Titlebook: An Introduction to Wavelet Analysis; David F. Walnut Textbook 2004 Springer Science+Business Media New York 2004 Fourier analysis.Fourier

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Burnout – Herausforderung für die KircheRecall that computing the DWT of a signal ..(.) involves recursevely applying the filtering operators . and . as in the diagram in Figure 6.1, where each node on the tree corresponds to a sequence.

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Fourier Series.. has period . > 0 .(. + .) = .(.) . ∊ .. . periodic.

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The Fourier TransformWe have seen that if .(.) is a function supported on an interval [−.] for some . > 0, then .(.) can be represented by a Fourier series as

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Signals and SystemsIn the previous chapter, we considered piecewise continuous functions with period 1 and showed that it is possible to represent such functions as an infinite superposition of exponentials ..(.) = .., . ∈ .. Each such exponential has period 1/. and hence completes . cycles per unit length (which we can interpret as measuring time).

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The Discrete Haar TransformRecall that a function .(.) defined on [0,1] has an expansion in terms of Haar functions as follows.

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Multiresolution AnalysisIn Section 5.5, we saw that if .(.) = ..(.) −.forms an orthonormal basis on ..

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Biorthogonal WaveletsIn Chapter 2, we considered the notion of orthonormal bases that have infinitely many elements and that can be used to represent arbitrary .. functions. In this section, we will consider nonorthogonal systems with many of the same properties. Such systems are referred to as ..

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