Wavelets and Signal Processing: An Application-Based by Hans-Georg Stark

By Hans-Georg Stark

As the functions of wavelet remodel have unfold to various parts of sign research and compression, scholars and practitioners want a sensible advent and evaluate. This textbook presents that concise and functional advent to the underlying foundations and critical functions. via a variety of examples and case stories from undefined, it demonstrates either the capability and the bounds of wavelet strategies, increasing the standard therapy past the discrete wavelet remodel to the continual rework. 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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As stated already after eq. 1, in order to keep notation simple we shall use the symbol f both to represent either a continuous-time signal or a discretely sampled signal. The actual meaning always shall be clear from the context. In sect. 1 we shall introduce some more continuous-time functions, which are uniquely connected with sequences. Here the same convention holds. 4) We shall see later that to each wavelet for which a DWT may be constructed, there belongs a unique scaling function. 4 belongs to the Haar-wavelet ψH (cf.

035226} The corresponding g-coefficients may be computed according to gk = (−1)k h1−k (cf. [7]). 28) 56 3 The Discrete Wavelet Transform The simplest member of the family of Daubechies-wavelets is db1. The corresponding scaling function and wavelet are given by the Haar-scaling function φH and the Haar-wavelet ψH , respectively. Indeed, the related filters have 2 coefficients √(cf. 20) and the g-coefficients satisfy the relations g0 = 22 = h1 and g1 = − 22 = −h0 in accordance with eq. 28. Likewise, for n = 2 eq.

19) ω Readers mainly interested in applications may skip the rest of this section and proceed from here to sect. 2 and then to sect. 4. The rest of this section and sect. 3 require basic knowledge of Fourier transforms and the discrete Fourier transform (cf. 2, respectively). 20) we obtain 1 Lψ f (a, t) = √ cψ 1 |a| ∞ ψa (t − u)f (u) du. −∞ Then from the convolution theorem (eq. 5) we may conclude that 1 Lψ f (a, t) ◦ − • √ cψ 1 |a| ψˆa (ω)fˆ(ω). 21) Similarly to eq. 5 in the STFT-case, eq. 21 is the key both for a fast CWT-computation-algorithm and for reconstructing the signal from the CWT.

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