Ambiguities in One-Dimensional Discrete Phase Retrieval from Fourier Magnitudes
Ambiguities in One-Dimensional Discrete Phase Retrieval from Fourier Magnitudes
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DOI：
10.1007/s00041-015-9405-2
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发表时间：
2015-12-01
影响因子：
1.2
通讯作者：
Plonka, Gerlind
中科院分区：
文献类型：
--
作者：
Beinert, Robert;Plonka, Gerlind
本文是一篇旨在描述离散相位恢复问题所有解的综述。把我们自己限制在有限支持的离散信号中，这个问题可以表述如下。我们想要恢复一个复值离散信号，它的离散时间傅里叶变换的模支持。我们将给出一个完整的分类所有琐碎和非琐碎的模糊的离散相位检索问题。在我们的分类中，微不足道的模糊性是由信号在空间中的移位、与旋转因子的乘法或信号的共轭和反射引起的。此外，我们证明了有限离散相位恢复问题的所有非平凡模糊性都可以用信号卷积来表征。在论文的第二部分，我们检查了通常使用的先验条件，关于它们减少相位检索问题的模糊性数量甚至确保唯一性的能力。对于相应的证明，我们可以利用我们在歧义分类上的发现。对歧义结构的考虑也清楚地表明，即使在理论上存在唯一性的情况下，相位检索问题也是病态的。
The present paper is a survey aiming at characterizing all solutions of the discrete phase retrieval problem. Restricting ourselves to discrete signals with finite support, this problem can be stated as follows. We want to recover a complex-valued discrete signal with support from the modulus of its discrete-time Fourier transform . We will give a full classification of all trivial and nontrivial ambiguities of the discrete phase retrieval problem. In our classification, trivial ambiguities are caused either by signal shifts in space, by multiplication with a rotation factor , , or by conjugation and reflection of the signal. Furthermore, we show that all nontrivial ambiguities of the finite discrete phase retrieval problem can be characterized by signal convolutions. In the second part of the paper, we examine the usually employed a priori conditions regarding their ability to reduce the number of ambiguities of the phase retrieval problem or even to ensure uniqueness. For the corresponding proofs we can employ our findings on the ambiguity classification. The considerations on the structure of ambiguities also show clearly the ill-posedness of the phase retrieval problem even in cases where uniqueness is theoretically shown.