Binary Discrete Fourier Transform and its Inversion

TL;DR

This paper explores how binary vectors can be uniquely reconstructed from a limited set of their Fourier coefficients, revealing conditions for uniqueness and stability, and demonstrating a super-resolution effect.

Contribution

It provides theoretical conditions for the minimal Fourier coefficients needed for unique and stable binary vector reconstruction, including prime and composite lengths.

Findings

Binary vectors are uniquely determined by two DFT coefficients if length is prime.

Additional DFT coefficients are needed for composite lengths to ensure uniqueness.

Stable inversion is achievable with about one-third of the total DFT coefficients, enabling super-resolution.

Abstract

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 $\sim 3$).