PaperlayerView exact revision on arXiv

Source-linked AI summary

Generic Optimization of Linear Precoding in Multibeam Satellite Systems

Gan Zheng, Symeon Chatzinotas, Bjorn Ottersten

arXiv:1109.0681v1cs.IT

TL;DR

Multibeam satellite systems must manage interbeam interference and nonlinear power constraints while meeting traffic demand. This paper develops convergent optimization-based precoding schemes and extends them to DPC and multiple-polarization terminals, achieving substantial gains over conventional and existing precoders with performance close to DPC.

  • Problem

    Interbeam interference and nonlinear satellite power constraints complicate multibeam transmission and require optimization beyond conventional linear power models.

  • Method

    The paper proposes a generic iterative optimization algorithm that alternates precoding-vector and power-allocation updates, with extensions to nonlinear DPC and terminals having co- or dual-polarization antennas.

  • Results

    The proposed schemes substantially outperform conventional processing and existing precoders for traffic matching, while optimized linear precoding is shown to be as effective as DPC precoding.

  • Takeaways & Limitations

    Jointly optimized linear precoding offers practical multibeam satellite gains while retaining performance close to DPC precoding.

  • Takeaways & Limitations

    The study assumes perfect feeder links and perfect channel state information at the gateway, which the authors identify as impractical and leave for future investigation.

Abstract

from arXiv · show

Multibeam satellite systems have been employed to provide interactive broadband services to geographical areas under-served by terrestrial infrastructure. In this context, this paper studies joint multiuser linear precoding design in the forward link of fixed multibeam satellite systems. We provide a generic optimization framework for linear precoding design to handle any objective functions of data rate with general linear and nonlinear power constraints. To achieve this, an iterative algorithm which optimizes the precoding vectors and power allocation alternatingly is proposed and most importantly, the proposed algorithm is proved to always converge. The proposed optimization algorithm is also applicable to nonlinear dirty paper coding. In addition, the aforementioned problems and algorithms are extended to the case that each terminal has multiple co-polarization or dual-polarization antennas. Simulation results demonstrate substantial performance improvement of the proposed schemes over conventional multibeam satellite systems, zero-forcing and regularized zero-forcing precoding schemes in terms of meeting the traffic demand. The performance of the proposed linear precoding scheme is also shown to be very close to the dirty paper coding.

I. INTRODUCTION

Multibeam joint precoding addresses interbeam interference by jointly designing transmissions at a shared gateway, avoiding the backhaul obstacle faced in terrestrial cooperation. This paper develops a generic, convergent optimization framework covering flexible power allocation, nonlinear constraints, DPC, and multiple receive polarizations.

  • Overlapping satellite beam patterns create interbeam interference that must be managed to maintain acceptable carrier-to-interference ratios.
  • Joint precoding sends each user’s precoded signal through all beams from a shared gateway, eliminating the need for expensive backhauling within one beam cluster.
  • The paper formulates linear precoding to optimize any function of individual user rates under general linear and nonlinear power constraints.
  • The framework extends from single-polarization terminals to multiple receive dimensions, including co-polarization and dual-polarization antennas.
  • An alternating-optimization algorithm separately updates power allocation and precoding vectors, with convergence proved and applicability to nonlinear DPC precoding.
  • Simulations report linear precoding close to optimized DPC, while flexible transmit power contributes more to performance improvement than nonlinear precoding.

C. Notations

The paper defines notation and models the fixed multibeam satellite channel, including free-space loss, rain fading, beam gains, and correlated fading assumptions.

  • The notation defines complex-vector and matrix conventions, Hermitian transpose, positive-sem definiteness, matrix products, expectations, and complex Gaussian variables.
  • The system is a fixed multibeam forward link with one gateway, perfect CSI, full frequency reuse, and one user served per beam per TDM slot.
  • Satellite channels above 10 GHz are modeled with free-space loss, rain fading, and beam gain patterns because atmospheric effects, especially rain attenuation, degrade performance.
  • Free-space loss varies across beams according to beam-center distance from the central beam and the wavelength.
  • Rain fading uses an ITU-R P.618-based lognormal model whose parameters depend on receiver location, frequency, polarization, and elevation angle.
  • Users within one beam share correlated rain fading, while fading is assumed independent across beams; beam gains depend on receiver angle and the beam’s center gain.

