研究一比特系数的噪声整形，实现逼近误差 $O(N^{-1})$
Research Report on Noise-Shaped One-Bit Coefficients in Discrete Polynomial Fourier Extension
- 利用噪声整形和离散求和分部法设计一比特量化方案
- 在紧凑参数集上达到 $O(N^{-1})$ 的逼近精度，且为紧界
- 适用于信号处理与压缩感知，尤其适合高阶光滑信号
本报告研究归一化离散多项式傅里叶扩展中噪声整形的一比特系数。对于一阶Sigma-Delta量化，误差可表示为 $e_k=u_k-q_k=Δv_k$，其中状态变量有界。通过离散求和分部法，得到复权重的变化率估计，并在紧凑参数集上获得 $O(N^{-1})$ 的逼近速率。对于抛物相位 $ϕ_{x,t}(ξ)=xξ+tξ^2$，该界由 $J(x,t)= int_0^1 |x+2tξ|dξ$ 表示，且在允许输入类上 $N^{-1}$ 精度为紧界。推导了高阶有限记录恒等式，保留所有端点迹；在端点兼容或显式边界修正后，$r$ 阶噪声整形误差 $e=Δ^r v$ 可实现 $C^{r,α}$ 光滑权重下 $O(N^{-r})$ 衰减，$C^{r-1,α}$ 权重下 $O(N^{-(r-1+α)})$ 衰减。还建立了精确 $L^2$ 正交恒等式、四阶矩公式、局部核估计与振荡转移界。拓展至多项式相位、多维参数族、增长观测区域及相关状态模型。
原文摘要 · Abstract (English)
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 $e_k=u_k-q_k=Δv_k$ 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 $ϕ_{x,t}(ξ)=xξ+tξ^2$, the bound is expressed through $J(x,t)=\int_0^1 |x+2tξ|dξ$, 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 $e=Δ^r v$ gives $O(N^{-r})$ decay for sufficiently smooth weights and $O(N^{-(r-1+α)})$ decay for $C^{r-1,α}$ 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.
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