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Downlink Transmit Design for Massive MIMO LEO Satellite Communications
Ke-Xin Li, Li You, Jiaheng Wang, Xiqi Gao, Christos G. Tsinos, Symeon Chatzinotas, Björn Ottersten
TL;DR
The paper studies how to design downlink transmission for massive MIMO LEO satellite systems when the transmitter has only slow-varying statistical channel information. It exploits channel structure to derive single-stream precoding, lower-complexity scalar optimization, and a learning-based implementation. The proposed methods achieve near-optimal performance with substantially reduced computational complexity and outperform existing schemes in simulations.
Problem
Practical LEO satellite propagation effects make instantaneous CSIT difficult to obtain, while limited payload resources demand efficient transmit designs that account for satellite-channel structure.
Method
The paper models massive MIMO LEO channels with uniform planar arrays, proves single-stream optimality, designs precoding vectors and scalar variables, and uses an MLP to learn the scalar optimization.
Results
5.6% of Algorithm 2’s complexity is required by the NN-based approach, while simulations show near-optimal performance and gains over existing schemes.
Takeaways & Limitations
Single-stream precoding simplifies transmit-covariance optimization, while scalar-variable and learning-based designs provide lower-complexity solutions for massive MIMO LEO downlinks.
Abstract
from arXiv · showhide
This paper investigates the downlink (DL) transmit design for massive multiple-input multiple-output (MIMO) low-earth-orbit (LEO) satellite communication systems, where only the slow-varying statistical channel state information is exploited at the transmitter. The channel model for the DL massive MIMO LEO satellite system is established, in which both the satellite and the user terminals (UTs) are equipped with uniform planar arrays. Observing the rank-one property of the channel matrices, we show that the single-stream precoding for each UT is the optimal choice that maximizes the ergodic sum rate. This favorable result simplifies the complicated design of transmit covariance matrices into that of precoding vectors without any loss of optimality. Then, an efficient algorithm is devised to compute the precoding vectors. Furthermore, we formulate an approximate transmit design based on the upper bound on the ergodic sum rate, for which the optimality of single-stream precoding still holds. We show that, in this case, the design of precoding vectors can be simplified into that of scalar variables, for which an effective algorithm is developed. In addition, a low-complexity learning framework is proposed for optimizing the scalar variables. Simulation results demonstrate that the proposed approaches can achieve significant performance gains over the existing schemes.
I. INTRODUCTION
The paper addresses downlink transmit design for massive MIMO LEO satellite communications under practical constraints on channel knowledge and onboard computation. It develops statistical-CSI-based precoding methods that exploit channel structure to simplify optimization while improving performance.
- Motivation: LEO satellite systems offer broad connectivity but require advanced processing to manage full-frequency-reuse interference and changing link conditions.LEO systems provide shorter delay and lower pathloss than geostationary satellites, while full frequency reuse increases the need for interference mitigation.
- System and objective: The proposed system uses digitally reconfigurable massive-MIMO arrays at the satellite and uniform planar arrays at both the satellite and user terminals.Digital reconfiguration is intended to adapt beamforming to dynamic user-terminal link conditions.
- Motivation: Instantaneous CSIT is difficult to acquire because satellite propagation delays and Doppler effects can make channel estimates outdated.The paper therefore focuses on slow-varying statistical CSIT for downlink transmission.
- Contributions: Single-stream transmission per user terminal maximizes ergodic sum rate and reduces transmit-covariance design to precoding-vector design without loss of optimality.The result applies even though each user terminal has multiple antennas.
- Contributions: An upper-bound-based design retains single-stream optimality and further reduces precoding-vector optimization to scalar-variable optimization.The paper develops an algorithm for the resulting scalar problem and a learning-based solution with lower onboard implementation complexity.
2) Propagation Delays:
The channel model accounts for propagation, array geometry, and OFDM transmission in LEO satellite links. After Doppler and delay compensation, the received signal is represented through an effective frequency-domain channel response.
- Propagation Delays: LEO propagation delay is more severe than in terrestrial channels because of the long satellite-to-user-terminal distance.At 1000 km altitude and 45° elevation, the round-trip delay is about 17.7 ms.
- Channel representation: The channel model represents each path using complex gain, Doppler shift, propagation delay, and satellite- and terminal-side array response vectors.These quantities describe the multipath channel associated with each user terminal.
- Array response vectors: Because scattering occurs near each user terminal, the satellite-side departure angles across paths are treated as nearly identical.This permits the satellite array response to be represented by a common vector for a user terminal and known slowly varying geometry.
- OFDM transmission: OFDM transmission uses subcarriers, a cyclic prefix, and symbol durations defined from the sampling period to support wideband signaling.The model specifies the subcarrier spacing and OFDM symbol timing before compensation.
