Foundations of Wavelets and Multirate Digital Signal Processing free videos and free material uploaded by IIT Bombay Staff .
Haar wavelet.
Origin of wavelets.
Dyadic wavelet.
Dilates and translates of Haar wavelet.
L2 norm of a function.
Piecewise constant representation of a function.
Ladder of subspaces.
Scaling function of Haar wavelet.
Demonstration: Piecewise constant approximation of functions.
Vector representation of sequences.
Properties of norm.
Parsevals theorem.
Equivalence of functions & sequences.
Angle between Functions & their Decomposition.
Additional Information on Direct-Sum.
Introduction to filter banks.
Haar Analysis filter bank in Z-domain.
Haar Synthesis filter bank in Z-domain.
Moving from Z-domain to frequency domain.
Frequency Response of Haar Analysis Low pass Filter bank.
Frequency Response of Haar Analysis High pass Filter bank.
Disqualification of Ideal Filter bank.
Ideal Two-band Filter bank.
Realizable Two-band Filter bank.
Demonstration: DWT of images.
Relating Fourier transform of scaling function to filter bank.
Fourier transform of scaling function.
Construction of scaling and wavelet functions from filter bank.
Demonstration: Constructing scaling and wavelet functions..Conclusive Remarks and Future Prospects.
COURSE OUTLINE: The word 'wavelet' refers to a little wave. Wavelets are functions designed to be considerably localized in both time and frequency domains. There are many practical situations in which one needs to analyze the signal simultaneously in both the time and frequency domains, for example, in audio processing, image enhancement, analysis and processing, geophysics and in biomedical engineering. Such analysis requires the engineer and researcher to deal with such functions, that have an inherent ability to localize as much as possible in the two domains simultaneously. This poses a fundamental challenge because such a simultaneous localization is ultimately restricted by the uncertainty principle for signal processing. Wavelet transforms have recently gained popularity in those fields where Fourier analysis has been traditionally used because of the property which enables them to capture local signal behavior.
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