For questions related to wavelets and wavelet theory

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9
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3answers
209 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 ...
1
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0answers
100 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 ...
0
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1answer
27 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 ...
0
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0answers
24 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 ...
0
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1answer
17 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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0answers
23 views

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 ...
0
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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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0answers
25 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 ...
1
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1answer
28 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)^* + a(x+\pi)h(x+\...
0
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1answer
57 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 ...
3
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1answer
63 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 $\...
1
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1answer
143 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 ...
0
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1answer
80 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?
2
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1answer
148 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 ...
1
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0answers
39 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 ...
2
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1answer
53 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 ...
1
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1answer
104 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 ...
0
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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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0answers
19 views

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 ...
1
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0answers
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 ...
0
votes
1answer
25 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 $Q(z)=1/2\sum_{k\in \mathbb{Z}}(-1)^k\overline{p}_{1-k}z^{k}$.
1
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0answers
121 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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0answers
750 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 ...
0
votes
1answer
429 views

Implementing 1D Discrete Wavelet Transform in Matlab

I'm trying to write my own version of the Discrete Wavelet Transform using the bior4.4 filters. I think my implementation is not properly working yet, because whenever I input a signal and a number ...
1
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1answer
83 views

How to prove these equations? ( theorem in multiresolution analysis)

Suppose $\left \{ V_{j} ; j\in \mathbb{Z} \right \}$ is a multiresolution analysis with scaling function $\varphi$ . then the following scaling relation hold: $ \varphi (x)=\sum_{k\in \mathbb{Z}} p_{...
4
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1answer
40 views

Admissability of wavelets

Can someone explain why the admissability of wavelets allows us to conclude the limit of the Fourier transform of a wavelet approaches 0 when $\omega $ approaches 0. Then if the Fourier transform of ...
1
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0answers
45 views

Can anyone provide a sketch as to what a wavelet transform of the dirac delta function look like?

I was trying to motivate the idea of a time frequency localization using wavelet transform to a peer today and I thought the impulse function would be a good example. In my mind I was thinking of a ...
1
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1answer
35 views

How do you obtain a connected (staircase looking) representation of the scaling and wavelet coefficients in Python

How do you obtain a connected (staircase looking) representation of the scaling and wavelet coefficients instead of the unconnected result in the image below? It looks nicer in Matlab than in Python? ...
1
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1answer
90 views

confusion about Morlet Wavelet: What is it exactly?

I was trying to follow and implment a method propose on the research paper. And currently, I have having some trouble to understand the wavelet transform. In particular, the paper I am looking at is "...
0
votes
1answer
80 views

Is fourier transform or Wavelet transform better for this applicaiton?

I am currently designing an alogirthm that is either based on Fourier Transform approach, or the Wavelet Transform Approach, or the combination of the two. Since Wavelet is new to me, I am having ...
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0answers
57 views

Frequency response of wavelets and scaling functions

I am getting started with wavelets! And I am having trouble going from scaling function to the frequency response of the scaling function. The scaling function and wavelet is defined on some axis (...
3
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1answer
139 views

Definition of “uniformly regular” signals (as used in the book “Wavelet Tour of Signal Processing”)

The author uses the term "uniformly regular" and I get the idea of it's meaning through the context, yet the phrase is used as if could also have a precise mathematical meaning. Is there a definition ...
0
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1answer
35 views

Confusion about space and multiresolution

I don't have a background in functional analysis so I find this hard to understand. How could it possible that each "dyadic" dilation of a function f(x) (where the dilation is f(2^i (x))) forms a ...
3
votes
2answers
5k views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
2
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0answers
45 views

Why the $L^1(R)$ space does not have a unconditional basis

Why the $L^1(R)$ space does not have a unconditional basis. It is a known fact that $L^p(R)$ ($1<p<+\infty$) has unconditional basis. A simple example is the dyadic wavelet or Haar system. I ...
2
votes
2answers
102 views

Is there an equivalent of Plancherel's theorem with wavelets transform?

For the Fourier transform, we know that: $$||f||_2=c||\hat{f}||_2$$ where $c$ depends on the normalization. Is there an equivalent with wavelet transform? Thanks.
1
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1answer
144 views

What are unconditional bases and which wavelets have this property?

What are unconditional bases and which wavelets have this property? Haar wavelet seems to be one but how common is it? Unconditional bases are related to Riesz basis condition (ordering of the terms ...
1
vote
1answer
508 views

Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
1
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1answer
98 views

How are generalized frames related to biorthogonal bases?

How are generalized frames related to biorthogonal bases? It seems like frames are a possible solution if neither orthonormal nor biorthogonal bases are available. I thought the generalized frames ...
0
votes
1answer
42 views

The use of wavelets in time series modelling ( feature extraction part)

I have been working on modelling a time series using wavelets for a long time. I am quite familiar with the wavelet theory and all...However, I have a big understanding issue and really appreciate it ...
0
votes
1answer
107 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
votes
1answer
35 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 \rangle=\int_{...
4
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0answers
152 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 ...
0
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1answer
31 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 limit$$\displaystyle\nu(\omega)=\lim_{n\to\infty}\frac{\omega_1}{N^n}+\frac{\omega_2}{N^{n-1}}+...+\frac{\...
0
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1answer
62 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 $\mathscr{H}$...
3
votes
2answers
244 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 ...
3
votes
0answers
95 views

Regularity of Daubechies wavelet

I am reading the book Wavelets: Theory and applications by A. K. Louis, D. Maass, A. Rieder (http://books.google.ca/books?id=58hpQgAACAAJ&dq=wavelets:+theory+and+application&hl=en&sa=X&...
3
votes
1answer
114 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 ...
2
votes
1answer
254 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 ...
3
votes
0answers
52 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 ...