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Basics

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

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Basics

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

  2. 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

  3. 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

  4. 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

  5. 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

  6. Inverse Wavelet Transform • Inverse wavelet transform from wavelet coefficients, • Uniqueness of an Inverse Transform

  7. 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

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