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About using cwt function

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Hi guys! I'm a little bit confused about using cwt function. I know, that some wavelets have a scaling function (phi), but some have only wavelet function (psi). But, as far as I know, cwt uses only psi. So, we have only detail coefficients. What about approximation coefficients, obtained with phi?
Here is the video to clarify what I mean Slide on 14:40.

Accepted Answer

Wayne King
Wayne King on 15 Aug 2017
Hi Alexander, For the CWT, the scaling functions are not commonly used. For discrete analysis wavelets are typically defined in terms of a multiresolution analysis. A multiresolution analysis is a nested series of subspaces and their orthogonal complements. The bases for the subspaces are the scaling functions (dilated versions of the scaling function for $V_0$) and the wavelets are derived from the scaling functions.
In the CWT, that is not the case. One usually starts with the wavelet, typically some rapidly-decreasing oscillatory function like a modulated Gaussian. You can however obtain the scaling function as the integral, $\int_{s'}^{\infty} \dfrac{\hat{\psi}(s \omega)}{s} ds$ where $s'$ is the maximum scale you have in the CWT. $\hat{\psi}$ here is the Fourier transform of the wavelet.
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Alexander Voznesensky
Alexander Voznesensky on 18 Aug 2017
Edited: Alexander Voznesensky on 18 Aug 2017
Thank you. I see, that we introduce the concept of scaling function in DWT. How the scaling func is connected with the wavelet func, is there any formula of connection? What is primary: scaling or wavelet? By the way, we use 2 filters in DWT: low-pass h[n] and high-pass g[n]. We can obtain these filters by dwt. But how these filters are connected with scaling and wavelet? I know 2 formulas (see the attachment), but i want to understand deeper.

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