In mathematics, Fourier analysis (/ˈfʊrieɪ,
Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then resynthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis often refers to the study of both operations.
The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis (see list of Fourierrelated transforms) has a corresponding inverse transform that can be used for synthesis.
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✪ But what is the Fourier Transform? A visual introduction.

✪ Fourier Series introduction

✪ Lecture 1  The Fourier Transforms and its Applications

✪ Fourier Series Part 1

✪ Fourier Series
Transcription
This right here is what we're going to build to, this video: A certain animated approach to thinking about a superimportant idea from math: The Fourier transform. For anyone unfamiliar with what that is, my #1 goal here is just for the video to be an introduction to that topic. But even for those of you who are already familiar with it, I still think that there's something fun and enriching about seeing what all of its components actually look like. The central example, to start, is gonna be the classic one: Decomposing frequencies from sound. But after that, I also really wanna show a glimpse of how this idea extends well beyond sound and frequency, and to many seemingly disparate areas of math, and even physics. Really, it is crazy just how ubiquitous this idea is. Let's dive in. This sound right here is a pure A. 440 beats per second. Meaning, if you were to measure the air pressure right next to your headphones, or your speaker, as a function of time, it would oscillate up and down around its usual equilibrium, in this wave. making 440 oscillations each second. A lowerpitched note, like a D, has the same structure, just fewer beats per second. And when both of them are played at once, what do you think the resulting pressure vs. time graph looks like? Well, at any point in time, this pressure difference is gonna be the sum of what it would be for each of those notes individually. Which, let's face it, is kind of a complicated thing to think about. At some points, the peaks match up with each other, resulting in a really high pressure. At other points, they tend to cancel out. And all in all, what you get is a waveish pressure vs. time graph, that is not a pure sine wave; it's something more complicated. And as you add in other notes, the wave gets more and more complicated. But right now, all it is is a combination of four pure frequencies. So it seems...needlessly complicated, given the low amount of information put into it. A microphone recording any sound just picks up on the air pressure at many different points in time. It only "sees" the final sum. So our central question is gonna be how you can take a signal like this, and decompose it into the pure frequencies that make it up. Pretty interesting, right? Adding up those signals really mixes them all together. So pulling them back apart...feels akin to unmixing multiple paint colors that have all been stirred up together. The general strategy is gonna be to build for ourselves a mathematical machine that treats signals with a given frequency... ..differently from how it treats other signals. To start, consider simply taking a pure signal say, with a lowly three beats per second, so that we can plot it easily. And let's limit ourselves to looking at a finite portion of this graph. In this case, the portion between zero seconds, and 4.5 seconds. The key idea, is gonna be to take this graph, and sort of wrap it up around a circle. Concretely, here's what I mean by that. Imagine a little rotating vector where each point in time its length is equal to the height of our graph for that time. So, high points of the graph correspond to a greater disance from the origin, and low points end up closer to the origin. And right now, I'm drawing it in such a way that moving forward two seconds in time corresponds to a single rotation around the circle. Our little vector drawing this wound up graph is rotating at half a cycle per second. So, this is important. There are two different frequencies at play here: There's the frequency of our signal, which goes up and down, three times per second. And then, separately, there's the frequency with which we're wrapping the graph around the circle. Which, at the moment, is half of a rotation per second. But we can adjust that second frequency however we want. Maybe we want to wrap it around faster... ..or maybe we go and wrap it around slower. And that choice of winding frequency determines what the wound up graph looks like. Some of the diagrams that come out of this can be pretty complicated; although, they are very pretty. But it's important to keep in mind that all that's happening here is that we're wrapping the signal around a circle. The vertical lines that I'm drawing up top, by the way, are just a way to keep track of the distance on the original graph that corresponds to a full rotation around the circle. So, lines spaced out by 1.5 seconds would mean it takes 1.5 seconds to make one full revolution. And at this point, we might have some sort of vague sense that something special will happen when the winding frequency matches the frequency of our signal: three beats per second. All the high points on the graph happen on the right side of the circle And all of the low points happen on the left. But how precisely can we take advantage of that in our attempt to build a