By CHARLES K. CHUI (Eds.)
An advent to Wavelets is the 1st quantity in a brand new sequence, WAVELET research AND ITS functions. this is often an introductory treatise on wavelet research, with an emphasis on spline wavelets and time-frequency research. one of the uncomplicated themes lined during this publication are time-frequency localization, essential wavelet transforms, dyadic wavelets, frames, spline-wavelets, orthonormal wavelet bases, and wavelet packets. furthermore, the writer offers a unified therapy of nonorthogonal, semiorthogonal, and orthogonal wavelets. This monograph is self-contained, the single prerequisite being a simple wisdom of functionality concept and genuine research. it truly is compatible as a textbook for a starting direction on wavelet research and is directed towards either mathematicians and engineers who desire to find out about the topic. experts may well use this quantity as a worthwhile supplementary analyzing to the large literature that has already emerged during this box.
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Extra resources for An Introduction to Wavelets
13) from two different points of view. First, we consider 1 2^ which says that, with the exception of the multiplicative term , the "window Fourier transform" of / with window function ga at t — b agrees with the "window inverse Fourier transform" of / with window function g1/4a at 77 — w. 1, the product of the widths of these two windows is S. 16). w is The Cartesian product of these two windows is called a rectangular time-frequency window. It is usually plotted in the time-frequency domain to show how a signal is localized.
Instead of windowing the Fourier and inverse Fourier transforms as the STFT does, the IWT windows the function (or signal) and its Fourier transform directly. This allows room for a dilation (or scale) parameter that narrows and widens the time-frequency window according to high and low frequencies. Inverting the IWT is required for reconstructing the signal from its decomposed local spectral information. Information on both continuous and discrete time observations will be considered. This leads to the study of frames and wavelet series in the last two sections of the chapter.
Hence, when a bi-infinite sequence is thought of as a digital signal, then its domain of definition, which is the set 2Z of integers, is called the discrete time-domain. In this case, its Fourier series again describes the spectral behaviour of the digital signal, and the domain of definition of a Fourier series is again the real line IR, which is the frequency domain. However, since Fourier series are 27-periodic, the frequency domain IR in this situation is usually identified with the unit circle.
An Introduction to Wavelets by CHARLES K. CHUI (Eds.)