Mathematical Sciences: Time Frequency Analysis & Synthesis of Non-Stationary Signals

在这个数学科学组的研究项目计划的工作旨在联合收割机的问题表示信号的离散函数集的扩展。一种是通过Gabor变换，另一种是使用小波展开。本研究的主要目标之一是使用称为乘法Zak变换的新技术将这两种方法联合收割机结合起来。这项工作的应用将集中在非平稳信号的特征。算法将被开发来计算所产生的双级数展开的系数和信号特性将来自扩展。由于乘法Zak变换可以解释为Weil映射。它出现在海森堡群表示论的另一个背景下。这个理论提供了深入了解特殊函数理论，包括θ函数和Krautchouk和Poisson-Charlier多项式。应用数论也计划。

The work planned in this mathematical sciences group research project seeks to combine two approaches to the problem of representing signals as expansions of discrete sets of functions. One is through the Gabor transform, the other uses the wavelet expansion. One of the primary goals of this research is to combine the two approaches using a new technique called the multiplicative Zak transform. Applications of this work will concentrate on characterizations of non-stationary signal. Algorithms will be developed to compute the coefficients of the resulting double series expansion and signal properties will be derived from the expansions. Since the multiplicative Zak transformation can be interpreted as the Weil map. It appears in another context within the representation theory of the Heisenberg group. This theory provides insight into special function theory including theta functions and Krautchouk and Poisson-Charlier polynomials. Applications to number theory are also planned.