique minimum-Frobenius-norm solution.Every feasible synthesis matrix differs from the minimum-norm solution by a component in the nullspace of the conjugate-transposed steering matrix.Source: A. Minimum-Energy Multiuser Synthesis · p. 5
  • As two steering vectors become nearly collinear, the minimum energy needed to separate their frequency translations grows without bound.The stated geometry link is that the minimum eigenvalue of the steering Gram matrix approaches zero.Source: A. Minimum-Energy Multiuser Synthesis · p. 5

B. Perfect Blockwise Multiuser Compensation

Perfect terminal-specific Doppler compensation requires spatial modes paired with terminal-matched translations. The theorem gives exact decoupling when the steering matrix has full column rank, and identifies how grouped compensation can relax that condition.

B. Perfect Blockwise Multiuser Compensation

  • The harmonic-domain identity separates spatial leakage from temporal rotation: each mode leaks according to H, while its frequency offset determines its residual rotation.Here H = A^HΨ, and the temporal offset is fD,k − νj for mode j at terminal k.Source: B. Perfect Blockwise Multiuser… · p. 5
  • For distinct terminal Dopplers, exact decoupling requires full column rank of the steering matrix, and that rank condition is sufficient for the theorem’s perfect compensation construction.The resulting terminal signal has zero residual Doppler and zero inter-mode leakage.Source: B. Perfect Blockwise Multiuser… · pp. 5–6
  • The theorem establishes a sufficient aperture field and rank condition, but finite-depth phase-only synthesis remains a separate question.The paper addresses whether a finite-depth phase-only stack can synthesize the field in Section IV.Source: B. Perfect Blockwise Multiuser… · p. 6
  • Grouping terminals to share one tone and multicast mode can permit decoupling even when the steering matrix lacks full column rank.The passage gives collinear terminals in one group as an example; G = K recovers terminal-specific translations, while G = 1 is a common translation.Source: B. Perfect Blockwise Multiuser… · p. 6

C. Irreducible Residual of a Common Frequency Translation

A common frequency translation cannot eliminate the worst-terminal residual when terminals span different Dopplers, while ST-SIM pairs translations with spatial modes. The section also relates residual Doppler to OFDM interference and identifies residual effects that carrier compensation does not remove.

C. Irreducible Residual of a Common Frequency Translation

  • Any common translation leaves a worst-terminal residual equal to half the Doppler span, whereas ST-SIM can pair a different translation with each spatial mode.In the evaluated hybrid mode, the feed uses the optimal common translation and the ST-SIM synthesizes only differential tones, reducing the largest synthesized tone by a factor of 1.05 in the baseline geometry and more for clustered terminals.Source: C. Irreducible Residual of a Common… · p. 6

D. OFDM Consequences and Residual Errors

  • Under perfect compensation, the carrier-Doppler-induced OFDM inter-carrier interference vanishes for every terminal.The CFO matrix is unitary, and zero residual gives C0(0) = 1.Source: D. OFDM Consequences and Residual Errors · p. 6
  • Carrier-Doppler compensation does not remove all residual effects: timing-scale ICI is common across carrier-Doppler schemes, while ephemeris or position errors leave a residual tone.The timing-scale effect is removed by OFDM receiver timing tracking.Source: D. OFDM Consequences and Residual Errors · p. 6
  • Residual frequency error produces OFDM interference whose scale becomes more critical as the carrier frequency-to-subcarrier-spacing ratio grows.The passage gives the imperfect-compensation ICI fraction as proportional to (δfD,k/∆f)^2, with a factor involving 1 − M^-2.Source: D. OFDM Consequences and Residual Errors · p. 6

IV. PHASE-ONLY ST-SIM SYNTHESIS

The phase-only synthesis problem fits a finite-depth stack’s field trajectory to desired fields over sampled control slots. It jointly optimizes layer phases and a common complex gain, with switching penalties available to discourage changes between slots.

