File Name: difference between wavelet transform and fourier transform .zip
- Wavelet versus Fourier Analysis
- Fractional Fourier transform
- Wavelets 4 Dummies: Signal Processing, Fourier Transforms and Heisenberg
Wavelet versus Fourier Analysis
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 a 32nd note. 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.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. Very roughly speaking: you can think of the difference in terms of the Heisenberg Uncertainty Principle , one version of which says that "bandwidth" frequency spread and "duration" temporal spread cannot be both made arbitrarily small. But by doing so you lose all control on spatial duration: you do not know when in time the signal is sounded.
Wavelets have recently migrated from Maths to Engineering, with Information Engineers starting to explore the potential of this field in signal processing, data compression and noise reduction. In doing this they are opening up a new way to make sense of signals, which is the bread and butter of Information Engineering. Because there are very few rules about what defines a wavelet, there are hundreds of different types. These little waves are shaking things up because now Wavelet Transforms are available to Engineers as well as the Fourier Transform. What are these transforms then and why are they so important? It ends by describing how wavelets can be used for transforms and why they are sometimes preferred because they give better resolution. This blog post does not have much maths in it, but it does deal with concepts that might be slightly beyond someone with no mathematical background.
Fractional Fourier transform
The advantages of wavelet analysis over Fourier analysis is the subject of Chapter 3. A comparison between frequency analysis, by means of the Fourier transform, and time—frequency representation, by means of the wavelet transform, is made. From an example of a nonstationary signal, the good extraction of the time and frequency characteristics of the wavelet transform is revealed. In addition, the properties of wavelet bases functions and WT signal processing applications will be described. Unable to display preview. Download preview PDF. Skip to main content.
Wavelets 4 Dummies: Signal Processing, Fourier Transforms and Heisenberg
Signal processing has long been dominated by the Fourier transform. However, there is an alternate transform that has gained popularity recently and that is the wavelet transform. The wavelet transform has a long history starting in when Alfred Haar created it as an alternative to the Fourier transform. In Norman Ricker created the first continuous wavelet and proposed the term wavelet. While the Fourier transform creates a representation of the signal in the frequency domain, the wavelet transform creates a representation of the signal in both the time and frequency domain, thereby allowing efficient access of localized information about the signal.
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