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By Hans-Georg Stark
As the purposes of wavelet rework have unfold to different parts of sign research and compression, scholars and practitioners want a sensible creation and review. This textbook offers that concise and useful advent to the underlying foundations and demanding functions. via a variety of examples and case stories from undefined, it demonstrates either the aptitude and the boundaries of wavelet suggestions, increasing the standard therapy past the discrete wavelet remodel to the continual remodel. offering the fundamentals of Fourier transforms and electronic filters within the appendix, the textual content is supplemented with end-of-chapter workouts, MATLAB code, and a brief creation to the MATLAB wavelet toolbox. scholars of electric engineering and engineers in can enjoy the focus on actual applications.
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Extra info for Wavelets and Signal Processing: An Application-Based Introduction
Example text
2 The Haar-System t a 45 t ω Fig. 1. Left: Under certain conditions the sampled signal f may be reconstructed from the corresponding CWT-matrix by selecting one suitable scale factor. Right: Under certain conditions the sampled signal f may be reconstructed from the corresponding STFT-matrix by selecting one suitable time value. of “frames” [7], for wavelets the most popular version is the “discrete wavelet transform” (DWT) described in this chapter. In contrast to the situation illustrated in the left part of Fig.
24 2 Continuous Analysis Reconstruction In this subsection we shall consider the question how to reconstruct the original continuous-time finite-energy signal f (t) from the CWT Lψ f (a, t). Similarly to the STFT-case we start from the correspondence pair eq. 21. An easy calculation shows how the Fourier transform of ψa (t) may be expressed in terms of the Fourier transform of ψ. 21 one obtains Lψ f (a, t) ◦ − • |a| ˆ ψ(ωa)fˆ(ω). 25) Recall (cf. sect. 1) that this relation is only a shorthand notation for +∞ Lψ f (a, t)e−jωt dt = −∞ |a| ˆ ψ(ωa)fˆ(ω).
3 Case Studies 33 that the members of this family are numbered with dbn (n = 1, 2, 3, . ). An important aspect for signal analysis purposes is that dbn-wavelets have n vanishing moments (cf. eq. 31)! 5 shows the wavelet transform of the above signal with the db2wavelet for three increasing scale factors (cf. figure caption; TS denotes the sampling distance). 5 4 −6 d 1 x 10 1 0 −1 −6 d 3 d 2 x 10 5 0 −5 −10 −5 x 10 4 2 0 −2 −4 1 Fig. 5. Top: Signal. Below: Wavelet transforms for a = TS , a = 2TS , a = 4TS (from above).