# Tagged Questions

For questions related to wavelets and wavelet theory

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 ...
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 ...
27 views

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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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+\...
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 ...
63 views

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 $\... 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 ... 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? 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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$Q(z)=1/2\sum_{k\in \mathbb{Z}}(-1)^k\overline{p}_{1-k}z^{k}$. 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 ... 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 ... 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 ... 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_{...
40 views

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 ...
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 ...
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? ...
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 "...
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 ...
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 (...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
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 ...
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 ...
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 ...
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 $$x_t = ( 5, 6)$$ $$X(z) = 5 + 6z$$ ...
35 views