- OFDM transmission: After joint Doppler and delay compensation, the model defines an effective channel frequency response and expresses each received subcarrier signal using a channel matrix and Gaussian noise.The resulting representation is used for the subsequent frequency-domain transmit design.
D. Statistical Properties of Channel
The paper models the statistical channel and formulates downlink transmit design under total-power constraints. Exploiting the channel structure, it establishes rank-one optimal transmit covariance matrices, reducing covariance optimization to precoding-vector design.
- Statistical channel model: The channel is modeled as Rician, with deterministic LoS and random scattering components whose parameters remain nearly unchanged within a specified movement range.The channel parameters must be updated when the satellite or a user terminal leaves that range.
- Channel correlation: The satellite-side channel correlation matrix is rank-one, indicating highly correlated signals across satellite antennas, while UT-side rank depends on the propagation environment.
- Rank-one optimality: The optimal transmit covariance matrix for every user terminal has rank no greater than one, so single-stream transmission is optimal even for multi-antenna terminals.
- Precoding formulation: Rank-one covariance matrices rewrite the covariance optimization as precoding-vector optimization without loss of optimality.The resulting vectors are collected in W and optimized under the reformulated sum-rate problem.
- Transmit model: The downlink model serves K user terminals simultaneously using Gaussian transmit signals with covariance matrices under a total transmit-power constraint.The received signal, noise, ergodic rates, and sum-rate maximization are defined from these quantities.
B. Optimal Linear Receivers
The paper derives optimal linear receivers after reducing transmission to one stream per user terminal. Each receiver aligns with its effective channel, while an MM algorithm addresses the resulting non-convex precoding-vector problem.
- Receiver structure: Because each terminal receives at most one stream, its linear receiver only needs to recover one data stream, with multiple antennas providing diversity gain.
- SINR formulation: The recovered symbol and SINR are formed from the desired signal, multiuser interference, and additive noise after linear combining.
- Optimal receiver: The optimal receiver is proportional to the effective channel vector, as established using the Cauchy-Schwarz inequality.Any nonzero complex scaling produces the same SINR.
- Precoding algorithm: The non-convex precoding-vector problem is solved with an MM algorithm that replaces each ergodic rate by a concave minorizing function.Each iteration then solves a convex program to obtain the next precoders.
- Computational consideration: Computing ergodic rates with exhaustive Monte Carlo sample averaging can become computationally demanding when many samples are used.The paper therefore introduces subsequent low-complexity designs that avoid sample averaging.
IV. TRANSMIT DESIGNS WITH ERGODIC SUM RATE UPPER BOUND
The paper approximates the ergodic sum rate with an upper bound to avoid exhaustive sample averaging, while retaining optimal single-stream precoding. This reduces covariance-matrix design to precoding-vector and then scalar-variable optimization.
- Rank-One Property: The upper-bound transmit design avoids exhaustive sample averaging and still has optimal rank-one transmit covariance matrices.Thus, single-stream precoding for each user terminal suffices to maximize the upper bound on the ergodic sum rate.
- Design Objective: The upper-bound formulation provides a lower-complexity alternative to exhaustive ergodic-sum-rate optimization.Its motivation is to avoid exhaustive sample averaging while preserving the single-stream structure.
- Precoding Reformulation: The upper-bound problem can be reformulated from transmit covariance matrices into precoding-vector design.The covariance matrix is written as Q_k = w_kw_k^H under the rank-one result.
- Required Information: The required channel parameters depend on user-terminal locations, average channel powers, and uplink sounding with statistical-channel reciprocity.With fixed planar-array placement, space-angle pairs can be derived from satellite and user-terminal location information obtained through GPS.
B. Precoding Vector Design
The precoding-vector design is further reduced to optimizing K scalar variables, from which the precoding vectors can be obtained in closed form. An MM-based algorithm solves the scalar problem, but its iterative complexity motivates a learning-based alternative.
- Scalar Reformulation: The high-dimensional precoding-vector design is transformed into the optimization of K scalar variables.The scalar variables determine the precoding vectors through a closed-form calculation.
- MM Algorithm: The scalar-variable problem is addressed with the minorization-maximization framework by sequentially solving convex subproblems.The method constructs concave minoring functions and iteratively updates the scalar variables.
- Algorithmic Cost: The resulting precoding vectors are computed after the scalar variables are determined.The stated total complexity of Algorithm 2 is N_iter(K^3+2K^2M)+K^3+K^2M.
- Complexity Limitation: Algorithm 2 uses complicated iterations that challenge real-time processing on satellites with limited payloads.A learning framework is therefore proposed to compute the scalar variables with reduced onboard implementation complexity.
C. Learning to Compute Scalar Variables {λk}K
The learning framework uses a multilayer perceptron to approximate the mapping from channel-related inputs to optimized scalar variables. Its low-dimensional input and output reduce onboard computation, especially as antenna and user-terminal dimensions grow.
