In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a shift-invariant space from their phaseless samples taken either on the whole line or on a discrete set with finite sampling density. We characterize all phase retrievable signals in a real-valued shift-invariant space using their nonseparability. For nonseparable signals generated by some function with support length L, we show that they can be well approximated, up to a sign, from their noisy phaseless samples taken on a discrete set with sampling density $2L-1$. In this paper, we also propose an algorithm with linear computational complexity to reconstruct nonseparable signals in a shift-invariant space from their phaseless samples corrupted by bounded noises.

在本文中，我们考虑了一个无限维相位检索问题，该问题可以从整行或有限采样密度的离散集上获取的无相位样本，重构生活在平移不变空间中的实值信号。我们使用它们的不可分性来表征实值移位不变空间中的所有相位可检索信号。对于由支持长度为L的某个函数生成的不可分离信号，我们表明，可以从在具有采样密度的离散集上获取的无噪无噪采样中很好地近似它们，直到一个符号$2\mathrm{大号}-\mathrm{1个}$。在本文中，我们还提出了一种具有线性计算复杂度的算法，用于从不变噪声空间中的无分离样本中，分离出受边界噪声破坏的不可分离信号，以重构它们。