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13 views

Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
2
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1answer
129 views

Best way to find magnitude and phase of a specific frequency in an empirical time series…

I've a discrete, univariate time series, and I'm interested in to investigate a specific frequency component. Assume I'm interested in a frequency with a cycle-time of $f$ samples - and I need to get ...
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0answers
13 views

use relevant wavelet basis for periodic components

i would like to understand how should i use following code for code for detection of uknown frequencies?let us consider following signal ...
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0answers
206 views

Wavelets: Cone Of Influence

While reading this paper I came across the term Cone of Influence which is described as ...
3
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1answer
702 views

How to intuitively interpret Gabor lambda param?

I have troubles understanding in an intuitive way (not by writing complicated math formulas) what is the meaning of the lambda parameter in the Gabor functions. (I have basic math understanding, grad ...
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1answer
372 views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
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0answers
14 views

Evaluating an integral involving Haar scaling function

Let $\phi(t)$ denote the Haar scaling function, and let $\psi(t)$ be the corresponding wavelet. My problem is to evaluate the integral $I = \int_{t} \phi(t - 1/2) \frac{\partial \psi(t)}{\partial t} ...
2
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1answer
50 views

Daubechies wavelet

I have two question about Daubechies wavelet are they really orthogonality? for example daub4 has the support equal $[0-3]$, and for scale function we have must: $$ \int_0^3\phi(x-n)*\phi(x-m)dx ...
2
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1answer
23 views

A relation for Fourier series

For $f$ and $f'$ in $L^2(0,1)$, define $e_k(x)=e^{2\pi ikx}$, $k \in \mathbb{Z}$. And define the Fourier series: $f=\sum _{k \in \mathbb{Z}}c_ke_k$, where $c_k=\left \langle f,e_k \right ...
2
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0answers
24 views

Matlab's dwt2 in publication quality notation?

I used Matlab's dwt2in my algorithm, what notation have you folks seen that describes the 4 resulting wavelets? How do I express ...
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0answers
39 views

Does these two operators commute?

It is an exercise, $z\in\mathbb T$, which is the unit circle $$m(z)=\sum_{k\in\mathbb Z}a_kz^k$$ set $$Sf(z)=\frac{1}{\sqrt N} m(z)f(z^N)$$ So the adjoint operator is ...
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1answer
23 views

How to show that the limit of $\frac{\omega_1}{N^n}+\frac{\omega_2}{N^{n-1}}+…+\frac{\omega_n}{N}$ exist

$$\omega=(\omega_1, \omega_2,...)\in \{0, 1, 2,...,N-1\}^\mathbb N$$ How to show that the ...
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1answer
28 views

Bessel Sequence proof check.

I have a similar question to definition of Bessel sequence, where it was solved using Banach-Steinhaus. A sequence $\{f_k\}_{k=1}^{\infty}$ is called a Bessel sequence in a Hilbert space ...
2
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2answers
36 views

In which case should a wavelet transform be applied instead of a Fourier transform?

I was wondering what are the advantages (and possibly the drawbacks) of using a wavelet transform instead of a Fourier transform for the signal processing, are there simple examples to illustrate that ...
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0answers
12 views

Are effective degrees of freedom after discrete wavelet transforms additive across scales?

I'm looking into effective degrees of freedom estimation with wavelets. Methods for estimating scale-specific effective degrees of freedom (EDOF) from the discrete wavelet transform of a signal are ...
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0answers
44 views

Regularity of Daubechies wavelet

I am reading the book Wavelets: Theory and applications by A. K. Louis, D. Maass, A. Rieder ...
2
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2answers
107 views

Morlet's wavelet reconstruction formula

The CWT (continuous wavelet transform) of a signal $x(t)$ is $$X_w(a,b)=\frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t)\psi^{\ast}\left(\frac{t-b}{a}\right)\, dt$$ In order to reconstruct the ...
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1answer
49 views

definition of Bessel sequence

A sequence $\{f_{k}\}_{k=1}^{\infty}$ is called a Bessel sequence in a Hilbert space $H$, if there exists $B>0$ such that $$\sum_{k=1}^{\infty}|\langle f,f_{k}\rangle|^{2}\leq B\|f\|^{2}$$ for all ...
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0answers
31 views