frequencyunmixing machine? Well, imagine this graph is having some kind of mass to it, like a metal wire. This little dot is going to represent the center of mass of that wire. As we change the frequency, and the graph winds up differently, that center of mass kind of wobbles around a bit. And for most of the winding frequencies, the peaks and valleys are all spaced out around the circle in such a way that the center of mass stays pretty close to the origin. But! When the winding frequency is the same as the frequency of our signal, in this case, three cycles per second, all of the peaks are on the right, and all of the valleys are on the left.. ..so the center of mass is unusually far to the right. Here, to capture this, let's draw some kind of plot that keeps track of where that center of mass is for each winding frequency. Of course, the center of mass is a twodimensional thing, and requires two coordinates to fully keep track of, but for the moment, let's only keep track of the x coordinate. So, for a frequency of 0, when everything is bunched up on the right, this x coordinate is relatively high. And then, as you increase that winding frequency, and the graph balances out around the circle, the x coordinate of that center of mass goes closer to 0, and it just kind of wobbles around a bit. But then, at three beats per second, there's a spike as everything lines up to the right. This right here is the central construct, so let's sum up what we have so far: We have that original intensity vs. time graph, and then we have the wound up version of that in some twodimensional plane, and then, as a third thing, we have a plot for how the winding frequency influences the center of mass of that graph. And by the way, let's look back at those really low frequencies near 0. This big spike around 0 in our new frequency plot just corresponds to the fact that the whole cosine wave is shifted up. If I had chosen a signal oscillates around 0, dipping into negative values, then, as we play around with various winding frequences, this plot of the winding frequencies vs. center of mass would only have a spike at the value of three. But, negative values are a little bit weird and messy to think about especially for a first example, so let's just continue thinking in terms of the shiftedup graph. I just want you to understand that that spike around 0 only corresponds to the shift. Our main focus, as far as frequency decomposition is concerned, is that bump at three. This whole plot is what I'll call the "Almost Fourier Transform" of the original signal. There's a couple small distinctions between this and the actual Fourier transform, which I'll get to in a couple minutes, but already, you might be able to see how this machine lets us pick out the frequency of a signal. Just to play around with it a little bit more, take a different pure signal, let's say with a lower frequency of two beats per second, and do the same thing. Wind it around a circle, imagine different potential winding frequencies, and as you do that keep track of where the center of mass of that graph is, and then plot the x coordinate of that center of mass as you adjust the winding frequency. Just like before, we get a spike when the winding frequency is the same as the signal frequency, which in this case, is when it equals two cycles per second. But the real key point, the thing that makes this machine so delightful, is how it enables us to take a signal consisting of multiple frequencies, and pick out what they are. Imagine taking the two signals we just looked at: The wave with three beats per second, and the wave with two beats per second, and add them up. Like I said earlier, what you get is no longer a nice, pure cosine wave; it's something a little more complicated. But imagine throwing this into our windingfrequency machine... ..it is certainly the case that as you wrap this thing around, it looks a lot more complicated; you have this chaos (1) and chaos (2) and chaos (3) and chaos (4) and then WOOP! Things seem to line up really nicely at two cycles per second, and as you continue on it's more chaos (5) and more chaos (6) more chaos (7) chaos (8), chaos (9), chaos (10), WOOP! Things nicely align again at three cycles per second. And, like I said before, the wound up graph can look kind of busy and complicated, but all it is is the relatively simple idea of wrapping the graph around a circle. It's just a more complicated graph, and a pretty quick winding frequency. Now what's going on here with the two different spikes, is that if you were to take two signals, and then apply this AlmostFourier transform to each of them individually, and then add up the results, what you get is the same as if you first added up the signals, and then applied this AlmostFourier transorm. And the attentive viewers among you might wanna pause and ponder, and... ..convince yourself that what I just said is actually true. It's a pretty good test to verify for yourself that it's clear what exactly is being measured inside this winding machine. Now this property makes things really useful to us, because the transform of a pure frequency is close to 0 everywhere except for a spike around that frequency. So when you add together two pure frequencies, the transform graph just has these little peaks above the frequencies that went into it. So this little mathematical machine does exactly what we wanted. It pulls out the original frequencies from their jumbled up sums, unmixing