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what are some real world application of Fourier series ? particularly the complex Fourier integrals

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advanced noise cancellation and cell phone network technology uses Fourier series where digital filtering is used to minimize noise ans bandwidth demands respectively. –  Randomizer Nov 25 '13 at 6:27
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up vote 12 down vote accepted

It turns out that (almost) any kind of a wave can be written as a sum of sines and cosines. So for example, if I was to record your voice for one second saying something, I can find its fourier series which may look something like this for example

$$\textrm{voice} = \sin(x)+\frac{1}{10}\sin(2x)+\frac{1}{100}\sin(3x)+...$$

and this interactive module shows you how when you add sines and/or cosines the graph of cosines and sines becomes closer and closer to the original graph we are trying to approximate.

The really cool thing about fourier series is that first, almost any kind of a wave can be approximated. Second, when fourier series converge, they converge very fast.

So one of many many applications is compression. Everyone's favorite MP3 format uses this for audio compression. You take a sound, expand its fourier series. It'll most likely be an infinite series BUT it converges so fast that taking the first few terms is enough to reproduce the original sound. The rest of the terms can be ignored because they add so little that a human ear can likely tell no difference. So I just save the first few terms and then use them to reproduce the sound whenever I want to listen to it and it takes much less memory.

JPEG for pictures is the same idea.

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Most audio and image CODECs (including JPEG and mp3) actually use DCTs, which are a subset of generalized Fourier transforms. –  fluffy Nov 24 '13 at 22:59
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I can say about these applications.

  1. Signal Processing. it may be the best application of Fourier analysis.

  2. Approximation Theory. We use Fourier series to write a function as a trigonometric polynomial.

  3. Control Theory. The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of differential equation. They are useful to find out the dynamics of the solution.

  4. Partial Differential equation We use it to solve higher order partial differential equation by the method of sepertion of variables.

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can you please be more elaborate. I have gone through entire wiki page and find Fourier series quite complicated i understand what it does graphically but want to understand where i can try to apply these equations. –  Randomizer Nov 24 '13 at 17:27
    
I shoul have added the FT-IR Spectrometry field, in that case this tool is so essential you won't be able to obtain the specific diagrams (in such a way it is almost possible to "see" the transformation of the functions). –  Andrea L. Nov 24 '13 at 22:36
    
I do not know Chemistry. Thank you for this addition. –  Dutta Nov 25 '13 at 1:39
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For a very specific example: One of our undergraduate students was taking data generated by a person running on a force plate. Since force exerted on your feet from running is for the most part periodic, she fit the data with a curve using Fourier analysis. The work that followed can be used to help develop better running shoes.

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I belive Shazam identifies music by comparing the Fourier decomposition of recorded sound to a data resource of Fourier decompositions of know songs.

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Here is another very-specific example that I do not know much about.

One of the big problems in bioinformatics/computational biology is "lining up" DNA sequences to reveal mutations, additions, and deletions between them. This becomes an astronomical task when dealing with a large number of long sequences. To date, the fastest and most accurate program for this task is MAFFT (which stands for Multiple Alignment by Fast Fourier Transform): http://mafft.cbrc.jp/alignment/software

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