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

在这个数学科学小组研究项目中计划的工作旨在结合两种方法来解决将信号表示为离散函数集的展开的问题。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一种称为乘法Zak变换的新技术将这两种方法结合起来。这项工作的应用将集中在非平稳信号的表征上。将开发算法来计算由此产生的双级数展开式的系数，并从展开式中导出信号性质。因为乘法Zak变换可以解释为Weil映射。它出现在海森堡群的表征理论的另一个语境中。该理论提供了对特殊函数理论的见解，包括函数和Krautchouk和泊松-查利尔多项式。还计划应用于数论。

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.