the mixed bucket of paint. And before continuing into the full math that describes this operation, let's just get a quick glimpse of one context where this thing is useful: Sound editing. Let's say that you have some recording, and it's got an annoying high pitch that you'd like to filter out. Well, at first, your signal is coming in as a function of various intensities over time. Different voltages given to your speaker from one millisecond to the next. But we want to think of this in terms of frequencies, so, when you take the Fourier transform of that signal, the annoying high pitch is going to show up just as a spike at some high frequency. Filtering that out, by just smushing the spike down, what you'd be looking at is the Fourier transform of a sound that's just like your recording, only without that high frequency. Luckily, there's a notion of an inverse Fourier transform that tells you which signal would have produced this as its Fourier transform. I'll be talking about inverse much more fully in the next video, but long story short, applying the Fourier transform to the Fourier transform gives you back something close to the original function. Mm, kind of... this is... ..a little bit of a lie, but it's in the direction of the truth. And most of the reason that it's a lie is that I still have yet to tell you what the actual Fourier Transform is, since it's a little more complex than this xcoordinateofthecenterofmass idea. First off, bringing back this wound up graph, and looking at its center of mass, the x coordinate is really only half the story, right? I mean, this thing is in two dimensions, it's got a y coordinate as well. And, as is typical in math, whenever you're dealing with something twodimensional, it's elegant to think of it as the complex plane, where this center of mass is gonna be a complex number, that has both a real and an imaginary part. And the reason for talking in terms of complex numbers, rather than just saying, "It has two coordinates," is that complex numbers lend themselves to really nice descriptions of things that have to do with winding, and rotation. For example: Euler's formula famously tells us that if you take e to some number times i, you're gonna land on the point that you get if you were to walk that number of units around a circle with radius 1, counterclockwise starting on the right. So, imagine you wanted to describe rotating at a rate of one cycle per second. One thing that you could do is take the expression "e^2π*i*t," where t is the amount of time that has passed. Since, for a circle with radius 1, 2π describes the full length of its circumference. And... this is a little bit dizzying to look at, so maybe you wanna describe a different frequency... ..something lower and more reasonable... ..and for that, you would just multiply that time t in the exponent by the frequency, f. For example, if f was one tenth, then this vector makes one full turn every ten seconds, since the time t has to increase all the way to ten before the full exponent looks like 2πi. I have another video giving some intuition on why this is the behavior of e^x for imaginary inputs, if you're curious 😉, but for right now, we're just gonna take it as a given. Now why am I telling you this you this, you might ask. Well, it gives us a really nice way to write down the idea of winding up the graph into a single, tight little formula. First off, the convention in the context of Fourier transforms is to think about rotating in the clockwise direction, so let's go ahead and throw a negative sign up into that exponent. Now, take some function describing a signal intensity vs. time, like this pure cosine wave we had before, and call it g(t). If you multiply this exponential expression times g(t), it means that the rotating complex number is getting scaled up and down according to the value of this function. So you can think of this little rotating vector with its changing length as drawing the wound up graph. So think about it, this is awesome. This really small expression is a superelegant way to encapsulate the whole idea of winding a graph around a circle with a variable frequency f. And remember, that thing we want to do with this wound up graph is to track its center of mass. So think about what formula is going to capture that. Well, to approximate it at least, you might sample a whole bunch of times from the original signal, see where those points end up on the wound up graph, and then just take an average. That is, add them all together, as complex numbers, and then divide by the number of points that you've sampled. This will become more accurate if you sample more points which are closer together. And in the limit, rather than looking at the sum of a whole bunch of points divided by the number of points, you take an integral of this function, divided by the size of the time interval that we're looking at. Now the idea of integrating a complexvalued function might seem weird, and to anyone who's shaky with calculus, maybe even intimidating, but the underlying meaning here really doesn't require any calculus knowledge. The whole expression is just the center of mass of the wound up graph. So... Great! Stepbystep, we have built up this kind of complicated, but, let's face it, surprisingly small expression for the whole winding machine idea that I talked about. And now, there is only one final distinction to point out between this and the actual, honesttogoodness Fourier