B. Signal Model

The signal model represents each user’s transmission with a linear precoding vector and separates its power from its normalized beamforming direction. It defines received interference, SINR, achievable rates, and general linear power constraints.

  • Each user’s unit-power data stream is weighted by a linear precoding vector t_k = √p_k w_k, separating transmit power p_k from normalized precoding vector w_k.
  • The satellite transmits the superposition of all users’ weighted signals, which each terminal receives through the multibeam channel with additive Gaussian noise.
  • The received SINR determines each user’s achievable Shannon rate, and the resulting rates form the vector r = [r_1, · · ·, r_K].
  • The model includes both linear and nonlinear power constraints on satellite antenna beams.
  • A general linear constraint bounds the summed quadratic powers t_k†Q_l t_k by q_l, where Q_l is a positive-semidefinite shaping matrix.
  • Per-beam constraints and flexible power-sharing constraints are special cases represented by diagonal shaping matrices.

D. General Nonlinear Power constraints

The paper incorporates nonlinear amplifier-related power constraints and supports general continuous rate objectives, including throughput, rate balancing, and rate matching.

  • Nonlinear power constraints model the relationship between satellite beam output power and costly onboard DC power consumed by high-power amplifiers.
  • Each beam’s output power z_k is linked to input power x_k through a continuous increasing one-to-one function g_k, allowing the inverse mapping g_k^-1.
  • The formulation permits J nonlinear power constraints with preset limits on beam antenna output powers.
  • The optimized objective is any continuous function f(r) of user rates, with traffic demand F_k specified for each user.
  • Supported objectives include throughput maximization, worst-user rate balancing normalized by demand, and rate matching via summed demand-rate errors.
  • The constraint r_k ≤ F_k prevents over-satisfying demand, while different objectives generally produce different optimized rate vectors chosen by the satellite operator.
  • The central problem minimizes f(r) over precoding vectors under general linear and nonlinear power constraints, despite non-convex objectives and potentially non-convex power constraints.

III. GENERIC PRECODING DESIGN FOR RATE ENHANCEMENTS

The proposed design alternates between precoding-vector optimization and power optimization to handle the non-convex formulation. A convex power-minimization subproblem and local power updates yield a convergent iterative procedure.

  • Alternating optimization first updates normalized precoding vectors w_k and then updates power allocations p_k with the vectors fixed.
  • The precoding-vector step solves per-beam power minimization subject to minimum-rate, general linear-power, and per-beam power constraints.
  • The auxiliary variable γ and parameters P̃_j connect the subproblem’s per-beam power limits to the beam powers.
  • The precoding subproblem is convex after transforming individual rate constraints and can be solved as a second-order cone programming problem.
  • At the optimum of the subproblem, every rate constraint is tight, so R_k = r_k for all users.
  • The optimized precoders provide normalized vectors and stored beam-power values for subsequent algorithmic updates.
  • Replacing minimum rates with traffic demands tests feasibility; the paper assumes demands are high enough that all users cannot be satisfied simultaneously.
  • With fixed precoding vectors, the power step has linear constraints in p_k but remains generally non-convex because of the objective and nonlinear power constraints.

C. The Proposed Generic Iterative Algorithm and Proof of Convergence

The paper proposes a generic iterative algorithm that alternates precoding-vector and power-allocation optimization, then proves that the algorithm always converges.

