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What are some real world applications 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. – Namit Sinha Nov 25 '13 at 6:27
up vote 24 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
Can you elaborate on how Fourier series converge fast (when they converge)? – AreaMan Jun 1 '15 at 23:11
@AreaMan I was talking about the decrease in the magnitude of fourier coefficients. Basically, the smoother the function is, the faster the fourier coefficients will decrease in magnitude and hence we need fewer terms to approximate the original function well. – Fixed Point Jun 1 '15 at 23:18
Another visualisation: – Den Feb 1 at 9:37

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 equations by the method of separation 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. – Namit Sinha 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

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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Another variation of the Fourier Series to compare DNA sequences is A Novel Method for Comparative Analysis of DNA Sequences which used Ramanujan-Fourier series. The idea is the same as the Fourier series, but with a different orthogonal basis (Fourier has a basis of trig functions, R-F uses Ramanujan sums). Other orthogonal basis are Walsh–Hadamard functions, Legendre polynomials, Chebyshev polynomial, etc.

Regardless of the orthogonal basis used, one of the practical applications is signal / data analysis. The transformation / decomposition into a sum of coefficients times basis functions, allows you to do either or both of:

  • "See" through the noise and highlight any non-obvious periodicity or patterning within the data / signal.
  • "Major on the majors" by focusing on or preserving the most important components of the signal. The most important components are precisely those components with the largest coefficients.

The basis determines what is highlighted in the signal / data. A Fourier series decomposition highlights sinusoidal components, A Walsh–Hadamard decomposition highlights components that are periodic square waves, an R-F decomposition highlights behaviors which are similar to the distribution of primes among integers.

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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):

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fourier series is broadly used in telecommunications system, for modulation and demodulation of voice signals, also the input,output and calculation of pulse and their sine or cosine graph.

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yea this is a short and precise answer. – kingsley Nov 12 '14 at 15:07

Although I'm not sure how much this has been used recently: shape analysis of closed curves for character recognition.

On the other hand, spherical harmonics, which are a Fourier series on the sphere, have been and still are used extensively for shape analysis in medical imaging.

Driver skill classification using frequency of steering wheel motion as an input feature.

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Hiss and pop in sound recordings can be cleaned up using Fourier analysis. What is static but a super-high frequency sound--higher than most sounds that normally appear in music and speech, etc. When a time-domain signal is represented in the frequency domain, i.e., as a sum of sine waves, you can cure the static by simply erasing all the highest frequencies and then reconstituting the sound.

Exactly the same trick works in removing speckles from photographs. The boundaries between the photo and the speckles are the highest frequency components of the image. Using Fourier analysis to you can drop all the highest frequency components. Then reconstitute the picture, and like magic, speckles are gone. Some minor features you might want will be gone too, but in practice it works very well.

The opposite works for finding outlines in a picture. Erase everything but the high frequencies, and when you reconstitute the picture, you will have all the outlines and the broad areas will all be erased.

Numerous image processing techniques are variations on this theme.

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