What is Fourier: Definition and 1000 Discussions

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.
The Fourier transform of a function of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation, as proven by the Fourier inversion theorem.

Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint. The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or Rn (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.

View More On Wikipedia.org
Recent contents Top users
  1. N

    Integrate sinx/x using fourier transform?

    I learned how to integrate it using the complex plane and semi circle contours but I was wondering if there is a way using Fourier transforms. I know that the Fourier transform of the rectangle wave form is the sinc function so I was thinking maybe i could do an inverse Fourier on sinc x and get...
  2. Y

    Fourier Transform, and the uncertainty principle

    Hello, Recently I've learned about Fourier Transform, and the uncertainty principle that is arose from it. According to Fourier Transform, if there is only one pulse in a signal, then it is composed from a lot more frequencies, compared to the number of frequencies that are building a...
  3. G

    Fourier series of complex function

    Homework Statement Hello guys, I have problem with the Fourier series, since we had only one lecture about it and I cannot find anything similar to my problem in internet. should we consider for the first f(x+1) integrated from -1 to 0 ? http://img819.imageshack.us/img819/3508/wbve.jpg when...
  4. U

    Complex Fourier series of sin (t)

    In finding Cn, I arrived at a different answer. I got an extra factor of (1/i) instead, which came when you do the integral of each exponential with respect to t; so you get a factor of 1/i(1-n) and 1/i(1+n) respectively.. Did they intentionally leave that out?
  5. U

    Fourier transform of RF signal with a prism ?

    Fourier transform of RF signal with a "prism"? We can use a prism to decompose visible light into components of different frequencies. This is a Fourier transform by nature. For an ideal prism, the energy is conserved in the process. How about RF signals? There is no fundamental difference...
  6. U

    Fourier Transform of this function

    Hi guys, I'm not sure how they got from first step to the second. Did they use integration by parts? I tried but I didn't arrive at the same result..
  7. L

    Trodimensional Fourier transform

    \mathcal{F}\{f(r)\}=\int e^{i\vec{k}\cdot \vec{r}}f(r)d\vec{r} in spherical polar coordinates \mathcal{F}\{f(r)\}=\int^{\infty}_0r^2dr\int^{\pi}_0\sin\theta d\theta\int^{\pi}_0d\varphi e^{ikr\cos \theta}f(r) Why could I take ##e^{ikr\cos \theta}## and to take that ##\theta## is angle which goes...
  8. M

    Fourier Transform using Transform Pair and Properties

    Homework Statement Finding the Fourier Transform using Transform Pair and Properties x(t) = 2[u(t+1)- t^{3}e^{6t}u(t)] Homework Equations The Attempt at a Solution For the first problem, I got u(t) \leftrightarrow ∏δ(ω)+\frac{1}{jω} F(at-t_{0}) \leftrightarrow...
  9. T

    Finding the Fourier Sine Series

    Find the Fourier SIne Series for f(x) = x on -L < x < L (Full Fourier) Ok, so my issue is in calculating the coefficients for the sine and cosine parts, more so an interpretation. So I have calulated the sine and cosine series to this point: let An: Cosine series Bn: sine series...
  10. M

    The windowed Fourier transform on R

    The windowed Fourier transform on R Defi nition-Proposition-Theorems (Plancherel formula-Parseval formula-inversion formula-Calderon's formula) http://www.4shared.com/office/b2Ho5n7H/The_windowed_Fourier_transform.html
  11. D

    How Do You Compute the Fourier Series for |sin(x)| Over the Interval (-1,1)?

    Hello, Find the Fourier serie of f(x)=|sin(x)| on the interval (-1,1) I'm just a little confused, does that mean that I have to integrate from -1 to 1 to find the coefficients ? Because the formula of the coefficients is in terms of the period T, for this function the period is pi. Or do I...
  12. U

    Continuous Periodic Fourier Series - Coefficients

    Homework Statement In the dirac notation, inner product of <f|g> is given by ∫f(x)*g(x) dx. Why is there a 1/∏ attached to each coefficient an, which is simply the inner product of f and that particular basis vector: <cn|f>? Homework Equations The Attempt at a Solution
  13. N

    What is the point of Fourier Series and what is it used for?

    I recently had an asignment where i calculated the Fourier series coefficients for f= 1+t for t= -1 to 0 f= 1-t for t=0-1 basically triangle looking. And as i summed more and more coefficients my function started looking more like this triangle (which was really cool). My question...
  14. T

    Fourier Series of Full Wave Rectifier

    Homework Statement Determine the Fourier series for the full-wave rectifier defined as f(t) = sinωt for 0 < ωt < pi -sinωt for -pi < ωt < 0Homework Equations The Attempt at a Solution This looks like an even function, so bm = 0 Ao = 1/pi∫sinωt from 0 to pi = 1/pi(-cos(ωt))/ω) from 0 to...
  15. D

    Fourier Series - Asymmetric Square Wave

    Good morning everyone, I am taking a signals and systems course where we are now studying the Fourier series. I understand that this is for signals that are periodic. But I get hung up when determining the Fourier coefficients. In the video by Alan Oppenheim, he derives the equation for...
  16. A

