For questions related to wavelets and wavelet theory

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Biorthogonal (discrete wavelet) noise bases?

I am slightly interested in discrete wavelet transforms (DWT), but so far I have mostly used already-derived and existing well-known wavelets, such as Daubechies, Cohen-Daubechies-Faveau, Symlets and ...
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7 views

Interpretation of diagonal detail in Haar Wavelet Transforms

I am a statistics grad student, and I have just begun exploring the topic of wavelet regression (specifically, Haar wavelets for discrete functions). I understand the generalization from a one ...
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12 views

Wavelet zero mean condition

I'm trying to understand more about wavelets so I went and read the wikipedia article and some other papers on the topic. I have learned that wavelet functions are compact supported and belong to the ...
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31 views

Scale resolution and frequency resolution in continuous wavelet transform

I'm reading the well known wavelets tutorial by Robi Polikar here. In part 3, about figure 3.7 and 3.8, it says "lower scales (higher frequencies) have better scale resolution (narrower in scale, ...
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What are the different type of Daubechies Wavelet transform?

Like Daub4 are there others named as Daub2, daub3 or we only have daub4 , daub8, daub16? What is the order of a transform(represented usually by N)? Does this order have any resemblance with the ...
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19 views

Invariance of stationary wavelet transform

Suppose we are given 64 points $x_1,\ldots ,x_{64}$ and divide them into two groups $x_1,\ldots, x_{32}$ and $x_{33},\ldots , x_{64}$. Then we apply stationary wavelet transform to both these groups ...
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1answer
94 views

Signal differentiation by wavelet transform

I have a noisy signal x(t) that I want to differentiate in time, in order to obtain x'(t). Though numerical finite difference approximantion does not work well on noisy signal, I would like to ...
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30 views

Wavelet through the lens of Harmonic analysis

I need some references for starting to study about wavelet. I have enough information about abstract Harmonic analysis. Thanks!
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24 views

Build polynomial orthogonal to set of other pre-defined polynomials

Basically. the question is simple. Is there any algorithm so I can build a polynomial, orthogonal for the set of pre-defined polynomials? I need the algorithm(like Gram-Schmidt) that would be ...
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25 views

How to orthogonalize piecewise polynomial with respect to set of piecewise polynomials?

Currently I'm working with multiwavelet basis(need it for solving the stochastic ODE system as described here) and can't understand how to build it. Basically, I have a set of functions $p_{i} = ...
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1answer
34 views

Absolute convergence related to Fourier analysis

If the Fourier transform of a function $f \in L^2$, whose frequency $\xi$ satisfies $|\xi| \leq \pi$, has compact support, it is famous that \begin{align*} f(x) = \sum_n ...
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1answer
61 views

Wavelet admissibility

This question is about Daubechies' book "Ten Lectures on Wavelets", section 2.4. This author says, if $\psi \in L^2(\mathbf{R})$, $\int_{\mathbf{R}}dx\ \psi(x) = 0$, and \begin{align*} \exists ...
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Why is the scaling and shifting variable for wavelet transform is chosen to be multiple of 2?

The continuos wavelet transform is $\gamma(s,\tau)=\int f(t)\psi(\frac{t-\tau}{s})dt$ where we choose the variables s and $ \tau $ to be discrete integers from -$\infty$ to +$\infty$. We choose their ...
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14 views

Why does multiplying by a wavelet equal to bandpass filtering?

The wavelet transform is given as $ \gamma(s,\tau) = \int f(t) \psi (\frac{t - \tau}{s})dt $ However when it comes to carrying out this transform it turns out that it has a "band pass" effect on ...
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1answer
47 views

What is time-frequency plane and when do we use it?

I have seen frequency-amplitude plane for fourier analysis however this time-frequency plane makes no sense to be. It lacks the amplitude information. What is the way to understand and interpret this ...
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1answer
57 views

What does it mean to “simultaneously localize a function in time and frequency domain”?

What does the statement "simultaneously localize a function in time and frequency domain" mean when it comes to signals and systems? What does it mean to localize?
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1answer
66 views

What does $L^2$(R) mean?

