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On the Closed-Form Solution for Robust Adaptive Beamforming

Licheng Zhao, Rui Zhou, Wenqiang Pu

TL;DR

A novel closed-form solution scheme containing three consecutive stages containing three consecutive stages: Diagonalization Transform, Phase Alignment, and KKT Solution is developed, specifically intended for the RAB problem and thus more efficient than MOSEK.

Abstract

In this paper, we consider the classical robust adaptive beamforming (RAB) problem. Conventionally, this problem is solved either with an off-the-shelf solver like MOSEK or through the well-known RMVB algorithm based on Lagrange multiplier approaches. The solver MOSEK is implemented with the general interior point method and RMVB is only limited to the full-rank covariance scenario. In order to improve the existing benchmarks, we develop a novel closed-form solution scheme containing three consecutive stages: Diagonalization Transform, Phase Alignment, and KKT Solution. The proposed scheme is specifically intended for the RAB problem and thus more efficient than MOSEK. Moreover, the derivation process is simpler than RMVB and the output solution can cover the rank-deficient covariance scenario in extra. Aside from a new solution, we manage to unveil the existence and uniqueness conditions, which have never been studied before. The simulation results show that the proposed solution improves the existing benchmarks in terms of computational time while maintaining optimality.

On the Closed-Form Solution for Robust Adaptive Beamforming

TL;DR

A novel closed-form solution scheme containing three consecutive stages containing three consecutive stages: Diagonalization Transform, Phase Alignment, and KKT Solution is developed, specifically intended for the RAB problem and thus more efficient than MOSEK.

Abstract

In this paper, we consider the classical robust adaptive beamforming (RAB) problem. Conventionally, this problem is solved either with an off-the-shelf solver like MOSEK or through the well-known RMVB algorithm based on Lagrange multiplier approaches. The solver MOSEK is implemented with the general interior point method and RMVB is only limited to the full-rank covariance scenario. In order to improve the existing benchmarks, we develop a novel closed-form solution scheme containing three consecutive stages: Diagonalization Transform, Phase Alignment, and KKT Solution. The proposed scheme is specifically intended for the RAB problem and thus more efficient than MOSEK. Moreover, the derivation process is simpler than RMVB and the output solution can cover the rank-deficient covariance scenario in extra. Aside from a new solution, we manage to unveil the existence and uniqueness conditions, which have never been studied before. The simulation results show that the proposed solution improves the existing benchmarks in terms of computational time while maintaining optimality.
Paper Structure (16 sections, 3 theorems, 54 equations, 8 figures, 1 table)

This paper contains 16 sections, 3 theorems, 54 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement and Closed-Form Solution
  3. Diagonalization Transform
  4. Phase Alignment
  5. KKT Solution
  6. Full Rank Quadratic Matrix
  7. Rank Deficient Quadratic Matrix
  8. Solution Summarization
  9. Existence and Uniqueness Analysis
  10. Existence Study
  11. Uniqueness Discussion
  12. Analysis Summary
  13. Numerical Simulations
  14. Full-Rank-Covariance RAB Formulation
  15. Rank-Deficient-Covariance RAB Formulation
  16. ...and 1 more sections

Key Result

Lemma 2

Suppose that there exists a solution to problem eq:opt prob 2. The optimal solution to the phase of $\mathbf{v}$ is given as To ensure mathematical consistency, we define the phases of zero-valued elements as 0.

Figures (8)

  • Figure 1: Existence and uniqueness conditions.
  • Figure 2: Constraint satisfaction versus problem dimension $N$ with full rank $\mathbf{R}$.
  • Figure 3: Optimality gap versus problem dimension $N$ with full rank $\mathbf{R}$.
  • Figure 4: Computational time versus problem dimension $N$ with full rank $\mathbf{R}$.
  • Figure 5: Constraint satisfaction versus problem dimension $N$ with rank deficient $\mathbf{R}$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Remark 1
  • Lemma 2
  • Lemma 3
  • Remark 4
  • Remark 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Example 10
  • ...and 3 more