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Reciprocity Calibration of Dual-Antenna Repeaters via MMSE Estimation

Shoma Hara, Takumi Takahashi, Hiroki Iimori, Hideki Ochiai, Erik G. Larsson

TL;DR

This work tackles reciprocity calibration in repeater-assisted MIMO under TDD by formulating a Bayesian MMSE framework that leverages detailed prior models for repeater-induced non-reciprocity and measurement noise. It introduces a von Mises denoiser to handle phase constraints on the unit circle and uses a MoM approach to adapt long-term priors, enabling robust, low-complexity calibration with fast convergence. Compared to state-of-the-art NLS methods, the MMSE algorithm delivers substantial accuracy gains, particularly for large-scale arrays, while maintaining comparable online complexity; simulations show improved RMSE across SNRs and rapid convergence (typically ~4 iterations). The practical impact is a more reliable, scalable calibration solution for repeater-enabled MIMO systems, facilitating high-quality beamforming with reduced pilot overhead and calibration latency.

Abstract

This paper proposes a novel Bayesian reciprocity calibration method that consistently ensures uplink and downlink channel reciprocity in repeater-assisted multiple-input multiple-output (MIMO) systems. The proposed algorithm is formulated under the minimum mean-square error (MMSE) criterion. Its Bayesian framework incorporates complete statistical knowledge of the signal model, noise, and prior distributions, enabling a coherent design that achieves both low computational complexity and high calibration accuracy. To further enhance phase alignment accuracy, which is critical for calibration tasks, we develop a von Mises denoiser that exploits the fact that the target parameters lie on the circle in the complex plane. Simulation results demonstrate that the proposed MMSE algorithm achieves substantially improved estimation accuracy compared with conventional deterministic non-linear least-squares (NLS) methods, while maintaining comparable computational complexity. Furthermore, the proposed method exhibits remarkably fast convergence, making it well suited for practical implementation.

Reciprocity Calibration of Dual-Antenna Repeaters via MMSE Estimation

TL;DR

This work tackles reciprocity calibration in repeater-assisted MIMO under TDD by formulating a Bayesian MMSE framework that leverages detailed prior models for repeater-induced non-reciprocity and measurement noise. It introduces a von Mises denoiser to handle phase constraints on the unit circle and uses a MoM approach to adapt long-term priors, enabling robust, low-complexity calibration with fast convergence. Compared to state-of-the-art NLS methods, the MMSE algorithm delivers substantial accuracy gains, particularly for large-scale arrays, while maintaining comparable online complexity; simulations show improved RMSE across SNRs and rapid convergence (typically ~4 iterations). The practical impact is a more reliable, scalable calibration solution for repeater-enabled MIMO systems, facilitating high-quality beamforming with reduced pilot overhead and calibration latency.

Abstract

This paper proposes a novel Bayesian reciprocity calibration method that consistently ensures uplink and downlink channel reciprocity in repeater-assisted multiple-input multiple-output (MIMO) systems. The proposed algorithm is formulated under the minimum mean-square error (MMSE) criterion. Its Bayesian framework incorporates complete statistical knowledge of the signal model, noise, and prior distributions, enabling a coherent design that achieves both low computational complexity and high calibration accuracy. To further enhance phase alignment accuracy, which is critical for calibration tasks, we develop a von Mises denoiser that exploits the fact that the target parameters lie on the circle in the complex plane. Simulation results demonstrate that the proposed MMSE algorithm achieves substantially improved estimation accuracy compared with conventional deterministic non-linear least-squares (NLS) methods, while maintaining comparable computational complexity. Furthermore, the proposed method exhibits remarkably fast convergence, making it well suited for practical implementation.
Paper Structure (29 sections, 2 theorems, 78 equations, 5 figures, 1 table, 3 algorithms)

This paper contains 29 sections, 2 theorems, 78 equations, 5 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. System Model
  3. State-of-the-Art Calibration Scheme
  4. NLS Algorithm
  5. Estimation of $\bm H$
  6. Estimation of $\bm Z$
  7. Estimation of $\bm A$ and $\bm B$
  8. Estimation of $\gamma$
  9. Alternating-Optimization NLS Algorithm
  10. Proposed MMSE Algorithm
  11. Prior Statistical Modeling of the Unknown Variables
  12. Complex-Valued Reciprocity Coefficients
  13. Measurement Noise
  14. Overview
  15. MMSE Algorithm
  16. ...and 14 more sections

Key Result

Lemma 1

When $\bm{A}(i,i)$ lies on the complex unit circle, the MMSE estimate of $\bm{A}(i,i)$ can be computed using the von Mises denoiser as follows: where $I_n(\cdot)$ denotes the modified Bessel function of the first kind of order $n$ and The corresponding posterior MSE is given by

Figures (5)

  • Figure 1: Two antenna arrays, $\mathrm{A}$ and $\mathrm{B}$, and a repeater ($\mathrm{R}$). $\bm{G}$ denotes the propagation channel from $\mathrm{A}$ to $\mathrm{B}$ when $\mathrm{R}$ is turned off. Radio channels are represented by solid lines, whereas repeater gains are represented by dashed lines. The figure is adapted from Larsson2024.
  • Figure 2: Illustration of the FG designed for the proposed MMSE algorithm.
  • Figure 3: Illustration of the tripartite FG for bilinear inference of $\bm{A}$ and $\bm{B}$ with $(M_\mathrm{A},M_\mathrm{B}) = (4,3)$.
  • Figure 4: RMSE of $\hat{\gamma}$ for different antenna configurations.
  • Figure 5: Iterative convergence behavior of NLS-based and MMSE-based reciprocity calibration algorithms.

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2