A. Trajectory Synthesis Problem

  • The design samples a predictable control horizon at transmit times separated by T_u and jointly optimizes the layer phases and common complex scaling α.The scaling absorbs the stack’s transmission gain because a passive stack cannot reach an arbitrary absolute amplitude.Source: A. Trajectory Synthesis Problem · p. 7
  • The finite-resolution problem includes a smooth, 2π-periodic penalty on phase changes between consecutive slots, and a hard slew limit can be imposed afterward if required.The penalty preserves the differentiability used in the method; its weight can be set to zero to decouple slots.Source: A. Trajectory Synthesis Problem · p. 7
  • Because cascaded phase products and the discrete phase alphabet make the design nonconvex, the method studies a continuous relaxation and then quantifies quantization effects.Source: A. Trajectory Synthesis Problem · p. 7

B. Optimal Scaling and Reusable Multilayer Gradient

The method eliminates the common complex gain in closed form and reuses forward and backward multilayer sweeps to compute phase gradients. Its wrapped Armijo descent is guaranteed to decrease the objective and converge to first-order stationary points, not necessarily a global optimum.

B. Optimal Scaling and Reusable Multilayer Gradient

  • For fixed phases, the common scaling α has a closed-form optimum, and the objective gradient with respect to phases is the gradient at that optimum.The phase derivatives use each layer’s incident field and transfer from that layer to the aperture.Source: B. Optimal Scaling and Reusable… · p. 7
  • One forward and one backward sweep provide all phase derivatives using 2(L − 1) N-sized matrix–vector products, rather than differentiating the full cascade for every meta-atom.Source: B. Optimal Scaling and Reusable… · p. 7

C. Algorithm and Convergence

  • Wrapped Armijo descent monotonically decreases the continuous-relaxation objective and converges to first-order stationary points under the stated step-size condition; discrete refinement is nonincreasing but does not guarantee global optimality.The algorithm first descends on continuous phases, projects onto the finite alphabet, then performs coordinate refinement.Source: C. Algorithm and Convergence · pp. 7–8
  • After 300 iterations, the objective F is 2.5 × 10^-5, 7.5 × 10^-6, 2.6 × 10^-6, and 9.2 × 10^-6 for L = 2, 3, 4, and 6, respectively.The single layer reaches the 10^-6 tolerance after 142 iterations but stalls at −5.9 dB at a stationary point.Source: C. Algorithm and Convergence · p. 8

D. Finite-Resolution Phase Quantization

The quantization bound gives a scaling law for field and synthesis error as phase resolution and stack depth vary. It is conservative, particularly for coarse quantization.

D. Finite-Resolution Phase Quantization

E. Control Rate, Switching Harmonics, and Control Bandwidth

The control-rate analysis characterizes the sampling and switching-harmonic costs of synthesizing Doppler tones with held layer states. The evaluation compares ST-SIM designs with space-only and digital benchmarks using worst-terminal broadcast metrics.

E. Control Rate, Switching Harmonics, and Control Bandwidth

  • Ideal interpolation requires fu > 2νmax, while zero-order-held control produces switching harmonics alongside the desired Doppler tone.The desired tone amplitude is scaled by sinc(fD,k/fu); images occur at −fD,k − pfu for nonzero integer p.Source: E. Control Rate, Switching Harmonics,… · p. 8
  • An in-band SIR of 30 dB for a synthesized 160 kHz tone requires a control update rate fu ≳ 9 MHz.When fu exceeds the signal bandwidth, switching images fall out of band and become an emission rather than an interference problem.Source: E. Control Rate, Switching Harmonics,… · p. 8
  • Streaming every state requires LNB_ϕfu bit/s, making aggregate control bandwidth a hardware constraint; one evaluated configuration requires 9.2 Gbit/s.That value is for LN = 400, B_ϕ = 3, and fu = 7.68 MHz.Source: E. Control Rate, Switching Harmonics,… · p. 8

A. Setup, Benchmarks, and Metrics

  • The evaluation uses medians over 50 random drops and compares space-only SIM, ideal digital per-terminal precompensation, and a single-layer time-modulated aperture.It reports worst-terminal SINR and broadcast spectral efficiency, with schemes compared at equal long-run radiated peak energy.Source: A. Setup, Benchmarks, and Metrics · p. 9

B. Realizability: Convergence, Quantization, and Control Rate

The paper evaluates ST-SIM realizability, link performance, and robustness across control settings, footprint radii, aperture sizes, terminal counts, and grouping. A four-layer stack can approach digital per-terminal precompensation at equal radiated peak energy, while performance depends on resolvability and the energy model.

