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
Source Excerpt
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 $N^{-1}$
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