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

关于离散多项式傅里叶展开中噪声整形一位系数的研究报告

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.

摘要： 本报告研究了归一化离散多项式傅里叶展开中的噪声整形一位（one-bit）系数。对于一阶 Sigma-Delta 量化，误差可表示为 $e_k=u_k-q_k=\Delta v_k$，且具有一致有界的状态。通过离散分部求和法，可以得出复权重的变分估计，并在紧参数集上获得 $O(N^{-1})$ 的逼近速率。

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.

对于抛物线相位 $\phi_{x,t}(\xi)=x\xi+t\xi^2$，其界限通过 $J(x,t)=\int_0^1 |x+2t\xi|d\xi$ 表示，并证明了该一致 $N^{-1}$ 速率在容许输入类上是精确的。文中推导了保留所有端点迹（endpoint traces）的高阶有限记录恒等式。

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.

在端点兼容或经过显式边界校正后，$r$ 阶噪声整形误差 $e=\Delta^r v$ 对于足够平滑的权重可提供 $O(N^{-r})$ 的衰减，对于 $C^{r-1,\alpha}$ 权重则提供 $O(N^{-(r-1+\alpha)})$ 的衰减。此外，报告还建立了精确的 $L^2$ 正交恒等式、四阶矩公式、局部核估计以及振荡传递界限。研究内容还涵盖了对多项式相位、多维参数族、增长观测区域以及相关状态模型的扩展。