I am reading about wavelets and it mentions something about "a function in $L^2(\mathbb{R})$". What does that even mean?
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30 views

Monte Carlo (or other) Approximation to Infinite Summations

This question is sort of paired with a similar post I wrote in Stack Overflow, but not a repeat because I am asking about using another mathematical method. StackOverflow I am trying to approximate a ...
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1answer
27 views

How does a wavelet help in compressing data

I have an understanding of how we carry out image compression by using DCT along with Huffman encoding. The next subject is wavelets. I understand that wavelets are small waves and there are different ...
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29 views

Haar wavelet transformation of a binary vector data over $GF(2)$

I am trying to perform Haar wavelet transformation on the following vector which is defined over $GF(2)$. $[1, 0, 1, 0, 1, 0, 1, 0]$ I am doing it as follows. $[1, 0, 1, 0, 1, 0, 1, 0]$ $\implies ...
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1answer
40 views

Wavelet transform and taking out of frequencies

We use a scaled wavelet and move it across the signal taking out frequencies so that they need not to be processed with a differently scaled wavelet. How does this show up in the math behind wavelet ...
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2answers
99 views

A polynomial that satisfies $x^pf(1-x) + (1-x)^pf(x) = 1$

The context of this question is the construction of the Daubechies wavelet. $f$ is a polynomial of degree $p-1$ which satisfies the equation: $$ x^pf(1-x) + (1-x)^pf(x) = 1 \tag{1} $$ Since $$ ...
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24 views

Directly compute image gradient from discrete wavelet or curvelet coefficients?

We know that (1) the discrete wavelet or curvelet transformation can be used in data compression, and (2) the Sobel operator can be used to compute the gradient of a 2D array of data. In the case of ...
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18 views

For wavelets, is a multiresolution analysis an example of a filtration? If so, is it a filtration on the algebra of L^2?

For wavelets, is a multiresolution analysis an example of a filtration? If so, is it a filtration on the algebra of $L^2$? Even elementary clarifications of basic concepts on the filtration side, in ...
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22 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
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135 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
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75 views

Compare between Short Time Fourier Transform and Wavelets

Fourier transform is localised in only frequency domain but Short time Fourier transform(STFT) is localised both in time and frequency domain same as in wavelets. I want to know How are STFT and ...
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29 views

Fourier transform of complex wavelet for negative frequencies?

One of the requirements that complex wavelets must satisfy is their Fourier transform must be real and vanish for negative frequencies ($f<0$). However, looking at the Fourier transform of Morlet ...
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1answer
22 views

About window function

In Charles.K.Chui's An introduction to wavelets, on Page54, the window function is a non-trivial function $w∈L^{2}(R)$ satisfying $tw(t)∈L^{2}(R)$. I want to ask how to understand the notion, and how ...
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23 views

Lipschitz property and wavelets

I'm trying to understand Nason et al(2000)'s proof of Proposition 3.3 in this paper: http://www.maths.bris.ac.uk/~guy/Research/papers/WavProcEWS1.pdf The expectation of the uncorrected periodogram is ...
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1answer
14 views

Filter output of a signal

So I have a filter $$H(z) = 0.5 + 0.5z^3 = (1/2, 0, 0, 1/2)$$ and need to find the output of it on a cyclical signal $$x = (..., 3, -1, 2, 1, 5, 2, 3,-1, 2, 1, 5, 2, 3,...) $$ Would the output be ...
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How to express a signal in terms of Riesz bases?

Fast discrete wavelet transform allows us to express any discrete signal in terms of wavelet bases by convolution with filter coefficients. How can one express a digital signal in terms of ...
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1answer
48 views

Equivalence of sums

I was hoping someone might be able to help me justify this sum equivalence I ran across in a proof. I'm sure it is something simple but never the less I am confused. I have the following for a ...
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23 views

What are the wavelet coefficients of a time series that is linear interpolated?

I want to know the relationship between the wavelet coefficients of a time series before and after linear interpolated. Suppose we have a time series $x(0),x(1),x(2),\cdots$. When this time ...
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1answer
26 views

Verification of a certain identity in wavelet basis lemma.