Integral involving wavelet

Let $\hat{\psi}_m$ be a Fourier transform of Daubechies wavelet of order $m$ and $\chi_I$ is a characteristic function of interval $I$. How to bound from above the following integral $$ ...
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0answers
12 views

Fast transpose of undecimated wavelets

I am using undecimated Daubechies wavelets and I need to compute the forward and adjoint of the wavelets several times. I am using the Rice Wavelet Matlab toolbox ...
2
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1answer
82 views

Need to learn wavelet, suggest steps and resources

I am looking for a good introduction to wavelets and wavelet transforms. that covers the following: Basics Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and ...
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1answer
31 views

Wavelets on $\mathbb{R}^{n}$

I want to study Wavelet theory on $\mathbb{R}^{n}$ and I can't find any book on this topic. Can you recommend me any good book that considers that?
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0answers
29 views

Error bounds in representing a vector using a truncated Moore-Penrose biorthogonal basis

I was reading and trying to reproduce the results in the arXiv preprint of Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression by Asaf ...
2
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1answer
44 views

Biorthogonal Projector Construction

I was trying to prove that the biorthogonal projector matrix $P$ given on the Wikipedia page on biorthogonal systems does in fact construct biorthogonal systems from input bases $\mathbf{u}$ and ...
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0answers
42 views

Wavelets, How can a zero padding lengh n signal be truncated to n coeficients from which a signal can be reconstructed

Wavelet transform is defined for infinite lengh signals, finite lengh signals must be extended in some way before they can be transformed. I know that periodic replication and zero padding are ...
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0answers
49 views

Continous wavelet transform and shannon Entropy.

Note: I have asked the same question on signal processing forum,but didn't get any answer. so it might be more like a math or physics question. Hope you don't consider it as cross-post. I am trying to ...
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0answers
37 views

Strange values of approximating coefficients returned by matlab's wavelets decomposition

I'm trying to get wavelet decomposition of arcsin(x) using, say, haar wavelets When using both Matlab's dwt or wavedec functions, I get strange values for approximating coefficients. Since applying ...
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1answer
57 views

Denoise using wavelet transform

My mathematical class task is to de-noise a function using wavelet transform. I am to select a function $f(x)$ and noise function with zero-mean $n(x)$. I am to add noise like this: $$f_{noise}(x) = ...
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2answers
43 views

Find support of a wavelet

I am a complete novice in wavelet analyses, but I was given a task to find a support set of a given wavelet (in 1D, e.g. db4). Please, point me a way of doing this. (Since, how I understood from some ...
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1answer
69 views

Linear algebra explanation of wavelet transform

Is there an easy way to explain wavelets / wavelet transform using only linear algebra? The discrete Fourier transform is a linear operator on $\mathbb C^N$ that simply changes basis to a special ...
1
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1answer
37 views

definition of wavelet in $L_p$ space

Function $\psi$ is called wavelet, if there is a dual $\widetilde{\psi}$ such that a function $f \in L_2(R)$ can be decomposed as $$ f(t)=\sum_{ j \in Z}\sum_{\nu \in Z} \langle f, ...
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88 views

wavelet galerkin method for solving differential equation by boundry condition

i want solve the differential equation by boundary condition with galerkin wavelet method.(for example:$$2y''+8y=cos(X),y(0)=0,y(1)=0 $$). i used the chebyshev wavelet and approximate the unknown ...
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1answer
43 views

Integrating a squared Haar wavelet function

I'm studying about wavelets and here is one derivation I couldn't understand: The constant that makes this orthogonal basis orthonormal is $2^{j/2}$. Indeed, from the definition of norm$^2$ in ...
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3answers
66 views

Orthogonality of Haar wavelet functions

I'm reading about wavelets and I bumped into the follwing: $\text{Haar wavelet is a step function}\; \psi(x), \text{which takes values 1 and -1, when}\; x \;\text{is in the ranges}\; [0, \frac{1}{2}) ...
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3answers
56 views

Reducing an infinite dimensional expansion to a finite dimensional algorithm

The following is a big-picture question about the interplay between infinite dimensional function expansions and finite dimensional algorithms. I feel like I have a good understanding of these ideas ...
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1answer
99 views

compare wavelet and Fourier transform

i would like to compare each other wavelet and Fourier transform on given signal,let us consider following signal ...
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1answer
65 views

Weak form for Linear Dynamic Wave Equation of Dirichlet/Neumann's boundaries?

