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Joint User Selection, Power Allocation, and Precoding Design with Imperfect CSIT for Multi-Cell MU-MIMO Downlink Systems

Jiwook Choi, Namyoon Lee, Song-Nam Hong, Giuseppe Caire

arXiv:1902.00824v1cs.ITmath.IT

TL;DR

Accurate CSI is important in MU-MIMO downlink systems, but computing exact mutual information is challenging. The paper proposes a joint linear preprocessing approach for user selection, power allocation, and precoding, and reports considerable ergodic sum-spectral-efficiency gains across system configurations.

  • Problem

    Accurate CSI is important in MU-MIMO downlink systems, while computing exact mutual information is very challenging.

    Source: A. Motivation · p. 2, B. A Lower Bound of Instantaneous… · p. 11

  • Method

    The paper proposes a linear preprocessing method that jointly designs user selection, power allocation, and precoding for multi-cell MU-MIMO downlink systems.

    Source: §VII · p. 32

  • Results

    Simulations show considerable gains in ergodic sum-spectral efficiencies over existing precoding methods across numerous system configurations.

    Source: §VII · p. 32

  • Takeaways & Limitations

    The resulting joint-design solution is sub-optimal and satisfies the first-order optimality condition.

    Source: §VII · p. 32

Abstract

from arXiv · show

In this paper, a new optimization framework is presented for the joint design of user selection, power allocation, and precoding in multi-cell multi-user multiple-input multiple-output (MU-MIMO) systems when imperfect channel state information at transmitter (CSIT) is available. By representing the joint optimization variables in a higher-dimensional space, the weighted sum-spectral efficiency maximization is formulated as the maximization of the product of Rayleigh quotients. Although this is still a non-convex problem, a computationally efficient algorithm, referred to as generalized power iteration precoding (GPIP), is proposed. The algorithm converges to a stationary point (local maximum) of the objective function and therefore it guarantees the first-order optimality of the solution. By adjusting the weights in the weighted sum-spectral efficiency, the GPIP yields a joint solution for user selection, power allocation, and downlink precoding. The GPIP is also extended to a multi-cell scenario, where cooperative base stations perform joint user selection and design their precoding vectors by sharing global yet imperfect CSIT within the cooperative BSs. System-level simulations show the gains of the proposed approach with respect to conventional user selection and linear downlink precoding.

I. INTRODUCTION

MU-MIMO downlink design must manage inter-user and inter-cell interference despite imperfect CSI, while jointly choosing users, power, and precoding is difficult. Prior approaches leave gaps in imperfect-CSIT support or joint design.

A. Motivation

  • Perfect CSI is infeasible in practice, and imperfect CSIT complicates optimization because exact achievable spectral efficiency is unavailable and its errors must be accounted for.In TDD, non-orthogonal uplink pilots can contaminate channel measurements across cells.Source: A. Motivation · p. 2
  • Joint user selection, power allocation, and precoding are challenging because each user’s SINR depends on other users’ scheduling and transmission choices.The joint weighted sum-spectral efficiency problem is described as NP-hard.Source: A. Motivation · p. 2
  • The paper proposes a computationally efficient heuristic framework for weighted sum-spectral efficiency maximization under imperfect CSIT.Source: A. Motivation · p. 2

B. Prior Works

  • Earlier studies assumed perfect CSIT, while the cited imperfect-CSIT studies focused only on precoding and did not jointly account for user selection and power allocation methods together with multi-user precoding.Source: B. Prior Works · p. 4
  • Some imperfect-CSIT strategies optimize degrees of freedom, which the authors describe as of limited use at practical finite SNR.Source: B. Prior Works · p. 4

C. Contributions

The paper reformulates joint user selection, power allocation, and precoding as a tractable non-convex optimization and develops GPIP for first-order optimality. It extends the framework to cooperative multi-cell transmission and reports simulation gains.

