Unique Bispectrum Inversion for Signals with Finite Spectral/Temporal Support
This addresses a signal processing problem for applications requiring precise signal retrieval from bispectrum data, but it is incremental as it builds on existing inversion methods with specific constraints.
The paper tackles the problem of accurately inverting a signal from its bispectrum, which is challenging with conventional methods, and presents an approach that uniquely recovers band-limited or time-limited signals using at least 3B measurements, where B is the bandwidth, with numerical experiments showing successful estimation even with undersampled observations.
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.