# What is Wavelet: Definition and 23 Discussions

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Generally, wavelets are intentionally crafted to have specific properties that make them useful for signal processing.

For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second. If this wavelet were to be convolved with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the Middle C note was being played in the song. Mathematically, the wavelet will correlate with the signal if the unknown signal contains information of similar frequency. This concept of correlation is at the core of many practical applications of wavelet theory.
As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including – but not limited to – audio signals and images. Sets of wavelets are generally needed to analyze data fully. A set of "complementary" wavelets will decompose data without gaps or overlap so that the decomposition process is mathematically reversible. Thus, sets of complementary wavelets are useful in wavelet based compression/decompression algorithms where it is desirable to recover the original information with minimal loss.
In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or frame of a vector space, for the Hilbert space of square integrable functions. This is accomplished through coherent states.

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1. ### Help with Haar wavelet transform problems?

Attempt: This is where I am having difficulties understanding and getting a start at a solution. I get a lot of definitions and yet no worked examples. The plot will be constant piece-wise functions, as I understand, but I am having trouble visualizing. Any guidance is much appreciated.
2. ### I Effect of impedance changes less than a wavelength

I am interested to know what is the impact of various length scales of impedance changes on wave propagation. From undergraduate physics (a few years ago for me) I roughly remember how to derive reflection and transmission coefficients for a wave traveling from one medium to another with a...
3. ### What is the basis for bessel function as we have for wavelet

Hi, I have recently studied about basis for wavelet function which is helpful to design any function. Likewise, what is the basis for bessel function and how can it be implemented for an image ( because image is also a function). Specifically, I am interested to know how bessel function can be...
4. ### A Regarding the Continous Wavelet Transform 'a' parameter

Hi there, I've recently been doing some studying into time-frequency analysis. I've covered some of the basic materials regarding the Short-Time Fourier Transform (STFT) along with the concepts of temporal and frequency resolution (along with the uncertainty principle of course). I've now...
5. ### A Fourier Transform of Piecewise linear spline wavelet

Fourier Transform of Piecewise linear spline wavelet is defined by 1-|t|, 0<t<1; 0, otherwise, is (sinc(w/2))^2. Can anyone please show me the steps. Thanks
6. ### How to use DWT when sample size is smaller thn filter length

Hi, I am new member over here. I wanted to know how to perform discrete wavelet transform, probably using symmlet 8 wavelet with 16 Low/High pass filter coefficients using matrix transformation method when decomposing to a certain level the approximate coefficients i.e. sample size becomes less...
7. ### How to find phase values at neighboring pixel in an image?

I have computed magnitude and phase spectrum of very famous image of cameraman using fft function in MATLAB.Here,we get magnitude and phase spectrum of the whole image. But I want to find phase values of the neighboring pixels . So if given gray scale image is of dimensions 256*256 and if I...
8. ### Wavelet coefficient based QRS complex classifier

Dear all, I am new to Wavelet field and I wanted to ask you for a help for an idea. I am supposed to create QRS complex (certain part of ECG signal wave) wave morphology classifier based on Wavelets in other words, I am supposed to create classifier which will separate waves with similar wave...
9. ### Morlet wavelet time-bandwidth product

Homework Statement I am trying to figure out σt and σf for the Morlet wavelet knowing that the time bandwidth product is equal to 2.5. Any suggestion ?
10. ### How to reconstruct a signal using the Discrete Haar Wavelet transform?

Homework Statement Hi, for a project for school, I need to implement the Discrete Haar Wavelet Transform to compress an audio signal. This would be fine and dandy, but I do not really understand how to use the the DHWT. Could anyone direct me towards some resources that would be very helpful...
11. ### What Is the Relationship Between Radius and Wavelength in Huygens' Wavelets?

