Unique Bispectrum Inversion for Signals with Finite Spectral/Temporal Support

TL;DR

This paper introduces a novel method for uniquely recovering band-limited and time-limited signals from bispectrum measurements using a two-step trust region algorithm, improving accuracy over conventional approaches.

Contribution

The paper presents a new approach for bispectrum inversion that guarantees unique signal recovery from minimal measurements, extending to both band-limited and time-limited signals.

Findings

Accurately recovers signals from bispectrum with minimal measurements

Effective for both complete and undersampled observations

Outperforms conventional bispectrum inversion methods

Abstract

Retrieving a signal from its triple correlation spectrum, also called bispectrum, arises in a wide range of signal processing problems. Conventional methods do not provide an accurate inversion of bispectrum to the underlying signal. In this paper, we present an approach that uniquely recovers signals with finite spectral support (band-limited signals) from at least $3B$ measurements of its bispectrum function (BF), where $B$ is the signal's bandwidth. Our approach also extends to time-limited signals. We propose a two-step trust region algorithm that minimizes a non-convex objective function. First, we approximate the signal by a spectral algorithm and then refine the attained initialization based on a sequence of gradient iterations. Numerical experiments suggest that our proposed algorithm is able to estimate band-/time-limited signals from its BF for both complete and undersampled observations.