transform. Namely, just don't divide out by the time interval. The Fourier transform is just the integral part of this. What that means is that instead of looking at the center of mass, you would scale it up by some amount. If the portion of the original graph you were using spanned three seconds, you would multiply the center of mass by three. If it was spanning six seconds, you would multiply the center of mass by six. Physically, this has the effect that when a certain frequency persists for a long time, then the magnitude of the Fourier transform at that frequency is scaled up more and more. For example, what we're looking at right here is how when you have a pure frequency of two beats per second, and you wind it around the graph at two cycles per second, the center of mass stays in the same spot, right? It's just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform, at that frequency. For other frequencies, though, even if you just increase it by a bit, this is cancelled out by the fact that for longer time intervals you're giving the wound up graph more of a chance to balance itself around the circle. That is...a lot of different moving parts, so let's step back and summarize what we have so far. The Fourier transform of an intensity vs. time function, like g(t), is a new function, which doesn't have time as an input, but instead takes in a frequency, what I've been calling "the winding frequency." In terms of notation, by the way, the common convention is to call this new function "ghat," with a little circumflex on top of it. Now the output of this function is a complex number, some point in the 2D plane, that corresponds to the strength of a given frequency in the original signal. The plot that I've been graphing for the Fourier transform, is just the real component of that output, the xcoordinate But you could also graph the imaginary component separately, if you wanted a fuller description. And all of this is being encapsulated inside that formula that we built up. And out of context, you can imagine how seeing this formula would seem sort of daunting. But if you understand how exponentials correspond to rotation... ..how multiplying that by the function g(t) means drawing a wound up version of the graph, and how an integral of a complexvalued function can be interpreted in terms of a centerofmass idea, you can see how this whole thing carries with it a very rich, intuitive meaning. And, by the way, one quick small note before we can call this wrapped up. Even though in practice, with things like sound editing, you'll be integrating over a finite time interval, the theory of Fourier transforms is often phrased where the bounds of this integral are ∞ and ∞. Concretely, what that means is that you consider this expression for all possible finite time intervals, and you just ask, "What is its limit as that time interval grows to ∞?" And...man, oh man, there is so much more to say! So much, I don't wanna call it done here. This transform extends to corners of math well beyond the idea of extracting frequencies from signal. So, the next video I put out is gonna go through a couple of these, and that's really where things start getting interesting. So, stay subscribed for when that comes out, or an alternate option is to just binge a couple 3blue1brown videos so that the YouTube recommender is more inclined to show you new things that come out... ..really, the choice is yours! And to close things off, I have something pretty fun: A mathematical puzzler from this video's sponsor, Jane Street, who's looking to recruit more technical talent. So, let's say that you have a closed, bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob. Now imagine taking every possible pair of points on that surface, and adding them up, doing a vector sum. Let's name this set of all possible sums D. Your task is to prove that D is also a convex set. So, Jane Street is a quantitative trading firm, and if you're the kind of person who enjoys math and solving puzzles like this, the team there really values intellectual curiosity. So, they might be interested in hiring you. And they're looking both for fulltime employees and interns. For my part, I can say that some people I've interacted with there just seem to love math, and sharing math, and when they're hiring they look less at a background in finance than they do at how you think, how you learn, and how you solve problems, hence the sponsorship of a 3blue1brown video. If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com/3b1b/
Contents
Applications
Fourier analysis has many scientific applications – in physics, partial differential equations, number theory, combinatorics, signal processing, digital image processing, probability theory, statistics, forensics, option pricing, cryptography, numerical analysis, acoustics, oceanography, sonar, optics, diffraction, geometry, protein structure analysis, and other areas.
This wide applicability stems from many useful properties of the transforms:
 The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality) (Rudin 1990).
 The transforms are usually invertible.
 The exponential functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones (Evans 1998). Therefore, the behavior of a linear timeinvariant system can be analyzed at each frequency independently.
 By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolutionbased operations such as polynomial multiplication and multiplying large numbers (Knuth 1997).
