Functional Multi-Target Detection via Bispectrum Inversion

2026-05-29

AI Summary

This study addresses the problem of recovering a compactly supported signal from a single noisy observation comprising multiple continuous (non-grid-aligned) translations of the signal, corrupted by stationary correlated Gaussian noise. The authors propose an initialization-free bispectral inversion method that first estimates the signal’s bispectrum via a debiased third-order empirical autocorrelation and then reconstructs the signal using either functional frequency recursion or Kotlarski-type deconvolution. This approach relaxes conventional assumptions of discrete grid alignment and white noise, accommodating continuous shifts and colored noise while providing non-asymptotic recovery guarantees without requiring bandlimitedness. Theoretical analysis shows that reconstruction error depends on signal smoothness and bispectrum estimation accuracy, and numerical experiments demonstrate high-fidelity recovery even at low signal-to-noise ratios.

Abstract

This paper develops a functional theory for multi-target detection, where a compactly supported signal is recovered from a single noisy observation containing many unknown translations of the signal. Our formulation allows continuous, off-grid translations and correlated stationary Gaussian process noise, extending beyond the discrete, grid-aligned, white-noise models common in prior work. We analyze two uninitialized recovery algorithms based on autocorrelation analysis; in particular, both algorithms first estimate the signal's bispectrum via a debiased third-order empirical autocorrelation. The signal is then recovered from the estimated bispectrum using either a functional frequency marching scheme or a Kotlarski-type deconvolution formula. For both algorithms, we prove non-asymptotic recovery guarantees for compactly supported signals without bandlimiting assumptions. The resulting error bounds depend on the smoothness of the signal and the accuracy of bispectrum estimation, with the latter governed by the noise characteristics and the number of signal occurrences. Numerical experiments validate our theory and demonstrate accurate recovery in low-SNR regimes.

Problem

• multi-target detection
• bispectrum inversion
• continuous translations
• correlated Gaussian noise
• signal recovery

Innovation

• bispectrum inversion
• multi-target detection
• off-grid translation
• correlated Gaussian noise
• non-asymptotic recovery