I have a linear problem with double derivate of space and time, which has Dirichlet boundary condition in $(1)_{2}$ and Neumann's boundary condition in $(1)_{3}$: \begin{equation} \frac{\delta^{2} ...
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0answers
38 views

Admissibility condition for n-dimensional wavelet

Theorem 14.2.1 [S. T. Ali, J.- P. Antoine, J.- P. Gazeau, Coheret States, Wavelets and Their Generalization] The operator family $U:SIM(n)=\mathbb{R}^n \rtimes(\mathbb{R}_*^+ \times SO(n)) ...
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0answers
99 views

Passing the singularities .

I need some information or detail with example to the following statements. Circumvent the singularity by a contour inside the wave-guide. Circumvent the singularity by a contour outside the ...
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0answers
62 views

Output of wavelet transforms

I am working on a time sensitive computer science and fluid dynamics project that requires me to find applications of wavelet analysis. I know that at its core, a wavelet transform simply takes a ...
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1answer
34 views

Show that the endpoints of a compactly supported function satisfying the scaling relation are integers

Suppose $ \phi \in C_0( \mathbb{R}) $ (compact support) satisfy the scaling relation $$ \phi(x) = \sum_{k \in \mathbb{Z}} p_k \phi(2x-k) , $$ with $$ p_k = 2^{1/2} \int_{- \infty}^\infty \phi(x) ...
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0answers
103 views

An exercise. A property of the Fourier transform of wavelet

In the book "An Introduction To Wavelet Analysis" by David F. Walnut, there is, Exercise 7.45. Show that if $\psi(x)$ is a wavelet, then $\sum\limits_{j}{\left|\hat{\psi}(2^j\gamma)\right|^2} = 1$ ...
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0answers
31 views

Finding correlations between many unknown functions.

Given an arbitrarily large number of black-box functions of one variable, is it possible to produce expressions that approximate their relationships to each other over their shared domain? Is it ...
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0answers
108 views

DWT (Discrete Wavelet) - Why are there different number of coefficients?

I have been studying wavelet theory and i am particularly interested in applying it to a discrete case. I found the following representation of a discrete function in one of the lectures: $s(n) = ...
2
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1answer
111 views

haar wavelet: unable to understand what limits they have used

Please explain what limits of integration they took for solving $p_i(x)$ and $q_i(x)$. like for 1st step of $p_i(x)$: $$\int_\alpha^x 1 \, \mathrm dx = x-\alpha$$ but for 2nd step of $p_i(x)$: ...
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0answers
97 views

Operators from $L^{\infty}$ to $L^{\infty}$, below bound of the norm

If $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded on $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le \|T\|_{L^{\infty}\rightarrow ...
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0answers
14 views

Fittig N-D periodic functions

Besides fourier series, are there any stable way to fit N-D nonlinear functions that demonstrate some degree of periodicity, sigmoid neural networks works poorly in this domain.
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2answers
5k views

What is a “vanishing moment”?

In this paper, Sweldens says about desireable properties of wavelets: To analyze and represent such signals we need wavelets that are local in space and frequency. Typically this is achieved by ...
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2answers
2k views

DFT and DWT difference?

what is the basic difference between the Discrete Fourier Transform and the Wavelet Transform ? and why does JPEG2000 preferred DWT over DCT or DFT ?
2
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0answers
57 views

Proving norm inequality with Schwarz's inequality

I'm stuck on the following problem: Suppose that $\{V_j : j \in \mathbb{Z}\}$ is a multiresolution analysis with scaling function $\phi$, and that $\phi$ is continuous and compactly supported. ...