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GPU-Accelerated Robust Beamforming for OTFS Massive MIMO with CSI Uncertainty

Haonan Zhang, Weihua Wu

arXiv:2609.34763v1cs.ITmath.IT

TL;DR

In high-mobility satellite links, imperfect CSI complicates reliable OTFS beamforming. The paper combines SVC-based uncertainty modeling with GPU-accelerated ADMM; simulations report lower power at prescribed reliability and faster computation at larger scales.

  • Problem

    Long propagation distances and high relative motion impair CSI acquisition, so practical satellite links cannot obtain perfectly accurate CSI.

    Source: B. OTFS 3D Delay-Doppler-Angle… · p. 5

  • Method

    The paper models CSI uncertainty with SVC-based sets in a 3D DDA channel framework, derives a robust counterpart, and solves the resulting convex programs with GPU-accelerated ADMM.

    Source: §I · p. 2, §VI · p. 14

  • Results

    Up to 50% power savings over the Box model at prescribed reliability; GPU ADMM achieves 15.6× speedup over CPU ADMM with 40 STs and Mt = 16, and 9.0× over CVX/SDPT3 with 16 STs and Mt = 32.

    Source: §VI · p. 15

  • Takeaways & Limitations

    SVC's data-fitted uncertainty set attains prescribed reliability across Γ, whereas AVG falls below it and Box and Ellipsoid remain below it despite consuming more power.

    Source: §V · p. 14

Abstract

from arXiv · show

This paper focuses on robust beamforming for orthogonal time frequency space (OTFS)-enabled massive multiple-input multiple-output (MIMO) systems under channel state information (CSI) uncertainty. In high-mobility satellite communications, uncertain CSI severely degrades beamforming accuracy and poses a major challenge to meeting users' quality of service (QoS) requirements. To address this challenge, we first formulate a chance-constrained optimization problem aiming to minimize the total transmit power while guaranteeing a predefined outage probability. Building on a 3D delay-Doppler-angle (DDA) channel representation, we propose a data-driven approach using support vector clustering (SVC) to model the uncertain CSI as an asymmetric uncertainty set. We then derive a robust counterpart that reformulates the intractable chance constraints into a deterministic semidefinite program. Finally, we design a graphics processing unit (GPU)-accelerated parallelizable alternating direction method of multipliers (ADMM) algorithm to address the computational complexity of large-scale antenna arrays. Simulation results show that the proposed SVC-based design reduces transmit power compared with conventional beamforming schemes, and that the GPU-accelerated ADMM achieves a speedup. These results confirm that the proposed framework achieves improved robustness, energy efficiency, and efficient computation in dynamic massive MIMO networks.

I. INTRODUCTION

High mobility and imperfect CSI make reliable, energy-efficient beamforming difficult in LEO satellite links, while conventional robust designs face modeling and computational challenges. The paper introduces a 3D OTFS channel framework and data-driven robust beamforming with GPU-accelerated computation.

I. INTRODUCTION

  • Imperfect CSI makes SINR random, creating outage risk when channel gain is overestimated and wasted power when it is underestimated.Long propagation delays and rapid channel variation make perfect transmitter CSI infeasible; robust design must control both risks.Source: §I · pp. 1–2
  • The paper models sparse satellite multipath in 3D delay-Doppler-angle coordinates and learns asymmetric convex CSI uncertainty sets from historical channel data.It reformulates chance constraints as a deterministic semidefinite program and proposes GPU-accelerated ADMM using independent, parallelizable subtasks.Source: §I · p. 2

II. RELATED WORKS

  • Prior robust designs use predefined symmetric sets or rigid worst-case formulations, while stochastic approaches may require frequent CSI acquisition or accurate assumed distributions.The related-work discussion identifies GPU-accelerated optimization for robust OTFS massive MIMO beamforming as an open challenge.Source: §II · p. 3

III. SYSTEM MODEL AND 3D OTFS CHANNEL

  • The system uses OTFS and a massive MIMO array to represent satellite links as sparse channels across delay, Doppler, and angle for beamforming under uncertainty.OTFS maps fast time-varying channels to a quasi-static sparse representation; the 3D form adds angular-domain structure.Source: §III · p. 3

A. High-Mobility MIMO Channel Model

The channel model represents each high-mobility satellite link as resolvable paths with delay, Doppler, direction, and complex gain. It uses a normalized ULA response and treats path parameters as constant within each OTFS frame.

