Advanced Digital Signal Processing

Overview

A Beginning with some practical situations, which call for multiresolution/ multiscale analysis - and how time-frequency analysis and wavelets arise from them. Examples: Image Compression, Wideband Correlation Processing, Magnetic Resonance Imaging, Digital Communication - Piecewise constant approximation - the Haar wavelet - Building up the concept of dyadic Multiresolution Analysis (MRA) - Relating dyadic MRA to filter banks - A review of discrete signal processing - Elements of multirate systems and two-band filter bank design for dyadic wavelets - Families of wavelets: Orthogonal and biorthogonal wavelets - Daubechies' family of wavelets in detail - Vanishing moments and regularity - Conjugate Quadrature Filter Banks (CQF) and their design - Dyadic MRA more formally - Data compression - fingerprint compression standards, JPEG-2000 standards - The Uncertainty Principle: and its implications: the fundamental issue in this subject - the problem and the challenge that Nature imposes - The importance of the Gaussian function: the Gabor Transform and its generalization; time, frequency and scale - their interplay - The Continuous Wavelet Transform (CWT) - Condition of admissibility and its implications - Application of the CWT in wideband correlation processing.

Journey from the CWT to the DWT: Discretization in steps - Discretization of scale - generalized filter bank - Discretization of translation - generalized output sampling - Discretization of time/ space (independent variable) - sampled inputs - Going from piecewise linear to piecewise polynomial - The class of spline wavelets - a case for infinite impulse response (IIR) filter banks - Variants of the wavelet transform and its implementational structures - The wavepacket transform - Computational efficiency in realizing filter banks - Polyphase components - The lattice structure - The lifting scheme - An exploration of applications (this will be a joint effort between the instructor and the class) - Examples: Transient analysis; singularity detection; Biomedical signal processing applications; Geophysical signal analysis applications; Efficient signal design and realization: wavelet based modulation and demodulation; Applications in mathematical approximation; Applications to the solution of some differential equations; Applications in computer graphics and computer vision; Relation to the ideas of fractals and fractal phenomena.