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Pilot Length Optimization with RS-LS Channel Estimation for Extremely Large Aperture Arrays

Mert Alıcıoğlu, Özlem Tuğfe Demir, Emil Björnson

TL;DR

This work addresses pilot-length optimization for RS-LS channel estimation in extremely large aperture arrays. By deriving an average SE and a statistically grounded lower bound that depends only on UE distribution statistics, it enables fast optimization of the pilot length without requiring UE-specific spatial correlations. The analysis reveals a unique optimal pilot length and provides a low-SNR closed-form, with simulations showing tight bounds and near-optimal performance, particularly as antenna density increases. Practically, the results indicate reduced pilot overhead in ELAA systems thanks to the RS-LS estimator's effective noise rejection in reduced subspaces, enhancing beamforming and spatial multiplexing capabilities.

Abstract

Extremely large aperture arrays can enable unprecedented spatial multiplexing in beyond 5G systems due to their extremely narrow beamfocusing capabilities. However, acquiring the spatial correlation matrix to enable efficient channel estimation is a complex task due to the vast number of antenna dimensions. Recently, a new estimation method called the "reduced-subspace least squares (RS-LS) estimator" has been proposed for densely packed arrays. This method relies solely on the geometry of the array to limit the estimation resources. In this paper, we address a gap in the existing literature by deriving the average spectral efficiency for a certain distribution of user equipments (UEs) and a lower bound on it when using the RS-LS estimator. This bound is determined by the channel gain and the statistics of the normalized spatial correlation matrices of potential UEs but, importantly, does not require knowledge of a specific UE's spatial correlation matrix. We establish that there exists a pilot length that maximizes this expression. Additionally, we derive an approximate expression for the optimal pilot length under low signal-to-noise ratio (SNR) conditions. Simulation results validate the tightness of the derived lower bound and the effectiveness of using the optimized pilot length.

Pilot Length Optimization with RS-LS Channel Estimation for Extremely Large Aperture Arrays

TL;DR

This work addresses pilot-length optimization for RS-LS channel estimation in extremely large aperture arrays. By deriving an average SE and a statistically grounded lower bound that depends only on UE distribution statistics, it enables fast optimization of the pilot length without requiring UE-specific spatial correlations. The analysis reveals a unique optimal pilot length and provides a low-SNR closed-form, with simulations showing tight bounds and near-optimal performance, particularly as antenna density increases. Practically, the results indicate reduced pilot overhead in ELAA systems thanks to the RS-LS estimator's effective noise rejection in reduced subspaces, enhancing beamforming and spatial multiplexing capabilities.

Abstract

Extremely large aperture arrays can enable unprecedented spatial multiplexing in beyond 5G systems due to their extremely narrow beamfocusing capabilities. However, acquiring the spatial correlation matrix to enable efficient channel estimation is a complex task due to the vast number of antenna dimensions. Recently, a new estimation method called the "reduced-subspace least squares (RS-LS) estimator" has been proposed for densely packed arrays. This method relies solely on the geometry of the array to limit the estimation resources. In this paper, we address a gap in the existing literature by deriving the average spectral efficiency for a certain distribution of user equipments (UEs) and a lower bound on it when using the RS-LS estimator. This bound is determined by the channel gain and the statistics of the normalized spatial correlation matrices of potential UEs but, importantly, does not require knowledge of a specific UE's spatial correlation matrix. We establish that there exists a pilot length that maximizes this expression. Additionally, we derive an approximate expression for the optimal pilot length under low signal-to-noise ratio (SNR) conditions. Simulation results validate the tightness of the derived lower bound and the effectiveness of using the optimized pilot length.
Paper Structure (8 sections, 2 theorems, 23 equations, 4 figures)

This paper contains 8 sections, 2 theorems, 23 equations, 4 figures.

Table of Contents

  1. Introduction
  2. System and Channel Modeling
  3. Channel Estimation
  4. Average Uplink Spectral Efficiency and Pilot Length Optimization
  5. Numerical Results
  6. Conclusions
  7. Proof of Lemma \ref{['lemma:infinite']}
  8. Proof of Lemma \ref{['lemma:concave']}

Key Result

Lemma 1

An achievable SE of a particular UE with the channel $\mathbf{h}\sim \mathcal{N}_{\mathbb{C}}(\mathbf{0},\beta\mathbf{R})$ with RS-LS channel estimation is

Figures (4)

  • Figure 1: The exact expression and proposed lower bound on the average SE versus the pilot length for different numbers of antennas and $-20$ dB SNR. The inter-antenna separation is $\Delta=\lambda/4$ when $M_{\rm H}=M_{\rm V}=12$ and $\Delta=\lambda/8$ when $M_{\rm H}=M_{\rm V}=24$, respectively.
  • Figure 2: CDFs of the SE when $M_{\rm H}=M_{\rm V}=24$, $\Delta=\lambda/8$, and $-20$ dB SNR using different pilot lengths.
  • Figure 3: Exact expression and proposed lower bound to the average SE versus the pilot length for different numbers of antennas and $-10$ dB SNR. The inter-antenna separation is $\Delta=\lambda/4$ when $M_{\rm H}=M_{\rm V}=12$ and $\Delta=\lambda/8$ when $M_{\rm H}=M_{\rm V}=24$, respectively.
  • Figure 4: CDFs of the SE when $M_{\rm H}=M_{\rm V}=24$, $\Delta=\lambda/8$, and $-10$ dB SNR using different pilot lengths.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof