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Simplified Information Geometry Approach for Massive MIMO-OFDM Channel Estimation -- Part II: Convergence Analysis

Jiyuan Yang, Yan Chen, Mingrui Fan, Xiqi Gao, Xiang-Gen Xia, Dirk Slock

TL;DR

This two-part paper proves the convergence of the simplified information geometry approach (SIGA) proposed in Part I and determines the range of the damping factor for massive MIMO-OFDM channel estimation by using the specific properties of the measurement matrices.

Abstract

In Part II of this two-part paper, we prove the convergence of the simplified information geometry approach (SIGA) proposed in Part I. For a general Bayesian inference problem, we first show that the iteration of the common second-order natural parameter (SONP) is separated from that of the common first-order natural parameter (FONP). Hence, the convergence of the common SONP can be checked independently. We show that with the initialization satisfying a specific but large range, the common SONP is convergent regardless of the value of the damping factor. For the common FONP, we establish a sufficient condition of its convergence and prove that the convergence of the common FONP relies on the spectral radius of a particular matrix related to the damping factor. We give the range of the damping factor that guarantees the convergence in the worst case. Further, we determine the range of the damping factor for massive MIMO-OFDM channel estimation by using the specific properties of the measurement matrices. Simulation results are provided to confirm the theoretical results.

Simplified Information Geometry Approach for Massive MIMO-OFDM Channel Estimation -- Part II: Convergence Analysis

TL;DR

This two-part paper proves the convergence of the simplified information geometry approach (SIGA) proposed in Part I and determines the range of the damping factor for massive MIMO-OFDM channel estimation by using the specific properties of the measurement matrices.

Abstract

In Part II of this two-part paper, we prove the convergence of the simplified information geometry approach (SIGA) proposed in Part I. For a general Bayesian inference problem, we first show that the iteration of the common second-order natural parameter (SONP) is separated from that of the common first-order natural parameter (FONP). Hence, the convergence of the common SONP can be checked independently. We show that with the initialization satisfying a specific but large range, the common SONP is convergent regardless of the value of the damping factor. For the common FONP, we establish a sufficient condition of its convergence and prove that the convergence of the common FONP relies on the spectral radius of a particular matrix related to the damping factor. We give the range of the damping factor that guarantees the convergence in the worst case. Further, we determine the range of the damping factor for massive MIMO-OFDM channel estimation by using the specific properties of the measurement matrices. Simulation results are provided to confirm the theoretical results.
Paper Structure (18 sections, 13 theorems, 127 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 13 theorems, 127 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Convergence of SIGA
  3. SIGA
  4. Convergence
  5. Application to Massive MIMO-OFDM Channel Estimation
  6. Simulation Results
  7. General Case
  8. Massive MIMO Channel Estimation
  9. Conclusion
  10. Calculation of \ref{['equ:update of SIGA']}
  11. Proof of Lemma \ref{['lemma:function tildeg']}
  12. Proof of Theorem \ref{['The-theta_1 conv']}
  13. Proof of Lemma \ref{['The-synchronous updates']}
  14. Proof of Lemma \ref{['Lemma-spectral radius']}
  15. Proof of Lemma \ref{['lemma:eigens of B star']}
  16. ...and 3 more sections

Key Result

Lemma 1

Given a finite initialization ${\boldsymbol\vartheta}\left(0\right) = \mathbf{f}\left( {\boldsymbol\theta}\left(0\right),{\boldsymbol\nu}\left(0\right)\right)$ with $-\frac{N-1}{\sigma_z^2}\mathbf{1}\le {\boldsymbol\nu}\left(0\right) \le \mathbf{0}$, then at each iteration, ${\boldsymbol\vartheta}

Figures (6)

  • Figure 1: Convergence of $\lVert {\boldsymbol\nu}\left(t\right) \rVert$ for different initializations and damping factors in general case.
  • Figure 2: Convergence and divergence of $\lVert {\boldsymbol\theta}\left(t\right) \rVert$ for different initializations and damping factors in general case.
  • Figure 3: Convergence of $\lVert {\boldsymbol\nu}\left(t\right) \rVert$ for different initializations and damping factors for general constant magnitude pilot.
  • Figure 4: Convergence and divergence of $\lVert {\boldsymbol\theta}\left(t\right) \rVert$ for different initializations and damping factors for general constant magnitude pilot.
  • Figure 5: Convergence of $\lVert {\boldsymbol\nu}\left(t\right) \rVert$ for different initializations and damping factors for APSPs.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 16 more