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.

翻译：从三重相关谱（又称双谱）中恢复信号是一系列信号处理问题中的常见需求。传统方法无法实现双谱到原始信号的精确反演。本文提出了一种方法，能够从至少 $3B$ 个双谱函数（BF）测量值中唯一恢复具有有限频谱支撑的信号（带限信号），其中 $B$ 为信号带宽。该方法同样适用于时域有限信号。我们提出了一种两步置信域算法，通过最小化非凸目标函数实现恢复：首先利用频谱算法近似信号，随后通过梯度迭代序列对初始化结果进行优化。数值实验表明，所提算法能够在完整观测和欠采样观测条件下，从双谱函数中有效估计带限/时域有限信号。