- Complexity: The neural network has lower implementation complexity than Algorithm 2 because it uses simple matrix-vector multiplications and activation functions.Its input and output dimensions are 3K+1 and K, respectively, independent of the number of satellite and user-terminal antennas.
- Learned Mapping: A neural network learns the nonlinear mapping from transmit power and channel parameters to the normalized scalar variables.The mapping has input dimension 3K+1 and output dimension K.
- Network Architecture: The proposed learner is a multilayer perceptron using ReLU activation functions.The ReLU function is G(x) = max{x, 0}, producing non-negative outputs at each layer.
- Training and Deployment: Training uses mean squared error, can be performed offline at the ground station, and is followed by normalization during testing.The predicted scalar variables are then used to calculate the precoding vectors.
V. SIMULATION RESULTS
The simulations evaluate convergence, sum-rate performance, and complexity for the proposed algorithms and neural-network approach under the stated channel and sampling setup.
- Simulation setup: The simulations generate user-terminal space-angle pairs with Poisson disk sampling and minimum distance ρmin = 0.037, ensuring at least 3 dB interference power decay.The channel simulations use 1000 samples, suburban propagation assumptions, and the specified noise model.
- Convergence and complexity: Algorithms 1 and 2 converge within about 20 iterations, so both simulations set Niter = 20.The NN-based approach has approximately 5.6% of Algorithm 2's complexity.
- Sum-rate performance: Algorithms 1 and 2, the NN-based approach, and perfect-iCSIT precoding are compared using downlink sum-rate performance.The comparison is presented in Fig. 5, with the perfect-iCSIT scheme derived from the MM algorithm.
- Sum-rate performance: The sCSIT-based proposed methods achieve performance close to perfect-iCSIT precoding, while Algorithms 1 and 2 have negligible sum-rate differences.The NN-based method achieves near-optimal performance with much lower computational complexity.
- Comparison with prior precoding: At P = 25 dBW, the ASLNR precoding vectors incur almost 1 dB performance loss relative to the NN-based approach.The proposed precoding designs rely only on slow-varying sCSI, which is independent of subcarriers and OFDM symbols within a stable sCSI period.
VI. CONCLUSION
The paper develops statistical-CSI downlink transmit designs for massive MIMO LEO satellite systems and verifies their effectiveness through simulations.
- Contributions: The paper derives a massive MIMO LEO satellite channel model with uniform planar arrays at the satellite and user terminals.The transmit design uses slow-varying statistical channel state information at the transmitter.
- Contributions: Single-stream precoding for each user terminal maximizes the ergodic sum rate for the linear transmitters.This result is established for both the direct ergodic-sum-rate design and the upper-bound-based design.
- Contributions: The direct design computes precoding vectors, while the upper-bound formulation reduces their optimization to scalar variables.An effective algorithm is developed for the scalar-variable problem.
- Contributions: A learning-based solution computes the scalar variables with much lower implementation complexity than iterative algorithms.Simulation results verify the effectiveness and performance gains of the proposed designs.
APPENDIX A PROOF OF THEOREM 1
The appendix proves that the optimal transmit covariance solution has rank one by combining KKT conditions, matrix-rank arguments, and complementary slackness.
- Proof strategy: The proof first establishes that the optimal solution to P is rank one, then applies the same proof steps to the upper-bound problem Pub.The gradient with respect to each transmit covariance matrix is used in the KKT analysis.
- KKT conditions: KKT conditions introduce nonnegative multipliers for the total-power and positive-semidefinite covariance constraints.The optimal covariance gradient must vanish under these conditions.
- Rank argument: The dual matrix has rank at least M − 1, while complementary slackness implies that its product with the optimal covariance matrix is zero.The rank inequalities are used to obtain the lower bound on the dual-matrix rank.
- Conclusion: The resulting rank bound is rank(Qk) ≤ 1, which concludes the rank-one optimality proof.This establishes the single-stream structure of each user's optimal transmit covariance.
A MINORIZING FUNCTION OF rk
The paper constructs a minorizing function for each virtual uplink rate from the MMSE representation and uses it to support scalar-variable optimization.
- Virtual uplink model: The virtual uplink models single-antenna users transmitting to an M-antenna base station with linear receivers and no successive interference cancellation.Each user transmits one data stream with nonnegative power λi.
- MMSE representation: The minimum virtual MSE is obtained with the specified linear receiver, yielding the corresponding virtual MMSE for each user.The rate is rewritten as rk = −log VMMSEk.
- Iterative surrogate: At each iteration, the scalar variables λk determine the virtual MMSE values used in the rate surrogate.The virtual MMSE is evaluated using the scalar variables from the current iteration.
- Iterative surrogate: Concavity of log(·) provides a minorizing function for rk, with equality enforced at the current scalar-variable iterate.The construction supports iterative optimization of the scalar-variable problem.