What is Wavelets: Definition and 19 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. F

    Wavelet transform (CWT and DWT)

    Hello, I recently got interested in wavelets. The main idea seems clear: we compute the inner product between the signal ##x(t)## and a chosen wavelet for different scale factors and translations of the wavelet over the signal. The inner product provides the coefficient for a wavelet with a...
  2. S

    I Young's double slit experiment -- Question about this wavefront diagram

    Hello, I have a question about the blue waves coming from sources S1 and S2 in de next picture. The blue waves from sources S1 and S2, are those two resulting waves (interference of all wavelets, Huygens Principle) or are those blue waves two wavelets?
  3. DoobleD

    I Why is the number of wavelets in a slit = slit length / lambda

    Hi folks, Huygens principle is not really new to me but I just realized there is something I don't understand with it. Take a single 1D slit with a coherent incident plane wave. It seems that the number of wavelets in the slits is ##n = l/\lambda##, where ##l## is the length of the slit, and...
  4. W

    I Vector-wavelet Galerkin projection of Navier-Stokes equation

    Hi, I am having a little trouble understanding a minor step in a paper by [V. Zimin and F. Hussain][1]. They define a collection of divergence-free vector wavelets as $$\mathbf{v}_{N\nu n}(\mathbf{x}) = -\frac{9}{14}\rho^{1/2}_N...
  5. R

    Wavelets vs disadvantages of Fourier Analysis and Synthesis?

    I am a beginner. The Fourier series, Fourier Transform and it's inverse play very important role in Fourier Analysis and Fourier Synthesis. I have read that Fourier transform is localised in only frequency domain.Also,it contains information about the signal in phase and frequency spectrum...
  6. R

    Why the Fourier transform is so important compared to other?

    I am engineering student and studying signal processing. The term Fourier transform comes in the discussion several times. There are many transforms like Laplace transform,Z transform,Wavelet transform.But as per my view ,Fourier transform is mostly used compared to others in general. My...
  7. A

    Application of wavelets in computational electromagnetics

    Hello everyone, I am a first year graduate student, wavelets is one of the courses that I am taking presently. As part of the course, I plan to explore application of wavelets in computational electromagnetics (CEM). The idea behind it is to study a few applications where wavelets are used in...
  8. S

    Looking for Material on Wavelets

    Hi, I'm looking for a good beginner text on Wavelets. Preferably an undergraduate or early graduate level. The background material that I'm missing most is infinite dimensional vector spaces/function spaces and Fourier analysis. Any advice would really be appreciated! Thanks!
  9. H

    The level of Daubechies wavelets

    Hi, I would like to know the technical approach that can be used to determine thses values: N Level As shown in the attachment: N=3 Level=6
  10. L

    A classical wave as sum of many coherent quantum wavelets?

    Merzbacher - Quantum mechanics, second edition, chapter 1 page 4,writes: "A classically observable wave will result only if the elementary wavelets representing the individual quanta add coherently. " Is this analysis correct? The context is the following: We may read (1.3) [ë =...
  11. T

    Discrete Wavelets - Question on decimation of high pass

    Hi, I'm trying to grasp the concept of discrete wavelets and can't seem to find an answer to my question. In the decomposition of a signal using wavelets filter banks, the signal goes through a low pass and high pass filter. The output of the low pass and high pass is decimated by 2. I can...
  12. A

    Wavelets question (Specifically Daubechies, understanding scaling functions)

    Hi all, I have a question about Daubechies Wavelets. I've recently been trying to teach them to myself from the pdf here...
  13. R

    What are the dimensions of huygen wavelets

    According to Wikipedia the formula for the field created by an aperture (the Kirchoff-Fresnel Integral) is: \Psi(r)\propto \int\!\!\!\int_\mathrm{aperture} E_{inc}(x',y')~ \frac{e^{ik | \bold r - \bold r'|}}{4 \pi | \bold r - \bold r' |} \,dx'\, dy'...
  14. G

    MATLAB Saving Plots & Data from Matlab Wavelets Toolbox

    Hello, Does anyone know how to save the plots and some of the data results generated by the Matlab Wavelets Toolbox with the GUI?? I would like to save that along with all the statistical results from the GUI when performing "Multisignal Analysis 1-D" I am using Wavelets 4.4 and it not as...
  15. E

    Find Filters for Wavelets Analysis Homework: p_k Coefficients & Relations

    Homework Statement Given the following p_k scaling coefficients and the following wavelet relations, find all four filters corresponding to these coefficients: low-pass decomposition, high-pass decomposition, low-pass reconstruction, and high-pass reconstruction.Homework Equations \phi (x)=...
  16. A

    Good resource for intro to wavelets?

    Two friends and myself are going to start doing research with a professor of ours on Haar wavelet transforms. I was curious if there are any texts or any online resources which those of PF would recommend? So far the three of us have completed Engineering Calc I, but the professor feels...
  17. L

    Wavelets and Fourier transform

    I am new to wavelets. I was reading about wavelets and Fourier transforms. So the main disadvantage of Fourier Transform is that you cannot use it on a non-uniform signal. Even though you use it you have to use a window and select your region of interest. If the window is small enough you can...
  18. R

    Huygens' theory, point on a wavefront, secondary wavelets

    Huygens' theory says that every point on a wavefront serves as a source of secondary wavelet. Doesn't that imply that if we consider any coherent source of light, and intercept its waves on a screen, we'll get a diffraction pattern, as all those secondary wavelets will interfere among themselves?
  19. K

    What Are Gabor Wavelets? A Synopsis

    what are gabor wavelets? a basic synopsis would help... i don't know what section to put this post under, so i apologize for that..