C. Contributions

  • The framework derives a lower bound on weighted sum-spectral efficiency using GMI and reformulates the joint design as a product of Rayleigh quotients, removing user-selection integer constraints.Source: C. Contributions · p. 4
  • GPIP treats the reformulated problem as a functional generalized eigenvalue problem and converges to a solution satisfying first-order optimality for joint user selection, power allocation, and precoding.Source: C. Contributions · p. 5
  • The framework extends to cooperative base stations sharing imperfect CSIT, with joint user scheduling, power allocation, and precoding under individual BS power constraints.The cooperative formulation accounts for inter-cell interference as colored noise.Source: C. Contributions · p. 5
  • The multi-cell reformulation is presented as applicable across different numbers of antennas per BS, users per cell, and cooperative BSs.Source: C. Contributions · p. 5
  • With perfect CSIT in single-cell MU-MIMO, the algorithm matches ZF-DPC sum-spectral efficiency at low SNR regardless of BS antennas or user count.The authors state this implies linear precoding can approach sum-capacity under a condition when selection and power allocation are jointly performed.Source: C. Contributions · p. 5
  • System-level simulations show noticeable spectral-efficiency gains over conventional user-selection and precoding methods under imperfect CSIT, including significant gains from cooperative transmission.Source: C. Contributions · pp. 5–6
  • Under imperfect CSIT, sum-spectral efficiency improves as users per cell increase for fixed BS antennas and as BS antennas increase for a fixed user count.Source: C. Contributions · pp. 5–6

II. SYSTEM MODEL

The system model describes a multi-cell MU-MIMO downlink and its spatially correlated channels, then specifies the assumptions used to represent imperfect CSIT.

A. Network and Channel Model

B. CSIT Assumption

  • The CSIT model uses an MMSE channel estimate and Gaussian estimation error independent of that estimate, with error covariance Φℓ,ℓ,k = Rℓ,ℓ,k − Rhatℓ,ℓ,k.Source: B. CSIT Assumption · p. 7
  • The framework applies to TDD estimation using uplink pilots and to FDD, where feedback-bit quantity determines CSIT error accuracy.Source: B. CSIT Assumption · p. 7

C. Downlink Ergodic Spectral Efficiency

The section models downlink transmission and rates under imperfect CSIT, then formulates weighted spectral-efficiency maximization over user selection, power, and precoding. Exhaustive user-subset search grows exponentially, while the precoding and power problem is non-convex.

C. Downlink Ergodic Spectral Efficiency

III. PROBLEM FORMULATION

  • With imperfect CSIT, the BS may overestimate instantaneous rates and transmit at an undecodable rate, although it can compute an instantaneous rate using the CSIT-error distribution.Source: A. Instantaneous Spectral Efficiency… · p. 9

A. Instantaneous Spectral Efficiency Maximization Problem with Imperfect CSIT

B. A Lower Bound of Instantaneous Spectral Efficiency with Imperfect CSIT

Because imperfect CSIT makes the interference-plus-noise non-Gaussian, the paper uses generalized mutual information to form a lower bound on instantaneous spectral efficiency. It then optimizes that bound jointly over scheduled users, precoding vectors, and power allocations.

IV. JOINT USER-SELECTION, PRECODING, AND POWER ALLOCATION ALGORITHM

The algorithm reformulates joint user selection, precoding, and power allocation as a product of Rayleigh quotients. It seeks a feasible first-order stationary solution efficiently, without guaranteeing a global optimum.

IV. JOINT USER-SELECTION, PRECODING, AND POWER ALLOCATION ALGORITHM

  • The proposed algorithm jointly finds the downlink user set, information-symbol power allocation, and precoding vectors while maximizing a lower bound on weighted instantaneous spectral efficiency.Source: §IV · p. 12

A. The Proposed Optimization Approach

P INK. (17)

  • Directly solving the first-order KKT condition is difficult because it entails NK nonlinear equations, especially when NK ≥ 1000.Source: P INK. (17) · p. 14
  • The proposed computationally efficient algorithm converges to a feasible point satisfying the first-order KKT condition, but that point may not be globally optimal.Source: P INK. (17) · p. 15

B. Generalized Power Iteration Precoding

The section introduces GPIP as a computationally efficient approach to the joint design problem, using a generalized first-order optimality condition. It also discusses the algorithm’s computational complexity and convergence behavior.

