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A Large-dimensional Analysis of ESPRIT DoA Estimation: Inconsistency and a Correction via RMT

Zhengyu Wang, Wei Yang, Xiaoyi Mai, Zenan Ling, Zhenyu Liao, Robert C. Qiu

TL;DR

This work analyzes ESPRIT DoA estimation in the large-array, limited-snapshot regime, showing that classical ESPRIT is inconsistent due to eigenspectral distortion of the sample covariance matrix when $N/T\to c\in(0,\infty)$. Leveraging random matrix theory, it derives a bias term $g_k$ that governs the inconsistency and introduces G-ESPRIT, a corrected method that yields consistent DoA estimates for both widely- and closely-spaced sources by debiasing the estimated signal subspace. A novel bound on eigenvalue differences for non-Hermitian matrices supports the theoretical framework, and simulations corroborate the asymptotic results, demonstrating reduced variance and near-CRB performance for G-ESPRIT under appropriate subarray configurations. The findings provide a principled path to reliable DoA estimation for large-scale arrays with limited observations and lay groundwork for second-order (CLT-type) analyses of these estimators.

Abstract

In this paper, we perform asymptotic analyses of the widely used ESPRIT direction-of-arrival (DoA) estimator for large arrays, where the array size $N$ and the number of snapshots $T$ grow to infinity at the same pace. In this large-dimensional regime, the sample covariance matrix (SCM) is known to be a poor eigenspectral estimator of the population covariance. We show that the classical ESPRIT algorithm, that relies on the SCM, and as a consequence of the large-dimensional inconsistency of the SCM, produces inconsistent DoA estimates as $N,T \to \infty$ with $N/T \to c \in (0,\infty)$, for both widely- and closely-spaced DoAs. Leveraging tools from random matrix theory (RMT), we propose an improved G-ESPRIT method and prove its consistency in the same large-dimensional setting. From a technical perspective, we derive a novel bound on the eigenvalue differences between two potentially non-Hermitian random matrices, which may be of independent interest. Numerical simulations are provided to corroborate our theoretical findings.

A Large-dimensional Analysis of ESPRIT DoA Estimation: Inconsistency and a Correction via RMT

TL;DR

This work analyzes ESPRIT DoA estimation in the large-array, limited-snapshot regime, showing that classical ESPRIT is inconsistent due to eigenspectral distortion of the sample covariance matrix when . Leveraging random matrix theory, it derives a bias term that governs the inconsistency and introduces G-ESPRIT, a corrected method that yields consistent DoA estimates for both widely- and closely-spaced sources by debiasing the estimated signal subspace. A novel bound on eigenvalue differences for non-Hermitian matrices supports the theoretical framework, and simulations corroborate the asymptotic results, demonstrating reduced variance and near-CRB performance for G-ESPRIT under appropriate subarray configurations. The findings provide a principled path to reliable DoA estimation for large-scale arrays with limited observations and lay groundwork for second-order (CLT-type) analyses of these estimators.

Abstract

In this paper, we perform asymptotic analyses of the widely used ESPRIT direction-of-arrival (DoA) estimator for large arrays, where the array size and the number of snapshots grow to infinity at the same pace. In this large-dimensional regime, the sample covariance matrix (SCM) is known to be a poor eigenspectral estimator of the population covariance. We show that the classical ESPRIT algorithm, that relies on the SCM, and as a consequence of the large-dimensional inconsistency of the SCM, produces inconsistent DoA estimates as with , for both widely- and closely-spaced DoAs. Leveraging tools from random matrix theory (RMT), we propose an improved G-ESPRIT method and prove its consistency in the same large-dimensional setting. From a technical perspective, we derive a novel bound on the eigenvalue differences between two potentially non-Hermitian random matrices, which may be of independent interest. Numerical simulations are provided to corroborate our theoretical findings.
Paper Structure (24 sections, 15 theorems, 133 equations, 8 figures, 2 algorithms)

