Irregular Sampling and Spectral Estimation: Statistical Analysis and Frequency-Refined Orthogonal Matching Pursuit
Abstract
The direct Fourier evaluation of irregularly sampled observations is a simple alternative to uniform-grid spectral estimation when the signal is composed of a limited number of sinusoids. This article examines its first-and second-order sampling statistics for both independent uniform random sampling and stratified sampling with one random point per temporal cell. Exact expressions are established for the mean and complex variance of the estimator for a single exponential and for a general deterministic multitone signal, retaining the cross-frequency covariance terms. The commonly used diagonal, sinc-weighted approximation is recovered only when the o!-diagonal covariance contributions are negligible. For real signals, conjugatefrequency coupling is explicitly accounted for. A frequency-refined orthogonal matching pursuit (OMP) procedure is then described, using the direct spectrum of successive residuals, local continuous-frequency refinement, and global least-squares amplitude estimation. Numerical illustrations reproduce the frequency configurations, observation counts and phase settings of the accompanying research presentation. The results clarify how irregular sampling trades coherent aliases for incoherent spectral fluctuations and how iterative subtraction helps expose weak sinusoids.
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BibTeX
@misc{badaoui2026irregular,
title = {{Irregular Sampling and Spectral Estimation: Statistical Analysis and Frequency-Refined Orthogonal Matching Pursuit}},
author = {Mohamed El Badaoui},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05844175v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Irregular Sampling and Spectral Estimation: Statistical Analysis and Frequency-Refined Orthogonal Matching Pursuit
AU - Mohamed El Badaoui
PY - 2026
DA - 2026/10/29
PB - HAL
UR - https://hal.science/hal-05844175v1
AB - The direct Fourier evaluation of irregularly sampled observations is a simple alternative to uniform-grid spectral estimation when the signal is composed of a limited number of sinusoids. This article examines its first-and second-order sampling statistics for both independent uniform random sampling and stratified sampling with one random point per temporal cell. Exact expressions are established for the mean and complex variance of the estimator for a single exponential and for a general deterministic multitone signal, retaining the cross-frequency covariance terms. The commonly used diagonal, sinc-weighted approximation is recovered only when the o!-diagonal covariance contributions are negligible. For real signals, conjugatefrequency coupling is explicitly accounted for. A frequency-refined orthogonal matching pursuit (OMP) procedure is then described, using the direct spectrum of successive residuals, local continuous-frequency refinement, and global least-squares amplitude estimation. Numerical illustrations reproduce the frequency configurations, observation counts and phase settings of the accompanying research presentation. The results clarify how irregular sampling trades coherent aliases for incoherent spectral fluctuations and how iterative subtraction helps expose weak sinusoids.
ER -