*WINNER* An Introduction to Wavelets and Multiresolution Analyses with an Application to Digital Signal Processing

Abstract

In this project we explore properties of the Haar wavelet and how it is used in multiresolution analysis. Using insights gained from the Haar wavelet, we look at wavelets in a more general sense, deriving crucial properties. To investigate these properties, we often look through the lens of the Fourier transform. Due to the convolutional structure of many of our identities, taking the Fourier transform creates simpler multiplicative identities that we can prove. The multiresolution analysis of a function is determined by the wavelet used. We will investigate an example of decomposing a signal using its multiresolution analysis via the Haar wavelet. Other types of wavelets will be discussed briefly. Wavelets also give rise to wavelet transforms. This project will then cover some basic properties of wavelet transforms and an introduction to their applications in digital signal processing.