 The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using Fast Fourier Transform (FFT) algorithms. (Conte & de Boor 1980)
In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum. The FT method is used to decode the measured signals and record the wavelength data. And by using a computer, these Fourier calculations are rapidly carried out, so that in a matter of seconds, a computeroperated FTIR instrument can produce an infrared absorption pattern comparable to that of a prism instrument.^{[2]}
Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses a variant of the Fourier transformation (discrete cosine transform) of small square pieces of a digital image. The Fourier components of each square are rounded to lower arithmetic precision, and weak components are eliminated entirely, so that the remaining components can be stored very compactly. In image reconstruction, each image square is reassembled from the preserved approximate Fouriertransformed components, which are then inversetransformed to produce an approximation of the original image.
Applications in signal processing
When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate narrowband components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fouriertransforming a signal, manipulating the Fouriertransformed data in a simple way, and reversing the transformation.^{[3]}
Some examples include:
 Equalization of audio recordings with a series of bandpass filters;
 Digital radio reception without a superheterodyne circuit, as in a modern cell phone or radio scanner;
 Image processing to remove periodic or anisotropic artifacts such as jaggies from interlaced video, strip artifacts from strip aerial photography, or wave patterns from radio frequency interference in a digital camera;
 Cross correlation of similar images for coalignment;
 Xray crystallography to reconstruct a crystal structure from its diffraction pattern;
 Fourier transform ion cyclotron resonance mass spectrometry to determine the mass of ions from the frequency of cyclotron motion in a magnetic field;
 Many other forms of spectroscopy, including infrared and nuclear magnetic resonance spectroscopies;
 Generation of sound spectrograms used to analyze sounds;
 Passive sonar used to classify targets based on machinery noise.
Variants of Fourier analysis
(Continuous) Fourier transform
Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, and it produces a continuous function of frequency, known as a frequency distribution. One function is transformed into another, and the operation is reversible. When the domain of the input (initial) function is time (t), and the domain of the output (final) function is ordinary frequency, the transform of function s(t) at frequency f is given by the complex number:
Evaluating this quantity for all values of f produces the frequencydomain function. Then s(t) can be represented as a recombination of complex exponentials of all possible frequencies:
which is the inverse transform formula. The complex number, S( f ), conveys both amplitude and phase of frequency f.
See Fourier transform for much more information, including:
 conventions for amplitude normalization and frequency scaling/units
 transform properties
 tabulated transforms of specific functions
 an extension/generalization for functions of multiple dimensions, such as images.
Fourier series
The Fourier transform of a periodic function, s_{P}(t), with period P, becomes a Dirac comb function, modulated by a sequence of complex coefficients:
for all integer values of k, and where ∫_{P} is the integral over any interval of length P.
The inverse transform, known as Fourier series, is a representation of s_{P}(t) in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients:
When s_{P}(t), is expressed as a periodic summation of another function, s(t):
the coefficients are proportional to samples of S( f ) at discrete intervals of 1/P:
 ^{[note 1]}
A sufficient condition for recovering s(t) (and therefore S( f )) from just these samples (i.e. from the Fourier series) is that the nonzero portion of s(t) be confined to a known interval of duration P, which is the frequency domain dual of the Nyquist–Shannon sampling theorem.
See Fourier series for more information, including the historical development.
Discretetime Fourier transform (DTFT)
The DTFT is the mathematical dual of the timedomain Fourier series. Thus, a convergent periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:
which is known as the DTFT. Thus the DTFT of the s[n] sequence is also the Fourier transform of the modulated Dirac comb function.^{[note 2]}
The Fourier series coefficients (and inverse transform), are defined by:
Parameter T corresponds to the sampling interval, and this Fourier series can now be recognized as a form of the Poisson summation formula. Thus we have the important result that when a discrete data sequence, s[n], is proportional to samples of an underlying continuous function, s(t), one can observe a periodic summation of the continuous Fourier transform, S( f ). That is a cornerstone in the foundation of digital signal processing. Furthermore, under certain idealized conditions one can theoretically recover S( f ) and s(t) exactly. A sufficient condition for perfect recovery is that the nonzero portion of S( f ) be confined to a known frequency interval of width 1/T. When that interval is [−1/2T, 1/2T], the applicable reconstruction formula is the Whittaker–Shannon interpolation formula.
