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Wavelets and Other Time-Frequency Methods, and their Applications

小波和其他时频方法及其应用

基本信息

批准号：
0070689
负责人：
Ingrid Daubechies
金额：
$ 15.67万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2000
资助国家：
美国
起止时间：
2000-09-01 至 2003-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070689&HistoricalAwards=false
关键词：
Wavelets Other Time Frequency Methods

项目摘要

AbstractA detailed mathematical study will be undertaken of so-called "Sigma-Delta"conversion schemes, which use simple and easily implementable algorithms to represent bandlimited functions by one-bit sequences; convolving such a one-bit sequence with an appropriately chosen filter leads to an approximation of the original bandlimited function. Different schemes, corresponding to different approximation orders, will be studied in detail; we expect the mathematical insights gained will lead to new variants that have better mathematical properties, and to generalizations to other settings than the approximation of bandlimited functions.Analog-to-digital and digital-to-analog conversion, present in a wide range of applications, some familiar to all of us (e.g., CD players), others more confined to the field of scientific instrumentation, makes use of schemes that start by taking highly redundant information, and then replacing it by one-bit or few-bit sequences, from which the original information can be reconstituted with good precision. Because of the physical constraints in these conversions, the schemes used in practice have very interesting mathematical properties, quite different from the standard representation of numbers by a decimal or binary notation. It is proposed to study these properties in more detail, with the goal of proposing variants on the schemes, and to identify a wider range of applications for these conversion schemes.

AbstractA详细的数学研究将进行所谓的“Σ-Δ“转换计划，它使用简单和易于实现的算法来表示由一位序列的带限函数;卷积这样一个一位序列与适当选择的过滤器导致原始的带限函数的近似。不同的方案，对应于不同的近似阶数，将被详细研究;我们期望所获得的数学见解将导致新的变种，具有更好的数学属性，并推广到其他设置比带限函数的近似。模数和数模转换，目前在广泛的应用中，一些熟悉我们所有人（例如，CD播放器），其他更局限于科学仪器领域，使用的方案是从获取高度冗余的信息开始，然后用一位或几位序列替换它，从中可以以良好的精度重建原始信息。由于这些转换中的物理约束，实际使用的方案具有非常有趣的数学性质，与十进制或二进制记数法的标准表示法截然不同。建议更详细地研究这些属性，目的是提出方案的变体，并确定这些转换方案的更广泛的应用。