Unique Bispectrum Inversion for Signals with Finite Spectral/Temporal Support

11 Nov 2021  ·  Samuel Pinilla, Kumar Vijay Mishra, Brian M. Sadler ·

Retrieving a signal from the Fourier transform of its third-order statistics or bispectrum arises in a wide range of signal processing problems. Conventional methods do not provide a unique inversion of bispectrum. In this paper, we present a 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. Then, we refine the attained initialization based upon 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.

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