Another reason to be interested in S_{1/T}( f ) is that it often provides insight into the amount of aliasing caused by the sampling process.
Applications of the DTFT are not limited to sampled functions. See Discretetime Fourier transform for more information on this and other topics, including:
 normalized frequency units
 windowing (finitelength sequences)
 transform properties
 tabulated transforms of specific functions
Discrete Fourier transform (DFT)
Similar to a Fourier series, the DTFT of a periodic sequence, s_{N}[n], with period N, becomes a Dirac comb function, modulated by a sequence of complex coefficients (see DTFT/Periodic data):
 where ∑_{n} is the sum over any sequence of length N.
The S[k] sequence is what is customarily known as the DFT of {{maths_{N}. It is also Nperiodic, so it is never necessary to compute more than N coefficients. The inverse transform is given by:
 where ∑_{k} is the sum over any sequence of length N.
When s_{N}[n] is expressed as a periodic summation of another function:
 and ^{[note 3]}
the coefficients are proportional to samples of S_{1/T}( f ) at discrete intervals of 1/P = 1/NT:
 ^{[note 4]}
Conversely, when one wants to compute an arbitrary number (N) of discrete samples of one cycle of a continuous DTFT, S_{1/T}( f ), it can be done by computing the relatively simple DFT of s_{N}[n], as defined above. In most cases, N is chosen equal to the length of nonzero portion of s[n]. Increasing N, known as zeropadding or interpolation, results in more closely spaced samples of one cycle of S_{1/T}( f ). Decreasing N, causes overlap (adding) in the timedomain (analogous to aliasing), which corresponds to decimation in the frequency domain. (see Sampling the DTFT) In most cases of practical interest, the s[n] sequence represents a longer sequence that was truncated by the application of a finitelength window function or FIR filter array.
The DFT can be computed using a fast Fourier transform (FFT) algorithm, which makes it a practical and important transformation on computers.
See Discrete Fourier transform for much more information, including:
 transform properties
 applications
 tabulated transforms of specific functions
Summary
For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components (Fourier series), and the transforms diverge at those frequencies. One common practice (not discussed above) is to handle that divergence via Dirac delta and Dirac comb functions. But the same spectral information can be discerned from just one cycle of the periodic function, since all the other cycles are identical. Similarly, finiteduration functions can be represented as a Fourier series, with no actual loss of information except that the periodicity of the inverse transform is a mere artifact.
We also note that it is common in practice for the duration of s(•) to be limited to the period, P or N. But these formulas do not require that condition.
Continuous frequency  Discrete frequencies  

Transform  
Inverse 
Continuous frequency  Discrete frequencies  

Transform 
 
Inverse 


Symmetry properties
When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a onetoone mapping between the four components of a complex time function and the four components of its complex frequency transform:^{[4]}
From this, various relationships are apparent, for example:
 The transform of a realvalued function (s_{RE}+ s_{RO}) is the even symmetric function S_{RE}+ i S_{IO}. Conversely, an evensymmetric transform implies a realvalued timedomain.
 The transform of an imaginaryvalued function (i s_{IE}+ i s_{IO}) is the odd symmetric function S_{RO}+ i S_{IE}, and the converse is true.
 The transform of an evensymmetric function (s_{RE}+ i s_{IO}) is the realvalued function S_{RE}+ S_{RO}, and the converse is true.
 The transform of an oddsymmetric function (s_{RO}+ i s_{IE}) is the imaginaryvalued function i S_{IE}+ i S_{IO}, and the converse is true.
Fourier transforms on arbitrary locally compact abelian topological groups
The Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact Abelian topological groups, which are studied in harmonic analysis; there, the Fourier transform takes functions on a group to functions on the dual group. This treatment also allows a general formulation of the convolution theorem, which relates Fourier transforms and convolutions. See also the Pontryagin duality for the generalized underpinnings of the Fourier transform.
More specific, Fourier analysis can be done on cosets,^{[5]} even discrete cosets.
