We consider the finite alphabet phase retrieval problem: recovering a signal whose entries lie in a small alphabet of possible values from its Fourier magnitudes. This problem arises in the celebrated technology of X-ray crystallography to determine the atomic structure of biological molecules. Our main result states that for generic values of the alphabet, two signals have the same Fourier magnitudes if and only if several partitions have the same difference sets. Thus, the finite alphabet phase retrieval problem reduces to the combinatorial problem of determining a signal from those difference sets. Notably, this result holds true when one of the letters of the alphabet is zero, namely, for sparse signals with finite alphabet, which is the situation in X-ray crystallography.
翻译：我们考虑了限定字母级的检索问题: 恢复一个信号, 其输入点在小字母上, 其可能值来自它的 Fourier 级数。 这个问题出现在用来确定生物分子原子结构的著名的X射线晶体学技术中。 我们的主要结果表明, 对于字母的通用值来说, 两个信号具有相同的四倍级, 如果并且只有在几个分区有相同的差异组数的话。 因此, 限制字母级的检索问题会降低到从这些差异组数中确定信号的组合问题。 值得注意的是, 当字母字母中的字母为零时, 也就是有限定字母的微量信号, 也就是X射线晶体学中的情况, 这个结果就是正确的 。
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