### Introduction to Fourier Analysis and Wavelets

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So in a way they are approximately localized as well in time as in frequency. But there is much more to it, e.

You should consult one of many primers on wavelet theory, e. Burrus et al. The answers by Willie Wong first part and vac, focussing on the localization in time and frequency are at least not the only property that's characteristic about the wavelet transform: the short-time Fourier transform is also localized in time and frequency, i. You will also find it being called the windowed Fourier transform, since it's essentially the Fourier transform together with window mask , that allows you to only look at a small portion of the signal.

While these answers cover the original question, I think it's important to know that this is just one property.

## Introduction to Fourier Analysis and Wavelets

What mathreadler says, is another very important perspective on this. For sounds, sinusoids seem the natural choice of what a "frequency" is, but for other signals especially in 2D and higher wavelets might be much more well-suited to represent a signal in a "rich" way. Meaning e. The Fourier Transform is often so predominant in science and engineering probably thanks to the invention of FFT right around when digital tech started to get big so we often think of "frequency" as sines and cosines without even being concious about it.

It is true that in Fourier cos and sin basis functions frequencies, the Fourier Transform is the one which contains only frequency and none "localisation" information. At least in the sense that a single sample will spread equal energy all over the Fourier spectrum irregardless of it's position or time. Example of signals which are good to model with FFT are pure musical tones in audio and music and how you would expect to model a smooth revolution of.. However if we are talking about non-harmonical or even non-continous or binary frequencies, like a digital clock tick, then the concept of frequency is more closely related to that of the Haar Wavelet or maybe a Hadamard Transform.

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A Haar wavelet function the "wave-lenght" of a second can much better "describe" or "encode" such localized discrete clock events and the Hadamard Transform can capture more global "frequency" like information of these discontinous signals but with less localization. So answering this question rather raises another question : What do we even mean with the word "frequency"? What kind of a frequency is important in this context? For example take the Gaussian function. This function is almost zero except at some neighborhood near zero.

Therefore the function is well-localized on some domain which is a neighborhood near the origin.

Such function is called well-localized in time and frequency. Again, the Gaussian function is an example of this kind. I hope the example helps. Sign up to join this community. The best answers are voted up and rise to the top.

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Home Questions Tags Users Unanswered. Difference between Fourier transform and Wavelets Ask Question. Asked 6 years, 8 months ago. Active 3 years, 8 months ago. Viewed 51k times. I did not understand what is meant here by "localized in time and frequency. TXC 5 5 silver badges 15 15 bronze badges. If we take the Fourier transform of the observed frequency, we can say that At some time the traffic light shows red.

We know frequency to infinite precision, and that the red part of the signal is non-zero. At some time the traffic light shows green.

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And if the traffic light malfunctions and shows both lights at the same time, we would still see from the Fourier transform At some time the traffic light shows red. This can be taken to mean that sometime around 1 o'clock sharp could be exactly 1 o'clock, could be 1 minute past, could be 30 second before the light showed a color that is more or less red could be a little bit purple, or maybe a little bit amber.

This can be taken to mean that at all the times around 1 o'clock say plus or minus 2 minutes the traffic light does not show any hint of green. This would indicate that around maybe , or the light shined greenish could have a tinge of teal or a bit of yellow in it. Willie Wong Willie Wong He 'just' put a practical i. Physics limit on the value in the real world. Wavelets, in a sense, work because they are on an exponential scale, rather than a linear scale, allowing their width-localisation product to be kept bounded the uncertainty principle.

That's why we can produce spectrum intensity vs frequency vs time using FFT. So I don't quite see the difference. Besides, the quote in the original question asked specifically about the "standard Fourier transform". As the first sentence in the wikipedia page states: A wavelet is a wave-like oscillation with an amplitude that starts out at zero 0 , increases, and then decreases back to zero.

Splinter 3 1 1 bronze badge. So Wavelets do not require interval to periodic is the difference? I learned more about wavelets from this book than from any other source. I should warn you, though, they're pretty fast and loose with the hypotheses of their theorems.

## Fourier analysis - Wikipedia

You'll be fine if you've studied advanced linear algebra, and especially fine if you already know some Fourier analysis. I haven't read very much of it, so I don't have a strong opinion on it yet.

I haven't read it myself, though. It seems to be out of print, unfortunately. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Introduction to wavelets? Ask Question.

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Asked 9 years, 10 months ago. Active 1 year, 5 months ago. Viewed 3k times. I have done 2 courses which covered parts of this book!.. Harald Hanche-Olsen.

Loop Space. Seriously; no, I don't find you annoying for saying so.

## Fourier analysis

I forgot that I have such SuperPowers. Alasdair McAndrew. Darsh Ranjan. Kim Greene. Mikhail Glushenkov.