A binary vector of length $N$ has elements that are either 0 or 1. We investigate the question of whether and how a binary vector of known length can be reconstructed from a limited set of its discrete Fourier transform (DFT) coefficients. A priori information that the vector is binary provides a powerful constraint. We prove that a binary vector is uniquely defined by its two complex DFT coefficients (zeroth, which gives the popcount, and first) if $N$ is prime. If $N$ has two prime factors, additional DFT coefficients must be included in the data set to guarantee uniqueness, and we find the number of required coefficients theoretically. One may need to know even more DFT coefficients to guarantee stability of inversion. However, our results indicate that stable inversion can be obtained when the number of known coefficients is about $1/3$ of the total. This entails the effect of super-resolution (the resolution limit is improved by the factor of $\sim 3$).

中文翻译：

---

二进制离散傅里叶变换及其反演

长度为$ N $的二进制向量具有0或1的元素。我们研究是否可以从有限的离散傅立叶变换（DFT）系数集中重建已知长度的二进制向量的问题。向量是二进制的先验信息提供了有力的约束。我们证明，如果$ N $是素数，则二进制向量由其两个复数DFT系数唯一定义（零，它给出popcount，第一个）。如果$ N $具有两个主要因子，则必须在数据集中包括其他DFT系数以保证唯一性，并且从理论上我们找到了所需系数的数量。可能需要知道更多DFT系数才能保证反演的稳定性。然而，我们的结果表明，当已知系数的数量约为总数的1/3 $时，可以获得稳定的反演。这带来了超分辨率的效果（分辨率限制提高了$ \ sim 3 $倍）。