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On Optimal MMSE Channel Estimation for One-Bit Quantized MIMO Systems

Minhua Ding, Italo Atzeni, Antti Tölli, A. Lee Swindlehurst

TL;DR

This work develops a new framework to compute the MMSE channel estimator that hinges on the computation of the orthant probability of a multivariate normal distribution and determines a necessary and sufficient condition for the BLMMSE channel estimator to be optimal and thus equivalent to the MMSE estimator.

Abstract

This paper focuses on the minimum mean squared error (MMSE) channel estimator for multiple-input multiple-output (MIMO) systems with one-bit quantization at the receiver side. Despite its optimality and significance in estimation theory, the MMSE channel estimator has not been fully investigated in this context due to its general non-linearity and computational complexity. Instead, the typically suboptimal Bussgang linear MMSE (BLMMSE) estimator has been widely adopted. In this work, we develop a new framework to compute the MMSE channel estimator that hinges on computation of the orthant probability of the multivariate normal distribution. Based on this framework, we determine a necessary and sufficient condition for the BLMMSE channel estimator to be optimal and equivalent to the MMSE estimator. Under the assumption of specific channel correlation or pilot symbols, we further utilize the framework to derive analytical expressions for the MMSE channel estimator that are particularly convenient for computation when certain system dimensions become large, thereby enabling a comparison between the BLMMSE and MMSE channel estimators in these cases.

On Optimal MMSE Channel Estimation for One-Bit Quantized MIMO Systems

TL;DR

This work develops a new framework to compute the MMSE channel estimator that hinges on the computation of the orthant probability of a multivariate normal distribution and determines a necessary and sufficient condition for the BLMMSE channel estimator to be optimal and thus equivalent to the MMSE estimator.

Abstract

This paper focuses on the minimum mean squared error (MMSE) channel estimator for multiple-input multiple-output (MIMO) systems with one-bit quantization at the receiver side. Despite its optimality and significance in estimation theory, the MMSE channel estimator has not been fully investigated in this context due to its general non-linearity and computational complexity. Instead, the typically suboptimal Bussgang linear MMSE (BLMMSE) estimator has been widely adopted. In this work, we develop a new framework to compute the MMSE channel estimator that hinges on computation of the orthant probability of the multivariate normal distribution. Based on this framework, we determine a necessary and sufficient condition for the BLMMSE channel estimator to be optimal and equivalent to the MMSE estimator. Under the assumption of specific channel correlation or pilot symbols, we further utilize the framework to derive analytical expressions for the MMSE channel estimator that are particularly convenient for computation when certain system dimensions become large, thereby enabling a comparison between the BLMMSE and MMSE channel estimators in these cases.
Paper Structure (31 sections, 7 theorems, 125 equations, 5 figures)

This paper contains 31 sections, 7 theorems, 125 equations, 5 figures.

Table of Contents

  1. Introduction
  2. System Model and Problem Statement
  3. System Model
  4. Problem Statement
  5. A Framework for the MMSE Channel Estimation of One-Bit Quantized MIMO Systems
  6. Preliminaries on the Orthant Probability of a Real-Valued MVN Distribution
  7. Framework for Computing the MMSE Channel Estimate for One-Bit Quantized MIMO Systems
  8. Condition for the Optimality of the BLMMSE Channel Estimator
  9. Spatially White Channel and Unitary Pilot Matrix
  10. Transmit-Only Channel Correlation
  11. Spatially White Channel and $\tau=2$ Real-Valued Pilot Symbols
  12. Spatially White Channel and $\tau=2$ Complex-Valued Pilot Symbols with $\pi/2$ Phase Difference
  13. SIMO with Real-Valued Channel Correlation, $N_R=2$, and $\tau=N_T=1$
  14. Cases with Nonlinear MMSE Channel Estimators
  15. SIMO Channel with Real-Valued Channel Correlation, $N_R\geq 3$, and $\tau=N_T=1$
  16. ...and 16 more sections

Key Result

Lemma 1

Let $\mathbf{q}=[q_1 \; q_2]^{\mathrm{T}}$, with $q_1, q_2\in\{\pm 1\}$, and let $\boldsymbol{\Lambda}_\mathbf{q} = \mathtt{diag}{{(}}\mathbf{q})$. Consider the $2\times 2$ covariance-type matrix with $\phi_{12}\in(-1, 1)$. Then, the vector is linear in $\mathbf{q}$.

Figures (5)

  • Figure 1: MSE comparison for the MISO case with $\tau=N_T \in \{16, 32\}$ using the exponential transmit correlation model and a tailored pilot matrix. In this scenario $\hat{\mathbf{h}}_{\rm MMSE}=\hat{\mathbf{h}}_{\rm BLM}$.
  • Figure 2: MSE comparison for the SIMO case with $N_R \in \{3, 4\}$ using the exponential channel correlation model.
  • Figure 3: MSE comparison for the SIMO case with $N_R \in \{ 4, 16, 32 \}$ and the same channel correlation coefficient $\rho=0.9$ between all antennas.
  • Figure 4: MSE comparison for the SIMO case with $N_R \in \{ 8, 16, 32, 64\}$ and the same channel correlation coefficient $\rho=0.9$ between all antennas.
  • Figure 5: MSE comparison for SISO or SIMO with no receive correlation, $\tau \in \{2, 16, 32\}$.

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Remark 4
  • Theorem 1
  • Remark 5
  • Remark 6
  • Corollary 1
  • ...and 6 more