Wavelets and Other Time-Frequency Methods, and their Applications

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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.