New Research Achieves O(N^{-1}) Rate for Noise-Shaped One-Bit Fourier Coefficients

In a new research report published on arXiv, Shengquan Wang investigates a technique called noise-shaped one-bit coefficients in the context of discrete polynomial Fourier extension. This method is pivotal for efficient signal representation, particularly in analog-to-digital conversion where only one bit per sample is used. The study focuses on Sigma-Delta quantization, a popular noise-shaping method that pushes quantization error to high frequencies. Wang shows that for first-order Sigma-Delta, the quantization error can be expressed as a first difference of a uniformly bounded state, leading to an O(N^{-1}) approximation rate on compact parameter sets—meaning the error decreases proportionally to 1/N as the number of samples N increases. This rate is proven sharp for a specific parabolic phase case, with the bound expressed via an integral of the phase gradient.

Building on this, the paper generalizes to higher-order noise shaping. Under endpoint compatibility or after explicit boundary correction, an rth-order error yields O(N^{-r}) decay for sufficiently smooth weights, and O(N^{-(r-1+α)}) for Hölder continuous weights. Beyond these core results, Wang derives exact L² orthogonality identities, fourth-moment formulas, and local kernel estimates that help characterize the error spectrum. The work also extends to polynomial phases, multidimensional parameter families, growing observation regions, and correlated state models. These contributions provide a rigorous mathematical foundation for using one-bit Sigma-Delta quantization in Fourier extension methods, which are critical for applications like super-resolution imaging, wireless communications, and neural network compression where high dynamic range and low bit-depth are constraints.