A. High-Mobility MIMO Channel Model

  • Each terminal’s channel comprises resolvable paths, each characterized by propagation delay, Doppler shift, direction of departure, and complex gain.The paths capture line-of-sight and reflected propagation in frequency- and time-selective links.Source: A. High-Mobility MIMO Channel Model · pp. 3–4
  • For a half-wavelength-spaced satellite ULA, the steering response depends on the sine of the direction-of-departure angle and is normalized to unit norm.The model defines Θ = sin θ to make the angular dependence explicit.Source: A. High-Mobility MIMO Channel Model · p. 4
  • Within one OTFS frame, the model treats all four path parameters as constant because their motion-induced changes are negligible relative to system resolution.The stated frame duration is 1.64 ms; the passage compares angular, Doppler, and delay drift with their respective resolutions.Source: A. High-Mobility MIMO Channel Model · p. 4

B. OTFS 3D Delay-Doppler-Angle Representation

OTFS represents paths as sparse delay-Doppler taps, while massive MIMO adds an angular dimension to form a three-dimensional channel view. Because practical per-path CSI is imperfect, channel estimation errors must also be represented.

B. OTFS 3D Delay-Doppler-Angle Representation

  • Each propagation path contributes a sparse tap at a delay-Doppler grid cell, with its spatial signature represented separately.The spatial vector captures the path’s complex gain and array steering response.Source: B. OTFS 3D Delay-Doppler-Angle… · pp. 4–5
  • The massive-MIMO channel extends the OTFS delay-Doppler plane with a spatial dimension, yielding a three-dimensional delay-Doppler-angle representation.The channel can also be represented as spatial vectors on a two-dimensional delay-Doppler grid.Source: B. OTFS 3D Delay-Doppler-Angle… · pp. 4–5
  • With full delay-Doppler reuse, paths are separable only through their spatial signatures, and sufficiently separated angles yield approximately orthogonal array responses.Angular separation is measured in Θ = sin θ; paths below the stated resolution share a layer and cannot be spatially separated.Source: B. OTFS 3D Delay-Doppler-Angle… · p. 5
  • The transmitter needs each path’s direction of departure and complex gain, but satellite propagation delay and motion make accurate per-path CSI acquisition impractical.The paper models the resulting channel inaccuracy with an additive estimation error.Source: B. OTFS 3D Delay-Doppler-Angle… · p. 5

C. Signal Model and Per-Path SINR

The signal model applies a beamforming vector to each terminal’s OTFS signal and defines received symbols and interference on the shared delay-Doppler grid. Per-path SINR accounts for desired and interfering signals, effective noise, and spatial beamforming matrices.

C. Signal Model and Per-Path SINR

  • Each terminal’s OTFS signal is spatially precoded by its beamforming vector, and total transmit power is the sum of the beamforming-vector norms squared.The model assumes unit average symbol energy.Source: C. Signal Model and Per-Path SINR · p. 5
  • The received per-path symbol model includes the desired and interfering data symbols, the path’s spatial channel vector, and effective additive noise.The noise variance accounts for AWGN and uncompensated fractional Doppler and delay interference.Source: C. Signal Model and Per-Path SINR · pp. 5–6
  • Because all terminals reuse the same delay-Doppler grid, the interference set comprises signals overlapping those cells, and user separation relies on angular information.The interference set follows the path separation principle.Source: C. Signal Model and Per-Path SINR · p. 6
  • The model expresses each path’s SINR using rank-one beamforming matrices and the path-specific channel and interference terms.The transmitted signal at each time instant aggregates the precoded terminal signals.Source: C. Signal Model and Per-Path SINR · pp. 5–6

D. Robust Beamforming Problem Formulation

The formulation minimizes transmit power while requiring per-path SINR targets to hold with a specified outage probability. It replaces chance constraints with constraints over channel uncertainty sets, then bounds the Taylor-expansion remainder to obtain a robust counterpart.

D. Robust Beamforming Problem Formulation

  • The design minimizes total transmit power subject to individual per-path SINR thresholds and a maximum outage probability.The joint chance constraint requires all paths for each terminal to meet their thresholds with probability at least 1 − ϵ.Source: D. Robust Beamforming Problem… · p. 6
  • Per-path uncertainty sets with confidence 1 − ϵ/P_n yield an outage bound of ϵ across the terminal’s P_n paths via the union bound.Enforcing each SINR constraint throughout its uncertainty set ensures the original chance constraints are satisfied.Source: D. Robust Beamforming Problem… · p. 6
  • A first-order Taylor expansion at the statistical mean separates the quadratic QoS constraint into a linearized term and an exact remainder.Omitting the remainder is unsafe because it can be positive or negative and may admit solutions below the required SINR.Source: D. Robust Beamforming Problem… · p. 6
  • The robust counterpart bounds the remainder conservatively, but remains non-convex because of its rank-one constraint, motivating semidefinite relaxation.The derivation bounds channel error by ρ_n,p and uses positive semidefinite beamforming matrices to bound interference terms.Source: D. Robust Beamforming Problem… · pp. 6–7

IV. GPU-ACCELERATED ROBUST BEAMFORMING OPTIMIZATION

The section addresses semi-infinite QoS constraints over uncertainty sets and non-convex rank-one constraints by developing a finite-dimensional conic formulation and GPU-oriented ADMM algorithm.

