Research Report on Noise-Shaped One-Bit Coefficients in Discrete Polynomial Fourier Extension
Quick Answer
This paper shows that This research explores noise-shaped one-bit coefficients in discrete polynomial Fourier extensions, demonstrating an O(N^{-1}) approximation rate for first-order Sigma-Delta quantization.
Quick Take
It establishes sharp bounds and decay rates for higher-order noise-shaped errors, contributing to the understanding of complex weights and their applications in computational models.
Key Points
- First-order Sigma-Delta quantization yields an O(N^{-1}) approximation rate.
- Sharp bounds for noise-shaped errors are established under specific conditions.
- Higher-order decay rates are shown for sufficiently smooth weights.
- Exact L^2 orthogonality identities and fourth-moment formulas are derived.
- Extensions include polynomial phases and multidimensional parameter families.
Paper Resources
📖 Reader Mode
~2 min readAbstract: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.
| Subjects: | Computation and Language (cs.CL) |
| Cite as: | arXiv:2607.24868 [cs.CL] |
| (or arXiv:2607.24868v1 [cs.CL] for this version) | |
| https://doi.org/10.48550/arXiv.2607.24868 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Shengquan Wang [view email]
[v1]
Sun, 26 Jul 2026 21:51:41 UTC (200 KB)
— Originally published at arxiv.org
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