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Percentile Optimization in Wireless Networks- Part II: Beamforming for Cell-Edge Throughput Maximization

Ahmad Ali Khan, Raviraj Adve

TL;DR

This second part tackles the significantly more challenging problems of optimizing SLqP rate via beamforming in a multiuser, multiple-input multiple-output (MU-MIMO) network to maximize cell-edge throughput and proposes an adaptation of the QFT algorithm that enables optimization of the complex-valued multidimensional beamforming weights for the SLqP rate utility function.

Abstract

Part I of this two-part paper focused on the formulation of percentile problems, complexity analysis, and development of power control algorithms via the quadratic fractional transform (QFT) and logarithmic fractional transform (LFT) for sum-least-qth-percentile (SLqP) rate maximization problems. In this second part, we first tackle the significantly more challenging problems of optimizing SLqP rate via beamforming in a multiuser, multiple-input multiple-output (MU- MIMO) network to maximize cell-edge throughput. To this end, we first propose an adaptation of the QFT algorithm presented in Part I that enables optimization of the complex-valued multidimensional beamforming weights for the SLqP rate utility function. We also introduce a new class of problems which we term as sum-greatest-qth-percentile weighted mean squared error (SGqP-WMSE) minimization. We show that this class subsumes the well-known sum-weighted mean squared error (WMMSE) minimization and max-WMSE minimization problems. We demonstrate an equivalence between this class of problems and the SLqP rate maximization problems, and show that this correspondence can be exploited to obtain stationary-point solutions for the aforementioned beamforming problem. Next, we develop extensions for the QFT and LFT algorithms from Part I to optimize ergodic long-term average or ergodic SLqP utility. Finally, we also consider related problems which can be solved using the proposed techniques, including hybrid utility functions targeting optimization at specific subsets of users within cellular networks.

Percentile Optimization in Wireless Networks- Part II: Beamforming for Cell-Edge Throughput Maximization

TL;DR

This second part tackles the significantly more challenging problems of optimizing SLqP rate via beamforming in a multiuser, multiple-input multiple-output (MU-MIMO) network to maximize cell-edge throughput and proposes an adaptation of the QFT algorithm that enables optimization of the complex-valued multidimensional beamforming weights for the SLqP rate utility function.

Abstract

Part I of this two-part paper focused on the formulation of percentile problems, complexity analysis, and development of power control algorithms via the quadratic fractional transform (QFT) and logarithmic fractional transform (LFT) for sum-least-qth-percentile (SLqP) rate maximization problems. In this second part, we first tackle the significantly more challenging problems of optimizing SLqP rate via beamforming in a multiuser, multiple-input multiple-output (MU- MIMO) network to maximize cell-edge throughput. To this end, we first propose an adaptation of the QFT algorithm presented in Part I that enables optimization of the complex-valued multidimensional beamforming weights for the SLqP rate utility function. We also introduce a new class of problems which we term as sum-greatest-qth-percentile weighted mean squared error (SGqP-WMSE) minimization. We show that this class subsumes the well-known sum-weighted mean squared error (WMMSE) minimization and max-WMSE minimization problems. We demonstrate an equivalence between this class of problems and the SLqP rate maximization problems, and show that this correspondence can be exploited to obtain stationary-point solutions for the aforementioned beamforming problem. Next, we develop extensions for the QFT and LFT algorithms from Part I to optimize ergodic long-term average or ergodic SLqP utility. Finally, we also consider related problems which can be solved using the proposed techniques, including hybrid utility functions targeting optimization at specific subsets of users within cellular networks.
Paper Structure (26 sections, 5 theorems, 45 equations, 8 figures, 1 table, 3 algorithms)

This paper contains 26 sections, 5 theorems, 45 equations, 8 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Overview
  3. Contributions of Part II
  4. Notation
  5. Organization
  6. Beamforming for Sum-Percentile Rate Optimization
  7. Problem Formulation
  8. Multidimensional Quadratic Fractional Transform (MQFT)
  9. Simulation Results
  10. Sum-Greatest $q^\mathrm{th}$ Weighted Mean Squared Error (SGqP-WMSE) Minimization
  11. Background
  12. Proposed Approach
  13. Discussion
  14. Simulation Results
  15. Hybrid Utility Functions
  16. ...and 11 more sections

Key Result

Lemma 1

Let $\mathbf{a}(\mathbf{V}):\mathrm{\mathbb{C}}^{d_{1}\times{1}}\mapsto\mathbb{C}^{d_{2}\times{1}}$, $\mathbf{B}(\mathbf{V}):\mathrm{\mathbb{C}}^{d_{1}\times{1}}\mapsto\mathbb{S}_{++}^{d_{2}\times d_{2}}$. Then we have where $\boldsymbol{\chi}\in\mathbb{C}^{d_{2}\times1}$ is an auxiliary variable.

Figures (8)

  • Figure 1: Convergence of SLqP rate objective for $K=35$, $K_{q}=2$.
  • Figure 2: Simultaneous convergence of SGqP-WMSE and SLqP-rate for $K=35$, $K_{q}=2$.
  • Figure 3: Convergence of SLqP utility for $K=14$ and $K_q=2$.
  • Figure 4: Tradeoff between sum-rate and cell-edge rate for different values of $w$.
  • Figure 5: Convergence of QFT algorithm for hybrid $10^\mathrm{th}$-percentile and $5^\mathrm{th}$-percentile utilities
  • ...and 3 more figures

Theorems & Definitions (24)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • proof
  • ...and 14 more