This paper contains 24 sections, 15 theorems, 133 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model and Preliminaries
  3. System Model
  4. The ESPRIT DoA Estimator
  5. Eigenspectral Inconsistency for Large-dimensional SCM
  6. Inconsistency of ESPRIT for Large Arrays
  7. Large-dimensional behavior of ESPRIT
  8. A novel bound on eigenvalue difference
  9. Proof of Theorem 2
  10. Consistent DoA Estimation with G-ESPRIT
  11. Numerical Simulations
  12. Validation of asymptotic theoretical results
  13. Comparison to other DoA approaches
  14. Conclusion
  15. Technical Lemmas
  16. ...and 9 more sections

Key Result

Theorem 1

Under the settings and notations of ass:large_array, we have, for ${\mathbf{X}} \in {\mathbb{C}}^{N \times T}$ defined eq:model_matrix_form and as $N,T \to \infty$ with $N/T \to c \in (0, \infty)$ that, with probability one, the empirical spectral measure in def:mu of the SCM $\hat{{\mathbf{C}}} = { with $E_\pm = (1 \pm \sqrt c)^2$ and $(x)^+ = \max(x,0)$. And its Stieltjes transform (see def:ST)

Figures (8)

  • Figure 1: System diagram for DoA estimation. A far-field signal with incident angle $\theta$ impinges on a ULA of $N$ sensors spaced $d$ apart.
  • Figure 2: MSEs of classical ESPRIT as a function of subarrary size $n$ with $n+\Delta=N$, $N = 80$, $T = 160$. The two closely-spaced sources at DoA $\theta_1 = 0$ and $\theta_2 = 0.8\times 2 \pi/N$, and the power matrix ${\mathbf{P}} =2\mathbf{I}_2$. Results are obtained by averaging over $500$ independent trials.
  • Figure 3: Approximation errors in spectral norm versus array length $N$, for $T = 2N$. Simulation results in blue are obtained by averaging over $200$ independent trials, versus polynomial fit in red. Left: widely-spaced DoAs with correlated sources at DoA $\theta_1 = 0$ and $\theta_2 = \pi/4$, power matrix ${\mathbf{P}} = (20.80.82)$, subarray size $n=N-1$, and distance $\Delta=1$. Right: closely-spaced DoAs ($\theta_1 = 0$ and $\theta_2 = 0.8\times 2 \pi/N$) having equal powers with ${\mathbf{P}} =2\mathbf{I}_2$, $n=2N/3$, and $\Delta=N/3$.
  • Figure 4: Left: comparison between DoA estimates $\hat{\theta}$ from ESPRIT in blue, $\bar{\theta}$ from \ref{['theo:main']} in green, $\hat{\theta}^G$ from G-ESPRIT of \ref{['prop:gesprit']} in red, and true DoAs $\theta$ in purple; in the same setting as in the left plot of \ref{['fig:spectral_error']}, for $N=80$ and $T = 160$. Right: MSEs and variances of DoA estimates for ESPRIT ($\hat{\theta}$) and the improved G-ESPRIT method ($\hat{\theta}^G$), as a function of the array length $N$. Results are obtained by averaging over $200$ independent trials.
  • Figure 5: Left: comparison between DoA estimates $\hat{\theta}$ from ESPRIT in blue, $\bar{\theta}$ from \ref{['theo:main']} in green, $\hat{\theta}^G$ from G-ESPRIT of \ref{['prop:gesprit']} in red, and true DoAs $\theta$ in purple; in the same setting as the right plot of \ref{['fig:spectral_error']}. Right: DoA estimation MSEs and variances of ESPRIT and the improved G-ESPRIT methods as the array length $N$ increases. Results are obtained by averaging over $200$ independent trials.
  • ...and 3 more figures

Theorems & Definitions (36)

  • Definition 1: Empirical spectral measure
  • Definition 2: Stieltjes transform
  • Theorem 1: Eigenspectral characterization of large-dimensional SCM marvcenko1967distributionsilverstein1995empiricalbaik2006eigenvaluesbenaych2011eigenvalues
  • Remark 1: $N$-consistent estimator
  • Remark 2: Widely versus closely-spaced DoAs
  • Theorem 2: Large-dimensional behavior of ESPRIT
  • Remark 3: Limiting cases: infinite snapshots or high SNR
  • Remark 4: Limiting case: small subarray
  • Remark 5: Special case: widely-spaced DoAs with uncorrelated sources
  • Remark 6: Special case: widely-spaced DoAs with correlated sources
  • ...and 26 more