An Abstract Formulation of Wavelets

小波的抽象公式

• 批准号: 9801658
• 负责人: Lawrence Baggett
• 金额: $ 10.4万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801658&HistoricalAwards=false
• 关键词: Abstract; Formulation; Wavelets

AbstractBaggettRecently, a kind of internal fine structure of wavelets has been discovered. That is, each wavelethas associated to it a "multiplicity function" that appears to carry some information about theredundancy, in the frequency domain, of the wavelet. Such redundancies should lead to morereliable reconstructions of signals from their wavelet transforms. This research focuses on thetheoretical nature of these multiplicity functions. That is, we will study the properties of thismultiplicity function, try to determine exactly which kinds of functions occur in this way, andattempt to classify wavelets according to their multiplicity functions.The way in which an unknown signal (radio wave, seismic shock, astronomic vibration,etc.) is typically analyzed is by numerically comparing it to a set of fixed standard signals.For instance, the signal can be reproduced (relatively accurately) from these numbers; it can be stored in a computer and re-examined later; or it can be designated as a "new" standard itself. The new idea that comes from Wavelet Theory is that this important set of fixed standards can be described in terms of a single standard, together with several expansions and shifts of it. This simplifies greatly the technology for making the comparisons with an arbitrary signal. Various of these single fixed standards (wavelets) are known, but their comparative virtues in signal analysis are still being developed. This research project deals with a newly discovered property of wavelets. Each one has associated to it a notion of "redundancy" of frequencies. It is thought that these redundancies might be an important feature of the wavelet, one that could further simplify and improve the accuracy and reliability of signal analysis.

最近，人们发现了小波的一种内在精细结构。也就是说，每个波都有一个“多重性函数”，它似乎携带了一些关于子波在频域中的不稳定性的信息。这样的冗余应该会导致从信号的小波变换中更可靠地重建信号。本研究的重点是这些多元函数的理论性质。也就是说，我们将研究这种多重函数的性质，试图准确地确定哪类函数以这种方式出现，并尝试根据它们的多重函数对小波进行分类。未知信号(无线电波、地震、天文振动等)通常是通过将其与一组固定的标准信号进行数字比较来分析的。例如，可以从这些数字中(相对准确地)再现信号；可以将其存储在计算机中并在以后重新检查；或者可以将其本身指定为“新的”标准。来自小波理论的新思想是，这套重要的固定标准可以用一个标准来描述，加上它的几个扩展和转变。这大大简化了与任意信号进行比较的技术。这些单一的固定标准(小波)中的各种都是已知的，但它们在信号分析中的相对优点仍在开发中。这项研究项目是关于小波的一个新发现的性质。每一个都与频率的“冗余”概念相关联。人们认为，这些冗余可能是小波的一个重要特征，它可以进一步简化和提高信号分析的精度和可靠性。