B. Generalized Power Iteration Precoding

  • GPIP identifies the principal eigenvector of a system matrix to address the computational difficulty of the optimization.The section describes the algorithm as simple and computationally efficient.Source: B. Generalized Power Iteration Precoding · p. 15

GENERALIZED POWER ITERATION PRECODING (GPIP) ALGORITHM

  • The GPIP framework generalizes a first-order optimality condition to use different weights for each Rayleigh quotient.The paper notes that GPIP was originally developed in prior work for a two-way relay channel with a multi-antenna relay.Source: GENERALIZED POWER ITERATION PRECODING… · p. 16
  • By adjusting the weights, the algorithm jointly designs user selection, power allocation, and precoding.The paper identifies this joint design as a different class of problem to which GPIP applies.Source: GENERALIZED POWER ITERATION PRECODING… · p. 16

COMPUTATIONAL COMPLEXITY OF THE GPIP ALGORITHM

D. No Channel Covariance Matrix Information

Without channel covariance information, the method uses an identity-based estimate of the channel-error covariance. The section also describes the resulting complexity reduction and illustrates how GPIP can deactivate a user through its precoding solution.

D. No Channel Covariance Matrix Information

  • When covariance information is unavailable, the estimate treats spatial covariance as identity and models error covariance as αℓ,ℓ,kI_N.The paper notes that estimating and tracking spatial channel covariance is challenging for large antenna arrays.Source: D. No Channel Covariance Matrix… · p. 17
  • A mismatch in the channel-error covariance estimate can reduce spectral efficiency.The section identifies this as a consequence of using imperfect channel-error covariance information.Source: D. No Channel Covariance Matrix… · p. 17
  • The diagonal structure of the estimated covariance enables Sherman–Morrison updates, replacing matrix inversion with vector multiplication and reducing computational complexity.The section says this reduction is particularly relevant for systems with many antennas.Source: D. No Channel Covariance Matrix… · pp. 17–18

E. Joint User Selection, Power Allocation, and Precoding

  • In the example, GPIP selects users 1 and 2 and deactivates user 3 by assigning it a near-zero precoding vector.The section presents this joint user-selection, power-allocation, and precoding behavior for (N, K) = (2, 3).Source: E. Joint User Selection, Power… · p. 19

V. EXTENSION TO MULTI-CELL COOPERATIVE TRANSMISSION

The paper extends GPIP to cooperative multi-cell downlink transmission, with cooperating base stations sharing global but imperfect CSIT. The resulting algorithm converges to a stationary point under the stated sum-power constraint.

A. BS Cooperation Model

  • Cooperating base stations share global imperfect CSIT over high-speed, error-free backhauls, but do not share downlink data.The model assumes each BS estimates CSIT for users in the cooperative cluster using orthogonal uplink pilots.Source: A. BS Cooperation Model · p. 20

B. Cooperative Precoding for Multi-Cell MU-MIMO

  • The cooperative precoding design is formulated as maximizing a product of Rayleigh quotients, and the non-convex problem is solved using a suboptimal first-order condition.The cooperative algorithm modifies GPIP and rescales its solution to satisfy each BS’s individual power constraint.Source: B. Cooperative Precoding for… · p. 23

GENERALIZED POWER ITERATION FOR MULTI-CELL PRECODING

  • The cooperative algorithm converges to a stationary point of the optimization problem with the sum-power constraint.The section also reports that its computational complexity scales quadratically with the number of cooperative BSs.Source: GENERALIZED POWER ITERATION FOR… · p. 24

VI. SIMULATION RESULTS

In single-cell link-level simulations, GPIP outperforms existing linear precoding methods under perfect CSIT, with gains over SUS-ZF at low and mid SNR. Under imperfect CSIT, performance depends on whether error-covariance information is available.