Huygens' belief was that every wave is made up of smaller wavelets, which are basically circles. I was wondering, is the radius of these wavelets equal to the length of the wavelength of its wave? I have another related question: If a wave is passing through a slit, what is the maximum...
12. ### Harr Wavelet Question: Proving Orthogonality of Psi_2,1 and Psi_2,0

Homework Statement I think this may be a simple problem, but I really have no idea if I did it right because it seemed to easy. Here's the question, consider the Harr Wavelet \psi^{}_n{}_,{}_k{}(x) = 2^n{}^/{}^2{}*\psi(2^n{}x-k) where \psi is the mother wavelet. Prove that...
13. ### Inquiry: Wavelet factorisation / Daubechies method /

I need to construct a non-standard wavelet filter using the factorisation technique / Daubechies method / applied to the wavelet family / any /. I need to obtain the wavelet coefficients. The problem is I can't even find a tutorial on how to perform the factorisation method. I've read Ingird...
14. ### Wavelet Theory: Solving ∫t^pψ(t)dt=0

I don't understand how to get the equation \sum_k (-1)^kk^pc_k=0 from ∫t^pψ(t)dt=0 from here on page 80. Can somebody explain it?
15. ### Wavelet transform for images cannot interpret it correctly

Wavelet transform for images...cannot interpret it correctly... Hello Forum, The discrete wavelet transform and the continuous wavelet transform are two different beasts. When we take the DWT of an image, we get a bunch of subimages that are different in size. Each subimage is like a...
16. ### Understanding Wavelet Transform for Time Series Data Analysis

I have a time series data, I want to use wavelet transform to de-noise the data and then use neural network to analysis the transform data. However, I quite not understand about the wavelet transform. Can tell me about how to use wavelet transform? How to determine the scale? Is it suitable use...
17. ### Getting Wavelet Basis functions

Hi all, Say that I have a 1D signal such that f=Bw where f is the signal B is the basis functions and w is the wave co-efficients. The question that I have is how do I find the B matrix in Matlab. I am looking through WaveLab and Rice Wavelet packages but simply cannot find an answer. As...
18. ### MATLAB How can I store wavelet values in an array while using a loop in MATLAB?

Hi all, I want to generate a ricker wavelet using MATLAB then I convolve it with the Earth reflictivity (e) to get a seismic trace I have (e) and I would like to generate ricker using the equation w=(1-(2*pi^2*f^2*t^2))*exp(-pi^2*f^2*t^2) between t:dt:tf I use loop in MATLAB to get...
19. ### Analytic expession for the width of a wavelet

Hello-- I've been working with a time-domain function to calculate the Ricker (Mexican Hat) wavelet for -0.2 \leq t \leq 0.2 . This function is given as s(t): s(t) = \left( 1 - \frac{1}{2} \omega_0^2 t^2 \right) \mbox{exp} \left ( -\frac{1}{4} \omega_0^2 t^2 \right ) In the...
20. ### Wavelet transform,STFT, freq. localization

hello Forum, taken a signal composed of 4 spectral components, with Fourier angular frequencies w1<w2<w3<w4. Take now a completely different signal, a chirp, made of the same 4 spectral components. IT is said that if the chirp increases in frequency with time (meaning its instantaneous...
21. ### MATLAB Daubechies wavelet - scaling and wavelet function in MATLAB

Hi everybody, I'm trying to reproduce daubechies basic building graph and daubechies wavelet function graph (φ(r)=0 if r≦0 or 3≦r). And i found this algorithm. I would appreciate if there is anybody could help me to understand the function defined below as function [s,w] = cascade(n,cs,cw)...
22. ### Understanding Wavelets: Definition and Functions Explained

Homework Statement What is a wavelet? Homework Equations Here is the Haar wavelet mother waveley function: <tex>\{ t \mapsto \psi(2^n t-k) ; n \in \N, 0 \leq k < 2^n\}</tex> The Attempt at a Solution Wavelets are used for analysis. There is a wavelet function and a scaling...
23. ### Wavelet Theory: An Overview

What is wavelet theory?