IV. GPU-ACCELERATED ROBUST BEAMFORMING OPTIMIZATION

  • The optimization problem has semi-infinite QoS constraints over uncertainty sets and a non-convex rank-one constraint.These are identified as the two main difficulties addressed in the section.Source: §IV · p. 7
  • The section converts the problem into a finite-dimensional conic program, using SVC-set LP duality and semidefinite relaxation with rank-one exactness conditions.It also standardizes the variable layout for the algorithm.Source: §IV · p. 7
  • The proposed ADMM updates are tailored for GPU computing to solve the resulting program efficiently.The section derives the detailed updates after specifying a standardized variable layout.Source: §IV · p. 7

A. SVC-Based Uncertainty Set and Robust Counterpart

The paper learns channel uncertainty from CSI samples using SVC in a real-valued embedding that preserves complex channel and beamforming structure. It converts worst-case QoS constraints over the learned set into a deterministic semidefinite program, then relaxes its rank-one constraint to obtain a convex SDP.

A. SVC-Based Uncertainty Set and Robust Counterpart

  • SVC maps channel vectors to real dimension D = 2Mt while preserving the complex Hermitian beamforming structure.The embedding preserves norms, keeping the channel-error radius unchanged and expressions affine in the beamforming variables.Source: A. SVC-Based Uncertainty Set and… · p. 7
  • The learned set contains at least (1 − ϵ/Pn) × 100% of the CSI samples at the chosen SVC penalty.At most ϵS/Pn samples fall outside the ball at the optimum.Source: A. SVC-Based Uncertainty Set and… · p. 7
  • A conservative upper bound on the maximum channel-error norm replaces costly exact maximization and can increase design conservatism.The radius enters the robust QoS constraint with a nonpositive coefficient, so using an upper bound preserves validity.Source: A. SVC-Based Uncertainty Set and… · p. 8
  • The SVC uncertainty set yields finite constraints through LP duality, making the robust counterpart a linear semidefinite program.The SDP relaxation omits the non-convex rank-one constraint and has a globally optimal solution.Source: A. SVC-Based Uncertainty Set and… · pp. 8–9

B. Analysis of Solution Properties Based on Matrix Transformations

This section analyzes the rank of the semidefinite relaxation using matrix transformations and KKT conditions. It bounds solution rank by active-path structure and gives conditions for exact rank-one solutions, with a recovery procedure when the certificate fails.

B. Analysis of Solution Properties Based on Matrix Transformations

  • The rank analysis assumes strict feasibility, which holds whenever the QoS targets are jointly achievable.Under this Slater condition, strong duality holds and KKT conditions are necessary and sufficient.Source: B. Analysis of Solution Properties… · p. 9
  • Under Assumption 1, every optimal relaxed solution has rank at most min(|Pact_n|, Mt), where |Pact_n| is the number of active paths.The stated bound also lower-bounds rank by one and bounds it by rank(Sn).Source: B. Analysis of Solution Properties… · p. 9
  • When the largest generalized eigenvalue is simple, the optimal relaxed beamforming matrix is rank one.A single active path or collinear worst-case channels for all active paths are identified as common cases satisfying the condition.Source: B. Analysis of Solution Properties… · pp. 9–10
  • If the rank-one certificate fails, a two-stage recovery scales the principal eigenvector to active QoS boundaries before supplementary Gaussian randomization.When the certificate holds, the solver guarantees an exact rank-one outcome.Source: B. Analysis of Solution Properties… · p. 10

C. Standard Form and Variable Layout

The robust SDP is mapped into a real conic-program vector form that separates beamforming matrices from path-level slack variables. Its objective selects the beamforming diagonal entries, while constraints are assembled as linear rows and cone requirements.