A. Link Level Simulations

  • Under perfect CSIT with (N, K) = (64, 64), GPIP provides higher sum-spectral efficiency than ZF, RZF, and SUS-ZF in all SNR regimes, gaining about 3 dB and 1.5 dB over SUS-ZF at low and mid SNR.Source: A. Link Level Simulations · p. 25
  • The gap between GPIP and ZF-DPC grows as SNR increases, which the paper attributes to the limitation of linear processing.The comparison reports near-optimal GPIP sum-spectral efficiency relative to ZF-DPC in the low-SNR regime.Source: A. Link Level Simulations · p. 26
  • With error-covariance matrix information, proposed GPIP provides about 3 dB gain over RRZF in all SNR ranges; when covariance information is available for both RZF and GPIP, their sum-spectral efficiencies decrease above 12 dB and 15 dB, respectively.Source: A. Link Level Simulations · p. 26

B. System Level Simulations

System-level simulations find that GPIP outperforms existing methods across antenna and user-count settings, while cooperative transmission and proportional-fair weights improve rate outcomes in their evaluated settings. Its precoding vector quickly converges to a stationary point, with four iterations sufficient for the stated stopping condition ε = 0.1.

Convergence

  • With ε = 0.1, four iterations suffice to end GPIP; its precoding vector quickly converges to a stationary point across the tested (N, K) settings.The convergence speed depends on the ratio between N and K.Source: B. System Level Simulations · p. 28, Figure 3 · p. 28

Rate distributions of non-cooperative transmission methods

  • For (N, K) = (64, 10), uniform-weight GPIP gains spectral efficiency over existing precoding methods, while PF weights improve cell-edge users' performance and degrade cell-center users'.Uniform-weight GPIP exhibits user selection with a water-filling power-control effect, as users with bad-channel conditions are not served.Source: B. System Level Simulations · p. 29, Figure 4 · p. 29

Rate distributions of multi-cell coordinated scheduling and precoding

Effects of the number BS antennas

Effects of the number users per cell

  • As K increases with N = 64, GPIP outperforms SUS-ZF, rank-adaptation with ZF, RRZF, and MRT; unlike ZF beyond K ≥40, its sum-spectral efficiency improves.The paper concludes user selection is essential when N/K < 2, with additional gain from its power allocation and precoding.Source: B. System Level Simulations · p. 31, Figure 7 · p. 32
  • Ergodic sum-spectral efficiency gains become larger as users per cell increase, particularly in the three-BS cooperation scenario, because multi-cell scheduling yields greater multi-user diversity (N = 64).Source: B. System Level Simulations · p. 31, Figure 7 · p. 32

VII. CONCLUSION

The paper reformulates weighted sum-spectral-efficiency maximization as a product of Rayleigh quotients and proposes an efficient algorithm for joint user selection, power allocation, and precoding. Simulations report considerable ergodic sum-spectral-efficiency gains over existing precoding methods across numerous system configurations.

VII. CONCLUSION

  • The framework reformulates weighted sum-spectral-efficiency maximization as maximizing a product of Rayleigh quotients.The objective remains non-convex.Source: §VII · p. 32
  • The proposed computationally efficient algorithm ensures the first-order optimality condition and finds a sub-optimal solution to the weighted sum-spectral-efficiency problem.The conclusion describes the method as a linear preprocessing method for downlink multi-cell MU-MIMO.Source: §VII · p. 32
  • The precoding solution jointly determines user selection, power allocation, and precoding for the cellular system.Source: §VII · p. 32
  • Simulations show considerable ergodic sum-spectral-efficiency gains over existing precoding methods for numerous system configurations.Source: §VII · p. 32
  • The authors identify extensions to mmWave systems with hybrid precoding, joint transmission for C-RANs, and non-orthogonal multiple-access downlink precoding.Source: §VII · p. 32
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