Time–frequency transforms
In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information.
As alternatives to the Fourier transform, in time–frequency analysis, one uses time–frequency transforms to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a tradeoff between these. These can be generalizations of the Fourier transform, such as the shorttime Fourier transform, the Gabor transform or fractional Fourier transform (FRFT), or can use different functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform.
History
A primitive form of harmonic series dates back to ancient Babylonian mathematics, where they were used to compute ephemerides (tables of astronomical positions).^{[6]}^{[7]}^{[8]}^{[9]}
The classical Greek concepts of deferent and epicycle in the Ptolemaic system of astronomy were related to Fourier series (see Deferent and epicycle: Mathematical formalism).
In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit,^{[10]} which has been described as the first formula for the DFT,^{[11]} and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string.^{[12]} Technically, Clairaut's work was a cosineonly series (a form of discrete cosine transform), while Lagrange's work was a sineonly series (a form of discrete sine transform); a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits.^{[13]} Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.^{[12]}
An early modern development toward Fourier analysis was the 1770 paper Réflexions sur la résolution algébrique des équations by Lagrange, which in the method of Lagrange resolvents used a complex Fourier decomposition to study the solution of a cubic:^{[14]} Lagrange transformed the roots x_{1}, x_{2}, x_{3} into the resolvents:
where ζ is a cubic root of unity, which is the DFT of order 3.
A number of authors, notably Jean le Rond d'Alembert, and Carl Friedrich Gauss used trigonometric series to study the heat equation,^{[15]} but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleur dans les corps solides by Joseph Fourier, whose crucial insight was to model all functions by trigonometric series, introducing the Fourier series.
Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory: Daniel Bernoulli and Leonhard Euler had introduced trigonometric representations of functions,^{[11]} and Lagrange had given the Fourier series solution to the wave equation,^{[11]} so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.^{[11]}
The subsequent development of the field is known as harmonic analysis, and is also an early instance of representation theory.
The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 by Carl Friedrich Gauss when interpolating measurements of the orbit of the asteroids Juno and Pallas, although that particular FFT algorithm is more often attributed to its modern rediscoverers Cooley and Tukey.^{[13]}^{[16]}
Interpretation in terms of time and frequency
In signal processing, the Fourier transform often takes a time series or a function of continuous time, and maps it into a frequency spectrum. That is, it takes a function from the time domain into the frequency domain; it is a decomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.
When the function f is a function of time and represents a physical signal, the transform has a standard interpretation as the frequency spectrum of the signal. The magnitude of the resulting complexvalued function F at frequency ω represents the amplitude of a frequency component whose initial phase is given by the phase of F.
Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze spatial frequencies, and indeed for nearly any function domain. This justifies their use in such diverse branches as image processing, heat conduction, and automatic control.
See also
 Generalized Fourier series
 Fourier–Bessel series
 Fourierrelated transforms
 Laplace transform (LT)
 Twosided Laplace transform
 Mellin transform
 Nonuniform discrete Fourier transform (NDFT)
 Quantum Fourier transform (QFT)
 Numbertheoretic transform
 Leastsquares spectral analysis
 Basis vectors
 Bispectrum
 Characteristic function (probability theory)
 Orthogonal functions
 Schwartz space
 Spectral density
 Spectral density estimation
 Spectral music
 Wavelet
Notes
 ^
 ^ We may also note that:
 ^ Note that this definition differs from the DTFT section by a factor of T.
 ^
References
 ^ "Fourier". Dictionary.com Unabridged. Random House.
 ^ Saferstein, Richard (2013). Criminalistics: An Introduction to Forensic Science.
 ^ Rabiner, Lawrence R.; Gold, Bernard (1975). Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ.
 ^ Proakis, John G.; Manolakis, Dimitri G. (1996), Digital Signal Processing: Principles, Algorithms and Applications (3 ed.), New Jersey: PrenticeHall International, p. 291, ISBN 9780133942897, sAcfAQAAIAAJ
 ^ Forrest, Brian. (1998). Fourier Analysis on Coset Spaces. Rocky Mountain Journal of Mathematics. 28. 10.1216/rmjm/1181071828.