This is from Lemma 7.1 in Mallat's Wavelet Tour 2nd edition. I am trying to show that $$ b(2x)h(x) + c(2x)g(x) = a(x) $$ when \begin{align*} b(2x) &= \frac{1}{2}\left[ a(x)h(x)^* + ...
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1answer
47 views

Uniform Approximation by Finite Wavelet Sum

Suppose $\psi:\mathbb{R}\rightarrow\mathbb{C}$ is a rapidly decreasing, bounded function with zero integral. For $j,k\in\mathbb{Z}$ define a function $\psi_{jk}(x):=\psi(2^{j}x-k)$. Suppose is the ...
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1answer
59 views

Maximal Function Estimate

Suppose $\psi$ is a rapidly decreasing function; i.e. for all $N>0$ there exists a constant $C_{N}$ such that $\left|\psi(x)\right|\leq C_{N}(1+\left|x\right|)^{-N}$. Define a family of functions ...
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1answer
97 views

Book recommendation for wavelet analysis

I am master student doing research in data mining, i read a paper about wavlet analysis for data mining, so i think it may help me in the future. But in my undergraduate degree the last course in ...
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1answer
64 views

Application of wavelet analysis in computer science

I am doing research in computer science (data mining), do you think wavelet analysis is useful for me?
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1answer
120 views

Wavelet Theory and Wavelet Series

I am new to Wavelet Theory. My mind came across one question. We learn about Fourier Series (FS) and then about Fourier Transform (FT). Then, why are we not dealing with "Wavelet Series" as FS and ...
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8 views

using wavelet transform for espisific frequencies

I want to analyze a signal in the time-frequency domain, at the frequency : 10,15 and 20 Hz, so should the input "scales" for matlab function 'cwt' be [.1 .15 .2], since the scale is the inverse of ...
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34 views

Continuous second derivative over the support of a Daubechies4 wavelet

I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly ...
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37 views

Significance of orthonormal basis in wavelet analysis

I've recently been looking into wavelet analysis and I have the question: What is the importance of wavelets having an orthogonal basis, say as opposed to a bi-orthogonal basis or otherwise? I'm ...
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1answer
76 views

Derivative of an L1 norm of transform of a vector.

I have to take derivative of the l-1 norm. L1 is the function R in the following expression: $$ R(\psi Fx) $$ where x is a vector, F is the inverse Fourier transform, and $\psi$ is a wavelet ...
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1answer
19 views

Let $x_{0}=1$ and $x_{1}=-1$ For $n\geq0$ inductively define $x_{n+2}=x_{n+1}+6x_{n}$

I am not so sure how to do this problem and would like some help here. How would you induct a relation given this information here? I mean I know what induction means but I'm not so sure what I'm ...
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Terminology with wavelets

I have seen in textbooks that the wavelet transform is stated as two different types of filters. When texts are defining the wavelet transform they call it a band pass filter. However when they talk ...
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27 views

Can wavelets be used for texture discrimination?

I've recently been studying wavelet analysis with a view to differentiating certain areas of texture images where the texture differs from the background pattern (which is quite random); for example a ...
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1answer
22 views

show that $Q(z)=1/2\sum_{k\in \mathbb{Z}}(-1)^k\overline{p}_{1-k}z^{k}$

Let $P(z)=\sum_{k\in \mathbb{Z}}p_{k}z^{k}$ and define $Q(z)=-z\overline{p(-z)}$. for $\left | z \right |=1$, show that $Q(z)=1/2\sum_{k\in \mathbb{Z}}(-1)^k\overline{p}_{1-k}z^{k}$.
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84 views

The Heisenberg uncertainty principle in the time-frequency plane

The Heisenberg uncertainty principle says that it is impossible to have a signal with finite support on the time axis which is at the same time band limited. Is the following reasoning correct: When ...
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610 views

How to calculate wavelet energy?

Part of my assignment about signal processing says the following: Compute the Discrete Wavelet Transform for the input signals Group the wavelet coefficients in trees growing across scales ...