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The Effect of Noise Correlation on MMSE Channel Estimation in One-Bit Quantized Systems

Minhua Ding, Prathapasinghe Dharmawansa, Italo Atzeni, Antti Tölli

TL;DR

This work addresses MMSE channel estimation for MIMO systems with one-bit receivers under spatially correlated noise. It derives a general nonlinear MMSE estimator that incorporates arbitrary channel and noise correlation through orthant probabilities, and then specializes to tractable SIMO configurations with single-parameter correlation models. The analysis reveals conditions under which the estimator remains linear despite nonzero channel and noise correlations and shows that noise correlation can improve performance at low-to-medium SNR for uncorrelated channels but degrade it when channels are strongly correlated. The results provide a unified framework and practical insights for designing one-bit ADC-based systems in the presence of correlated disturbances, with asymptotic behavior indicating independence from noise correlation when SNR is high.

Abstract

This paper analyzes the impact of spatially correlated additive noise on the minimum mean-square error (MMSE) estimation of multiple-input multiple-output (MIMO) channels from one-bit quantized observations. Although additive noise can be correlated in practical scenarios, e.g., due to jamming, clutter, or other external disturbances, the effect of such correlation on the MMSE channel estimator in this setting remains unexplored in prior work. Against this backdrop, we derive a novel analytical expression for the general MIMO MMSE channel estimator, which is inherently nonlinear in one-bit observations, and accommodates arbitrary channel and noise correlation structures. To further characterize the impact of noise correlation, we subsequently specialize the general MMSE expression to certain tractable multi antenna configurations in which both the channel and the noise assume single-parameter constant correlation structures. Our analyses reveal nontrivial, noise-correlation-induced scenarios in which the estimator remains linear despite non-zero channel and noise correlation parameters. Moreover, the results indicate that, at low-to-medium signal-to-noise ratio, noise correlation improves the MMSE performance when channels are uncorrelated, but degrades performance when channels are strongly correlated.

The Effect of Noise Correlation on MMSE Channel Estimation in One-Bit Quantized Systems

TL;DR

This work addresses MMSE channel estimation for MIMO systems with one-bit receivers under spatially correlated noise. It derives a general nonlinear MMSE estimator that incorporates arbitrary channel and noise correlation through orthant probabilities, and then specializes to tractable SIMO configurations with single-parameter correlation models. The analysis reveals conditions under which the estimator remains linear despite nonzero channel and noise correlations and shows that noise correlation can improve performance at low-to-medium SNR for uncorrelated channels but degrade it when channels are strongly correlated. The results provide a unified framework and practical insights for designing one-bit ADC-based systems in the presence of correlated disturbances, with asymptotic behavior indicating independence from noise correlation when SNR is high.

Abstract

This paper analyzes the impact of spatially correlated additive noise on the minimum mean-square error (MMSE) estimation of multiple-input multiple-output (MIMO) channels from one-bit quantized observations. Although additive noise can be correlated in practical scenarios, e.g., due to jamming, clutter, or other external disturbances, the effect of such correlation on the MMSE channel estimator in this setting remains unexplored in prior work. Against this backdrop, we derive a novel analytical expression for the general MIMO MMSE channel estimator, which is inherently nonlinear in one-bit observations, and accommodates arbitrary channel and noise correlation structures. To further characterize the impact of noise correlation, we subsequently specialize the general MMSE expression to certain tractable multi antenna configurations in which both the channel and the noise assume single-parameter constant correlation structures. Our analyses reveal nontrivial, noise-correlation-induced scenarios in which the estimator remains linear despite non-zero channel and noise correlation parameters. Moreover, the results indicate that, at low-to-medium signal-to-noise ratio, noise correlation improves the MMSE performance when channels are uncorrelated, but degrades performance when channels are strongly correlated.
Paper Structure (10 sections, 4 theorems, 37 equations, 3 figures)

This paper contains 10 sections, 4 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model and Estimation Problem
  3. System Model
  4. Problem Statement
  5. The General MIMO MMSE Channel Estimator with Noise Correlation
  6. Effect of Noise Correlation on SIMO Systems
  7. $n=2$ with Complex Correlation Structures
  8. Arbitrary $n$ with Real Correlation Structures
  9. Numerical Results
  10. Conclusions

Key Result

Theorem 1

The MIMO MMSE channel estimator in the presence of noise correlation is given by where, for $k=1, \ldots, \tau n$, the $k$-th element of the vector $\mathbf{g}(\mathbf{r}, \mathbf{B}_{\rm c})\in\mathbb{C}^{\tau n}$ is given by Furthermore, for $i=1, \ldots, 2\tau n$, $\mathbf{E}_i\in\mathbb{R}^{(2\tau n-1)\times (2\tau n)}$ denotes the matrix obtained by removing the $i$-th row of $\mathbf{I}_{2

Figures (3)

  • Figure 1: Illustration of the parametric dependence of MMSE in a SIMO system with $n=2$, highlighting the effects of SNR $\gamma$ as well as complex-valued channel correlation $\phi$ and noise correlation $\xi$.
  • Figure 2: MMSE versus $\gamma$ for a SIMO system with $n=16$, evaluated for different values of $\xi\in\{0, 0.5, 0.9\}$ and $\phi\in\{0,0.9\}$.
  • Figure 3: Illustration of the extended linearity region induced by noise correlation $\xi$ and the corresponding MMSE for a SIMO system with $n=4$ (see \ref{['eq: MMSE_realB_interplay']}).

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 1
  • Proposition 1
  • Proposition 2