Basics

• Course Outline, Discussion about the course material, reference books, papers, assignments, course projects, software packages, etc.

Introductory Material

• Introductory Remarks about Wavelets, Wavelet-based Signal Processing

• Review of historical trend in signal analysis: From Fourier transform to short-time FT, Gabor transform to wavelet transform

• Why wavelets: comments about some of the main features of wavelets

• Illustration of some of the commonly used wavelets• Illustrative examples of wavelets in signal analysis, some

illustrative demos• Application areas of wavelets

Introduction

• 3. Background material in signal processing and signal decomposition – Fourier transform(FT), Discrete-time Fourier

transform(DTFT) and discrete-Fourier Transform(DFT), complex exponential bases functions

– Main stages in signal decomposition: analysis, coding and manipulation, and synthesis (signal reconstruction) stage

– Wavelets as bases for signal/ function decomposition•

Introduction

• Wavelet function: definition and conditions of a function to be a wavelet function– Examples of wavelets function– Examples of different types of wavelet functions– Parameterization of wavelet functions, Shift and

scale in a wavelet function– Two alternatives values for translation and scale

parameters, continuous or discrete( integer) values– Interpretation of scale as a parameter for

frequency

Introduction, Wavelet Transform

• Wavelet Transform of a given L2 Norm function, definition– Physical interpretation of a wavelet transform, Correlation

of a function with a given analyzing wavelet function– Two alternative wavelet transform, Continuous Wavelet

Transform( CWT), Discrete Wavelet Transform( DWT)– Definition of dyadic wavelet transform, other alternative

wavelet transform structure– Representation of a function in wavelet domain, two

dimensional space of wavelet parameters

Inverse Wavelet Transform

• Inverse wavelet transform from wavelet coefficients,

• Uniqueness of an Inverse Transform

Vector and Function Space• Mathematics of function expansion/signal decomposition and

wavelets– Linear Function space, definition and properties– Dimension of a space, finite and infinite dimensional spaces, examples– Basis in a space, linear independent or orthogonal basis set– Nonuniqueness of basis set of given space– Inner product space, Banach and Hilbert spaces, completeness in a

space, properties of inner product– Linear function space, orthogonal, biorthogonal and Riesz bases– Construction of orthogonal/biorthogonal functions from a given

wavelet function( mother function by sequential changes in wavelet scale and translation parameters

• Frames and redundant signal expansion