  • Algorithm: The proposed generic iterative algorithm jointly optimizes precoding vectors and power allocation through alternating optimization.The algorithm initializes feasible precoding vectors and powers, updates rates, solves a precoding subproblem, then optimizes power allocation iteratively.
  • Algorithm: The algorithm initializes small powers satisfying linear power and maximum-rate constraints, then evaluates each user’s achievable rate.
  • Algorithm: The precoding update solves a per-beam power minimization problem with minimum-rate, general linear, and per-beam power constraints.The resulting vectors and powers are used to initialize the subsequent power-allocation update.
  • Algorithm: The power-allocation update uses a gradient-based algorithm with the previous power solution as its initial point, and iterations continue until convergence.
  • Convergence proof: The proposed Algorithm 1 always converges because its objective value decreases monotonically and is lower bounded by zero.The Step 3 update uses no more power per beam than Step 2, while Step 4 further optimizes power from the Step 3 solution.
  • Algorithm design: The Step 3 objective can use maximum beam power, maximum user power, or total transmit power depending on the power constraints and the desired emphasis.

IV. AN EXTENSION TO DPC AND A SPECIAL CASE FOR FAIRNESS MAXIMIZATION

The paper extends the optimization framework to dirty paper coding and develops a special fairness-maximization case under convex power constraints using auxiliary-variable reformulation and bisection.

  • DPC extension: Given a decoding order, the proposed MISO algorithm can be applied to dirty paper coding because the received SINR has a similar structure.
  • DPC extension: The optimal encoding order is computationally expensive, so the paper investigates heuristics that encode users according to channel conditions and rate requirements.Users with good channels are encoded first, while users with high rate requirements are decoded last.
  • Fairness maximization: The fairness special case extends worst-user weighted-rate maximization from total-power constraints to nonlinear but convex power constraints.
  • Fairness maximization: The method introduces an auxiliary variable γ to reformulate the fairness problem into an optimization problem that is convex except for γ.
  • Fairness maximization: Bisection search finds the optimum γ by repeatedly checking constraint feasibility, increasing γ when feasible and decreasing it otherwise.Restricting γ̃ ≤ 1 guarantees that r_k ≤ F_k for every user.

V. MULTIPLE ANTENNAS AT RECEIVE TERMINALS

The paper extends linear precoding to terminals with multiple receive antennas, examining co-polarization and dual-polarization configurations under channel models with polarization and rain-fading effects.

  • Scope: The multiple-antenna extension studies co-polarization and dual-polarization receive terminals in fixed multibeam satellite forward links.
  • Receive strategies: Four receive strategies are considered: co-polarization or cross-polarization antennas combined with receive beamforming or antenna/polarization selection.Receive beamforming is generally optimal, while selection has lower computational and implementation cost.
  • Receive strategies: Cross-polarization antennas have lower receive SNR because of power imbalance but better decorrelation due to propagation characteristics.The paper therefore evaluates the alternatives using realistic channel parameters rather than assuming one strategy is universally best.
  • Channel model: Each terminal’s two-antenna channel is modeled as a 2 × K matrix with full transmit-side correlation, while co-polarization receive antennas are also assumed fully correlated.
  • Channel model: The dual-polarization model uses a Kronecker correlation structure with transmit and receive correlation, polarization power imbalance, and rain-fading coefficients.The model also includes a uniformly distributed phase and a polarization correlation matrix.
  • Channel model: The polarization power-imbalance factor is related to Cross Polarization Discrimination, and channel matrices are normalized for fair comparisons across scenarios.

B. Receive Strategy I: Antenna Selection

The antenna-selection strategy shows that fully correlated co-polarization antennas reduce to the previously studied single-antenna case, while selection configurations yield equivalent multiuser MISO channels.

  • Antenna selection: Two fully correlated co-polarization receive antennas reduce to the single-antenna terminal scenario studied earlier.
  • Antenna selection: For the cross-polarization case, one option selects alternating satellite-feed polarizations, while another also selects the best terminal polarization.
  • Antenna selection: Selecting the best terminal polarization produces a 1 × N channel matrix.
  • Antenna selection: Both antenna-selection cases produce equivalent multiuser MISO channels, allowing the generic MISO algorithm to optimize their precoding vectors.

