AI 中文总结
研究提出基于连续斯莱皮安多窗函数，将带能量估计转化为有限样本近似问题，构造可计算直接估计器，推导误差界及相关理论，经数值实验验证，该方法相比传统方法性能优异。
AI 中文摘要
在这项工作中，我们提出了一种在明确的时域和频域正则性及集中性假设下，利用有限采样数据进行直接带能量估计的方法。标准的两阶段方法总是先估计功率谱密度，然后在目标频段上积分，从而通过频谱模糊、泄漏和数值频段积分积累误差。相反，我们将带能量估计表述为一个有限样本近似问题。然后，我们在多窗函数框架下，基于采样的连续长球波函数（PSWFs）构造了一个可计算的直接估计器，它将连续时间/频率假设与离散观测联系起来。该估计器通过首先构造一个频段\(L^2\)范数估计器，然后对其平方来估计带能量。我们推导了确定性和非渐近误差界，明确区分了四种近似误差来源：时间限制、采样、频段限制和有限项截断。此外，还推导了几个理论，以说明估计精度如何依赖于长度、带宽、采样步长、时频平滑度和集中度。我们还建立了加性噪声下的确定性稳定性，分析了计算复杂度，并表明该方法通过离线重用与PSWF相关的计算允许高效的多频段实现。通过对多个观测长度、感兴趣频段和噪声水平进行数值实验，结果表明与积分周期图类型的方法相比，所提出的方法具有优异的性能。
英文摘要
In this work, a direct band-energy estimation approach using finite sampled data under explicit time- and frequency-domain regularity and concentration assumptions is proposed. Standard two-stage methods always estimate the power spectral density and then integrate it over the target band, thereby accumulating error through spectral smearing, leakage, and numerical band integration. Instead, we formulate the band-energy estimation as a finite-sample approximation problem. We then construct a computable direct estimator based on sampled continuous prolate spheroidal wave functions (PSWFs) in a multitaper-type framework, which links continuous-time/frequency assumptions to discrete observations. The estimator is obtained by first constructing a band $L^2$-norm estimator and then squaring it to estimate the band energy. We derive deterministic and non-asymptotic error bounds that explicitly separate four sources of approximation error: time limitation, sampling, band limitation, and finite-term truncation. In addition, several theories are derived to show how the accuracy of the estimation depends on the length, bandwidth, sampling step, time-frequency smoothness, and concentration. We also establish deterministic stability under additive noise, analyze computational complexity, and show that the method admits an efficient multiple-band implementation through offline reuse of PSWF-related computations. Numerical experiments across multiple observation lengths, bands of interest, and noise levels are conducted to show that the proposed approach achieves excellent performance compared to integrated-periodogram-type methods.
Comments44 pages, 9 figures, 1 table
