What are some real world applications of Fourier series? Particularly the complex Fourier integrals?
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5advanced 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 06:27

2Besides the answers below I would add Fourier Transform infrared and FTRaman spectroscopy, nuclear magnetic resonance ( a basic tool in chemistry but more familiar in medicine via MRI imaging), and xray diffraction from crystals, the ultimate tool for determining molecular structure. – porphyrin May 19 '18 at 09:52

and also computed tomography or CT xray scans used in every hospital. – porphyrin Oct 09 '21 at 09:28
13 Answers
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
$$\textrm{voice} = \sin(x)+\frac{1}{10}\sin(2x)+\frac{1}{100}\sin(3x)+\cdots$$
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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8Most 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

2Can you elaborate on how Fourier series converge fast (when they converge)? – Elle Najt Jun 01 '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 01 '15 at 23:18

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1@Ooker, Well it happens at different levels. First, you need periodic functions with a finite period. Then you need piecewise continuous functions, functions with left and right limits existing at all points for the fourier expansion to even exist. $y=1/x$ on $[\pi,\pi]$ is one example for which the fourier expansion doesn't exist. Then if you restrict to point or jump discontinuities, a fourier expansion may exist but it won't converge to the original function at the point of discontinuity. – Fixed Point Sep 26 '17 at 21:21

@Ooker I am also sure that there are pathological examples for which fourier series exist but converge nowhere instead of just on a finite number of points. – Fixed Point Sep 26 '17 at 21:23

@LorenzoNajt the most important continuous convergence result appears to be on L2: https://en.wikipedia.org/w/index.php?title=Riesz%E2%80%93Fischer_theorem&oldid=903779889#Modern_forms_of_the_theorem where everything has a series. Speed is likely just a practical observation without proof and linked to resonance of physical systems. – Ciro Santilli Путлер Капут 六四事 Jan 11 '20 at 09:47

"when fourier series converge, they converge very fast." What?????????? Could you maybe say exactly what you mean by this? I can't come up with a precise version that's actually true... – David C. Ullrich Feb 17 '21 at 13:00
I can say about these applications.
Signal Processing. It may be the best application of Fourier analysis.
Approximation Theory. We use Fourier series to write a function as a trigonometric polynomial.
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.
Partial Differential equation. We use it to solve higher order partial differential equations by the method of separation of variables.

5can 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

1I shoul have added the FTIR 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). – TheVal Nov 24 '13 at 22:36