 ^ Prestini, Elena (2004). The Evolution of Applied Harmonic Analysis: Models of the Real World. Birkhäuser. p. 62. ISBN 9780817641252.
 ^ Rota, GianCarlo; Palombi, Fabrizio (1997). Indiscrete Thoughts. Birkhäuser. p. 11. ISBN 9780817638665.
 ^ Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2nd ed.). Dover Publications. ISBN 9780486223322.
 ^ BrackBernsen, Lis; Brack, Matthias (2004). "Analyzing shell structure from Babylonian and modern times". International Journal of Modern Physics E. 13 (1): 247. arXiv:physics/0310126. Bibcode:2004IJMPE..13..247B. doi:10.1142/S0218301304002028.
 ^ Terras, Audrey (1999). Fourier Analysis on Finite Groups and Applications. Cambridge University Press. p. 30. ISBN 9780521457187.
 ^ ^{a} ^{b} ^{c} ^{d} Briggs, William L.; Henson, Van Emden (1995). The DFT: An Owner's Manual for the Discrete Fourier Transform. SIAM. p. 4. ISBN 9780898713428.
 ^ ^{a} ^{b} Briggs, William L.; Henson, Van Emden (1995). The DFT: An Owner's Manual for the Discrete Fourier Transform. SIAM. p. 2. ISBN 9780898713428.
 ^ ^{a} ^{b} Heideman, M. T.; Johnson, D. H.; Burrus, C. S. (1984). "Gauss and the history of the fast Fourier transform". IEEE ASSP Magazine. 1 (4): 14–21.
 ^ Knapp, Anthony W. (2006). Basic Algebra. Springer. p. 501. ISBN 9780817632489.
 ^ Narasimhan, T. N. (February 1999). "Fourier's heat conduction equation: History, influence, and connections". Reviews of Geophysics. 37 (1): 151–172. Bibcode:1999RvGeo..37..151N. CiteSeerX 10.1.1.455.4798. doi:10.1029/1998RG900006. ISSN 19449208. OCLC 5156426043.
 ^ Terras, Audrey (1999). Fourier Analysis on Finite Groups and Applications. Cambridge University Press. p. 31. ISBN 9780521457187.
Further reading
 Conte, S. D.; de Boor, Carl (1980). Elementary Numerical Analysis (Third ed.). New York: McGraw Hill, Inc. ISBN 9780070662285.
 Evans, L. (1998). Partial Differential Equations. American Mathematical Society. ISBN 9783540761242.
 Howell, Kenneth B. (2001). Principles of Fourier Analysis. CRC Press. ISBN 9780849382758.
 Kamen, E. W.; Heck, B. S. (2 March 2000). Fundamentals of Signals and Systems Using the Web and Matlab (2 ed.). PrentissHall. ISBN 9780130172938.
 Knuth, Donald E. (1997). The Art of Computer Programming Volume 2: Seminumerical Algorithms (3rd ed.). AddisonWesley Professional. Section 4.3.3.C: Discrete Fourier transforms, pg.305. ISBN 9780201896848.
 Müller, Meinard (2015). The Fourier Transform in a Nutshell (PDF). Springer. In Fundamentals of Music Processing, Section 2.1, p. 40–56. doi:10.1007/9783319219455. ISBN 9783319219448.
 Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations. Boca Raton: CRC Press. ISBN 9780849328763.
 Rudin, Walter (1990). Fourier Analysis on Groups. WileyInterscience. ISBN 9780471523642.
 Smith, Steven W. (1999). The Scientist and Engineer's Guide to Digital Signal Processing (Second ed.). San Diego: California Technical Publishing. ISBN 9780966017632.
 Stein, E. M.; Weiss, G. (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press. ISBN 9780691080789.
External links
 Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
 An Intuitive Explanation of Fourier Theory by Steven Lehar.
 Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 6 is on the 1 and 2D Fourier Transform. Lectures 7–15 make use of it., by Alan Peters
 Moriarty, Philip; Bowley, Roger (2009). "∑ Summation (and Fourier Analysis)". Sixty Symbols. Brady Haran for the University of Nottingham.