Research Report on Noise-Shaped One-Bit Coefficients in Discrete Polynomial Fourier Extension

Abstract:This report studies noise-shaped one-bit coefficients in normalized discrete polynomial Fourier extension. For first-order Sigma-Delta quantization, the error is written as $e_k=u_k-q_k=\Delta v_k$ with a uniformly bounded state. Discrete summation by parts then yields variation estimates for complex weights and an $O(N^{-1})$ approximation rate on compact parameter sets. For the parabolic phase $\phi_{x,t}(\xi)=x\xi+t\xi^2$, the bound is expressed through $J(x,t)=\int_0^1 |x+2t\xi|d\xi$, and the uniform $N^{-1}$ rate is shown to be sharp over the admissible input class. Higher-order finite-record identities are derived with all endpoint traces retained. Under endpoint compatibility, or after explicit boundary correction, an $r$th-order noise-shaped error $e=\Delta^r v$ gives $O(N^{-r})$ decay for sufficiently smooth weights and $O(N^{-(r-1+\alpha)})$ decay for $C^{r-1,\alpha}$ weights. Exact $L^2$ orthogonality identities, fourth-moment formulas, local kernel estimates, and oscillatory transfer bounds are also established. Extensions to polynomial phases, multidimensional parameter families, growing observation regions, and correlated state models are included.

Computation and Language (cs.CL)

arXiv:2607.24868 [cs.CL]

(or arXiv:2607.24868v1 [cs.CL] for this version)

https://doi.org/10.48550/arXiv.2607.24868 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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