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 believe 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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@theonlygusti maybe seeing how audio synthesis is best done through frequencies rather than amplitudes will serve as a good argument: https://stackoverflow.com/questions/732699/howisaudiorepresentedwithnumbersincomputers/36510894#36510894 It all likely comes down to resonance in the ear/vocal chords. – Ciro Santilli Путлер Капут 六四事 Jan 11 '20 at 09:43
Here is another veryspecific 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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Another variation of the Fourier Series to compare DNA sequences is A Novel Method for Comparative Analysis of DNA Sequences which used RamanujanFourier series. The idea is the same as the Fourier series, but with a different orthogonal basis (Fourier has a basis of trig functions, RF 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 nonobvious 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 RF decomposition highlights behaviors which are similar to the distribution of primes among integers.
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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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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 superhigh frequency soundhigher than most sounds that normally appear in music and speech, etc. When a timedomain 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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My answer is not about real world application, but about a real mechanical machine that was capable of graphing Fourier Transforms. In both senses, analysing and synthetizing a signal.
Harmonic Analyzer Mechanical Fourier Computer  Hackaday
I was amazed to discover it, and it was exactly on 2017/14/3, which is the $\pi$ day!
The video playlist explains it detailed and there is even a book Albert Michelson’s Harmonic Analyzer for sale Albert Michelson's Harmonic Analyzer: A Visual Tour of a Nineteenth Century Machine that Performs Fourier Analysis about it.
I also discovered there is an instructables project on http://www.instructables.com/id/PlywoodMathMachine/ about building one
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Example of solving a partial differential equation with separation of variables and the Fourier series
This application was mentioned at https://math.stackexchange.com/a/579457/53203 but I'd like to give it a bit more emphasis and a high level motivation.
This application is important not only for practical reasons (PDEs are important for physics obviously), but it also has particular historical relevance as it is apparently what initially motivated Fourier, so we can imagine that it will be simple and intuitive.
I will be skipping over a few minor details, but the full derivation can be found here: https://en.wikipedia.org/wiki/Heat_equation#Solving_the_heat_equation_using_Fourier_series This technique is called separation of variables.
What we will find out is that solving the heat equation is equivalent to calculating the Fourier transform of the initial condition $F$.
Consider the heat equation for a one dimensional rod of length $L$:
$$ \frac{\partial f(t, x)}{\partial t} = \frac{\partial^2 f(t, x)}{\partial x^2} $$
with boundary conditions:
$$ f(t, 0) = 0 \\ f(t, L) = 0 $$
and known initial condition:
$$ f(0, x) = F(x) $$
The method of separation of variables starts with a guess that the solution has the form:
$$ f(t, x) = T(t)X(x) $$
thus "separating" $t$ and $x$.
Trying a solution of this form is just a guess, and initially we have no reason to be sure it will work out.
By plugging the guess that into the equations, and doing simple manipulations, we determine that $T$ and $X$ can be any solution of:
$$ T(t) =  \frac{n \pi}{L} \frac{d T(t)}{dt} \\ X(x) =  \frac{n \pi}{L} \frac{d^2 X(x)}{dx^2} $$
where $n$ is an integer $n > 0$. This means that picking $n = 1$ gives one possible $X$ and $T$ (we could call them $X_1$ and $T_1$), $n = 2$ give another one, and so one.
These are just simple ordinary differential equations that we know how to solve! Given the boundary conditions, the solutions are:
$$ T(t) = e^{\frac{n^2 \pi^2 t}{L^2}} \\ X(x) = \sin \left(\frac{n\pi x}{L}\right) $$
We could then plug those back in the equation to confirm that our separation of variables guess does give possible answers.
Because the equation is linear, multiplying a solution by a constant or adding two solutions together is another solution, so we reach a general solution form of:
$$ f(x,t) = \sum_{n = 1}^{\infty} D_n \sin \left(\frac{n\pi x}{L}\right) e^{\frac{n^2 \pi^2 \alpha t}{L^2}} $$
where we are free to chose the constants $D_n$. If each $D_n$ is chosen, we have a specific solution.
But remember that for $t = 0$, we must have $F(x) = f(0, x)$, and therefore we have to chose those $D_n$ such that:
$$ F(x) = \sum_{n = 1}^{\infty} D_n \sin \left(\frac{n\pi x}{L}\right) $$
which is basically the Fourier series decomposition of $F(x)$!
Therefore, all we need to do to solve the differential equation is calculate the weights $D_n$ of the Fourier series of the initial condition, and then just plug them into the general $f(x,t)$ formula and we are done!
Completeness
A big question that comes up then is that of completeness: can we approximate any initial condition $F$ well enough by just a bunch of sines? If we couldn't, then this approach wouldn't be very useful!
But luckily the answer is yes for a very wide class of functions that satisfies most of our needs.
I think in $L^p$ this is guaranteed by Carleson's theorem:
 "Every function can be represented as a Fourier series"?
 Proving that the Fourier Basis is complete for C(R/$2*\pi$ , C) with $L^2$ norm
Other PDEs
Previously, we discussed the heat equation. But separation of variables and the Fourier transform can be used solve other very important PDEs as well, which makes this method even more important. E.g.:
wave equation: see https://web.archive.org/web/20200621205928/https://courses.maths.ox.ac.uk/node/view_material/1720 section 4.4.1 "Application of Fourier series".
In that case, due to the second time derivative, we have two functions for the initial condition: position and speed of each point. And it turns out that for the solution we have to take the Fourier transform of each separately. Neat!
Schrodinger equation in cartesian coordinates: analogous to the heat equation, since we have a time derivative of order 1. For example, for a quantum free particle, the solution is a sum of complex exponentials.
2D Laplace equation: see https://web.archive.org/web/20200621205928/https://courses.maths.ox.ac.uk/node/view_material/1720 5.2.1 "Application of Fourier series"
3D Laplace equation and the Schrodinger equation of the hydrogen atom in polar coordinates: in this case, separation of variables can also be used (twice in a row!), see e.g.: https://en.wikipedia.org/wiki/Spherical_harmonics#Laplace's_spherical_harmonics but they serve as an example that "you don't always get a Fourier series" out. But you get something a bit analogous, with associated legendre polynomials rather than sines and cossines: https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
 when solving a 2D wave equation in spherical coordinates, we have another well known case where you don't get a Fourier series, but rather Bessel functions with an analogous motivation.
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We radio hams have developed new modes of digital communication that use discrete Fourier transforms to extract signals that are so deeply buried in noise that the human ear would not notice them. This allows communication with very low power (at reduced speed, of course). This is similar to what the SETI program does in its search for bug eyed monster civilizations in outer space. (So far they haven't admitted to finding anything.)
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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.

1Can you give some examples about learning the dynamics from the Fourier series? – Elle Najt Feb 02 '17 at 03:57