B. Realizability: Convergence, Quantization, and Control Rate

  • For L ≥3, Algorithm 1 reaches −50 to −61 dB after 300 iterations, while the single layer stalls at −5.9 dB and L = 2 near −26 dB.Adam reaches −68 and −73 dB but is nonmonotone; quantization-aware refinement improves four-layer projection error by more than one bit-equivalent, with refined curves following the 6-dB-per-bit slope.Source: B. Realizability: Convergence,… · pp. 9–10

C. What Time Modulation Adds to a SIM

  • At 200 km, continuous-phase ST-SIM stays within 0.2 dB of Benchmark 2 from 100 km on, but Benchmark 3 reaches 20–25 dB at every radius and ST-SIM pays diffraction loss at equal feed power.Along the pass, Benchmark 1 has average SIR of −32 to +3 dB, while ST-SIM maintains 35–49 dB with continuous phases and 21–27 dB with three bits.Source: C. What Time Modulation Adds to a SIM · pp. 10–11

D. Aperture Size and Number of Terminals

  • From N = 36 to 256, ST-SIM gains 8.4 dB and tracks Benchmark 2 within 0.1 dB for every aperture size; at K = 4, its spectral efficiency tracks Benchmark 2 within 0.05 bit/s/Hz.Three-bit phases cost 0.8–1.4 dB across aperture sizes and at most 0.4 bit/s/Hz across terminal counts.Source: D. Aperture Size and Number of Terminals · p. 11

E. Grouped Compensation and Robustness

  • At 50 km, adaptive grouping reaches 5.1 dB median SINR, 9.2 dB above per-terminal ST-SIM; it follows the static benchmark at small footprints and per-terminal ST-SIM at large ones.For K = 8, terminal-side correction yields 5.2 bit/s/Hz at G = 1 and 4.7 bit/s/Hz at G = 2, while reducing the frequency search from 17 to 9 hypotheses.Source: E. Grouped Compensation and Robustness · p. 12

VI. CONCLUSION

  • Initial acquisition of an unregistered terminal is outside the ST-SIM’s scope; at equal feed power, normalized-cascade diffraction loss favors two or three layers.The conclusion reports about 4 dB loss per added layer in that model.Source: E. Grouped Compensation and Robustness · p. 12, §VI · p. 12
  • Above about 45 km, a four-layer ST-SIM removes the static SIM’s Doppler floor and approaches ideal digital per-terminal precompensation within 0.5 dB at equal radiated peak energy.The conclusion reports −58 dB synthesis error with continuous phases and −13 dB with three-bit phases; one time-modulated layer stalls at −6 dB.Source: §VI · p. 12

APPENDIX A LIGHT-TIME DOPPLER MODEL

The appendix defines the retarded transmit time through the one-way light-time equation and uses it to derive the instantaneous carrier Doppler frequency. For LEO motion, the relative correction is small enough that the first-order form is used.

APPENDIX A LIGHT-TIME DOPPLER MODEL

  • The light-time equation relates transmit time u to reception at t = u + τk(u), with τk determined by the satellite-terminal distance.The retarded transmit time is the inverse mapping from receive time to transmit time.Source: Appendix A · p. 12
  • Because the satellite speed is below c, the light-time equation has a unique solution and the transmit-time mapping is strictly increasing.Source: Appendix A · p. 12
  • For LEO motion, the relative correction is at most ∥˙rs∥₂/c = 2.5 × 10^-5, so the paper uses the first-order Doppler form throughout.Source: Appendix A · p. 12

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