Wavelets and Other Time-Frequency Methods, and their Applications

摘要：本文将对所谓的“Sigma-Delta”转换方案进行详细的数学研究，该转换方案使用简单且易于实现的算法来用一位序列表示带宽有限的函数；将这样的一位序列与适当选择的滤波器进行卷积，可以得到原始带限函数的近似值。我们将详细研究对应不同近似阶数的不同方案；我们期望获得的数学见解将导致具有更好数学性质的新变体，并将其推广到其他设置，而不是近似带宽限制函数。模数转换和数模转换存在于广泛的应用中，其中一些为我们所有人所熟悉（例如，CD播放机），另一些则更多地局限于科学仪器领域，它们利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，从中可以很精确地重建原始信息。由于这些转换中的物理限制，实际使用的方案具有非常有趣的数学性质，与用十进制或二进制表示数字的标准表示有很大不同。建议更详细地研究这些特性，以提出方案的变体，并确定这些转换方案的更广泛的应用范围。

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.