C. Receive Strategy II: RBF and The Proposed Generic MIMO Algorithm

The proposed generic MIMO algorithm alternates updates of receive beamforming, precoding, and power allocation to satisfy rate and power constraints. Simulations compare polarization strategies and precoding baselines using throughput, traffic matching, and power metrics.

  • Receive Beamforming: For fixed precoding vectors, the optimal RBF vector maximizes each user’s received SINR.The resulting RBF updates distinguish the MIMO procedure from the MISO algorithm.
  • Proposed Generic MIMO Algorithm: The generic MIMO algorithm initializes feasible receive beamforming, precoding, and power variables, then alternates rate evaluation, optimization, and receive-beamformer updates.Rate requirements prevent over-satisfaction during the receive beamforming update.
  • Algorithmic Extension: The algorithm extends the generic MISO method to terminals with multiple receive dimensions and is proved to always converge.The convergence proof uses the same argument as the MISO theorem.
  • MISO Comparison: 40% lower l2 norm is achieved by the proposed generic optimization scheme than by ZF, indicating substantially better traffic matching.The proposed scheme also achieves slightly higher throughput than ZF in the reported MISO comparison.
  • Power and Efficiency: The proposed scheme consumes less power than ZF while adapting allocation more closely to traffic demand within the per-beam power limit.ZF uses about 60% of the conventional scheme’s total power, while the proposed scheme prioritizes adaptive demand matching.
  • Polarization Strategies: Dual-polarization satellite feeds with RBF at dual-polarization terminals achieve the best performance, at the cost of increased hardware complexity.Co-polarization receive antennas significantly reduce the l2 norm relative to the MISO case.

VII. CONCLUSIONS

The paper develops convergent optimization methods for linear and nonlinear precoding in fixed multibeam satellite forward links, including multi-antenna terminals. Simulations demonstrate the effects of polarization choices, while practical deployment remains bounded by idealized feeder-link and CSI assumptions.

  • VII. CONCLUSIONS: The paper proposes a generic iterative algorithm for arbitrary individual-rate objectives and general linear and nonlinear power constraints, with proved convergence.The method is also extended to DPC precoding with fixed encoding order.
  • VII. CONCLUSIONS: The algorithms extend to terminals with two co-polarization or dual-polarization receive antennas, and simulations demonstrate co- and dual-polarization effects.The conclusion identifies polarization impact as an evaluated aspect of the proposed design.
  • VII. CONCLUSIONS: The study assumes perfect feeder links and perfect CSI at the gateway, which the authors identify as impractical and leave for future investigation.Future work also includes nonlinear THP precoding in a DVB-S2 system.

APPENDIX EFFICIENT ALGORITHM TO SOLVE (12)

The appendix solves the per-beam power minimization subproblem through a dual formulation and iterative updates of dual variables, virtual uplink powers, and downlink precoders.

  • Dual Formulation: The dual problem of (12) provides the basis for an efficient solution to the per-beam power minimization subproblem.The appendix explicitly introduces the dual problem before describing the iterative solver.
  • Dual Updates: With fixed λ and µ, the method solves for α and obtains primary solutions {t_k}, then updates λ and µ.This alternating dual-update strategy exploits the analytical structure of the subproblem.
  • Power and Precoding Recovery: The optimal solution is interpreted as virtual uplink power, while optimal precoding follows the resulting direction.Downlink power is recovered because the rate constraints hold with equality.
  • Iterative Procedure: The efficient algorithm initializes nonnegative λ and µ, repeatedly applies fixed-point updates, and updates downlink power and precoding vectors.The downlink vectors are scaled as t_k = √δ_k t̃_k.
  • Convergence Procedure: Subgradient updates adjust λ and µ using projected steps until the algorithm converges.The multipliers remain in their feasible set through projection.
Loading 1109.0681v1…