C. Standard Form and Variable Layout

  • The custom svec mapping represents each Hermitian matrix with real diagonal entries and real and imaginary upper-triangle parts, preserving inner products and norms.The inverse mapping smat lets the beamforming update return to matrix form for eigendecomposition.Source: C. Standard Form and Variable Layout · p. 10
  • The vector x stacks N beamforming blocks first, followed by one slack block per path.The slack variables turn the robust inequalities into equalities; path-indexed blocks include support-vector-related variables.Source: C. Standard Form and Variable Layout · p. 10
  • The assembled linear system stacks three constraint groups path by path, with the right-hand side containing −Γtσ2 in each path's (54b) row.The matrix A is assembled offline from the linear constraints to avoid redundant online computation.Source: C. Standard Form and Variable Layout · pp. 10–11
  • The conic objective selects the diagonal entries of the N beamforming matrices because Tr(Vn) is the sum of those entries.The corresponding coefficient vector has ones at the beamforming diagonal positions and zeros elsewhere.Source: C. Standard Form and Variable Layout · p. 10

D. GPU-Accelerated ADMM Solution

The GPU-oriented ADMM solver splits the conic program into cone and affine updates, with beamforming blocks updated independently and in parallel. It uses matrix eigendecompositions for positive-semidefinite projection and a linear-system projection for affine feasibility.

D. GPU-Accelerated ADMM Solution

  • Duplicating a consensus variable separates positive-semidefinite cone enforcement from the linear equations, the main computational bottleneck.The beamforming update decouples into N matrix problems, while slack entries are clipped elementwise at zero.Source: D. GPU-Accelerated ADMM Solution · p. 11
  • Each beamforming update projects W − β^-1I onto the positive-semidefinite cone by eigendecomposition and clipping negative eigenvalues to zero.The matrix update is performed separately for each user's beamforming block.Source: D. GPU-Accelerated ADMM Solution · p. 11
  • The affine update projects onto Az = b using z = t − AT M(At − b), where M = (AAT)^-1.A and M are assembled once offline before being transferred to GPU memory.Source: D. GPU-Accelerated ADMM Solution · pp. 11–12
  • ADMM stops when primal and dual residuals meet tolerance criteria, and adapts the penalty periodically before holding it constant for stability.The scaled dual variable is rescaled when the penalty changes.Source: D. GPU-Accelerated ADMM Solution · pp. 11–12
  • Per-iteration work includes N Hermitian eigendecompositions and dense matrix-vector products, with the latter identified as memory-bandwidth-bound and GPU-suited.The projection uses products with A and M; remaining steps are elementwise.Source: D. GPU-Accelerated ADMM Solution · p. 12

V. SIMULATION RESULTS

Simulations show that SVC-based uncertainty modeling reduces transmit power while meeting the prescribed reliability, and GPU-accelerated ADMM improves computational efficiency as system scale grows.

  • The SVC set follows the data’s asymmetric shape and encloses the same sample fraction in less volume than the Box and Ellipsoid sets.The symmetric sets extend into directions with no samples and only partially cover separated modes and curved tails.Source: Figure 5 · p. 13, §V · p. 13
  • At Γ = 0.5, AVG often misses the SINR threshold and WCCC exceeds it for nearly all samples, while SVC tracks the threshold and reduces WCCC resource waste.The SVC SINR distributions are also shown across varying predefined QoS thresholds Γ.Source: Figure 6 · p. 13, Figure 7 · p. 13, §V · p. 13
  • Up to 50% lower total transmit power than the Box model, while maintaining reliability at 1−ϵ across QoS thresholds Γ.The Box and Ellipsoid methods use more power yet remain below the prescribed reliability level.Source: §V · pp. 13–14, Figure 8 · p. 14, Figure 9 · p. 14
  • At 40 STs and Mt = 16, GPU ADMM is 15.6× faster than CPU; at 16 STs and Mt = 32, it is 9.0× faster than CVX-SDPT3.At 8 STs, GPU execution takes 5.90 s, or 0.74× the CVX-SDPT3 time, because launch and data-transfer overhead is not yet amortized.Source: Figure 10 · p. 14, Figure 11 · p. 14, §V · p. 14

VI. CONCLUSIONS

The paper formulates chance-constrained robust beamforming under 3D DDA-domain CSI uncertainty, using empirical SVC sets and GPU-accelerated ADMM. Simulations report power savings at prescribed reliability and computational speedups, with overhead in the small-scale regime.

  • The method minimizes total transmit power under chance constraints, uses a Taylor expansion and conservative residual bound, and constructs empirical SVC uncertainty sets for a robust counterpart.A GPU-accelerated ADMM algorithm solves the resulting convex programs.Source: §VI · p. 14
  • Up to 50% power savings over the Box model while holding achieved reliability at the prescribed level 1−ϵ.Source: §VI · p. 15
  • GPU-accelerated ADMM achieves 15.6× speedup over CPU ADMM with 40 STs and Mt = 16, and 9.0× over CVX/SDPT3 with 16 STs and Mt = 32.Its advantage grows with array and user population, at the price of small-scale overhead.Source: §VI · p. 15
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