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