    Fourier component crystal potential - physical significance

    I have done several exercises concering periodic potentials in crystal. Especially I did one, where I had to show that the Fourier component of the shortest reciprocal lattice vector (call this vector a) in the z-direction was zero. Now solving the problem was just about writing up the right...
  17. P

    Proof of Fourier Series Coeffecients

    Homework Statement #35 on this page Homework Equations Integral of a series can be assumed to be the sum of integrals The Attempt at a Solution Picture of Work I am not sure where to proceed from here, advice?
  18. H

    Using Fourier Sine basis to write x(L-x) [0,L]

    Homework Statement A function F(x) = x(L-x) between zero and L. Use the basis of the preceding problem to write this vector in terms of its components: F(x)= \sum_{n=1}^{\infty}\alpha _{n}\vec{e_{n}} If you take the result of using this basis and write the resulting function outside the...
  19. B

    Fourier Transform - how to increment in time?

    I'm working on some research with a professor, and we're looking at data collected by an x-band radar array looking at ocean waves as they approach the coast (the radar is on land, and we can see about 3 miles out). What we're trying to do is perform an fft on the signal using Matlab, and...
  20. skate_nerd

    MHB Fourier transform of a gaussian

    I'm given a Gaussian function to apply a Fourier transform to. $$f(x)=\frac{1}{\sqrt{a\sqrt{\pi}}}e^{ik_ox}e^{-\frac{x^2}{2a^2}}$$ Not the most appetizing integral... $$g(k)=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{a\sqrt{\pi}}}\int_{-\infty}^{\infty}e^{ik_ox}e^{-\frac{x^2}{2a^2}}e^{-ikx}dx$$...
  21. J

    Fourier Mistake: Help Jenny Out

    can someone spot my mistake, I'm stuck thanks, jenny
  22. S

    Understanding Completeness of Fourier Basis

    So the other day in class my teacher gave a proof for the completeness of \phi_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx} in L^2([-\pi,\pi]) . And I'm trying to convince my self I understand it at least a little. He defined Frejer's Kernel K_n(x) = \frac{1}{2\pi(n+1)}...
  23. Radarithm

    QM: Need Help Applying Fourier Transform

    I understand the Fourier transform conceptually, but I am unable to reproduce it mathematically; I am very familiar with calculus and integration, but I am taking a QM course and I need to know how to apply it. No websites or videos are able to give me a good explanation as to how I can use it...
  24. S

    Why Doesn't the Fourier Series of a Dirac Comb Match Pointwise Values?

    http://en.wikipedia.org/wiki/Dirac_comb Please have a look at the Fourier Series section, and its last equation. Let T = 1. After expanding the Equation x(t) = 1 + 2cos(2∏t) + 2cos(4∏t) + 2cos(6∏t) ... Now this does not give the original Dirac Comb. Eg: at t = 1/2 x(1/2) = 0 But RHS =...
  25. H

    Solving Heat Equation by Fourier Transform

    When the rod is infinite or semi-infinite, I was taught to use Fourier transform. But I don't know when should the full Fourier transform or sine/cosine transform be used. how's the B.C. related to the choice of the transform ?
  26. D

    Fourier coefficients and partial sum of Fejer

    Homework Statement f(t) a continuously differentiable function twice over the circle T1 cr its Fourier coefficients and σn(f,t) partial sum of Fejer. a.Demonstrate that http://imageshack.us/a/img94/5992/ds35.png b. Consider k as -n≤k≤n , using cr coefficients calculate...
  27. N

    MHB Fourier Series (simple?)

    Is there some properties I should be aware of? after making the relevant substitutions, I ended up with $2 = 1 + \sum\nolimits_{m=0}^\infty \frac{4}{(2m+1)\pi}\sin(\frac{(2m+1)\pi}{2})$ but I can't get past this
  28. M

    Understanding Fourier series

    I've just started learning Fourier series and I'm having trouble understanding it. What do they actually do? And what does the amplitude-frequency show me? I'm asking as a rookie in signal analysis, so if you could explain it to me as simple as you can it will be of great help. Thanks!
  29. A

    Fourier Series and Absorption Questions?

    Sorry if I am posting in the wrong place. I'm really interested in the Fourier series, but I'm not an expert on it yet. I am very well aware yoy can do it with sound waves, but can you manipulate any other waves? What about light waves? And for absorption, how can you measure the...
  30. A

    Fourier Series and Absorbtion questions?

    I have to do a science fair, and I am really interested in physics. I spent forever trying to think of a topic, and I got myself stuck doing something with waves. After weeks of thinking of an expirement that will compete well without being extremely difficult, I came up with one. Does using...
  31. dexterdev

    A doubt related to infinitesimals in continuous fourier transform.

    Hi all, Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.) Now I have a doubt on CTFT (continuous time Fourier transform). This is a concept I got from my...
  32. L

    Fourier Transform of propagators

    I've been assigned the following homework: I have to compute the spectral density of a QFT and in order to do so I have to compute Fourier tranform of the following quantity (in Minkowsky signature, mostly minus) \rho\left(p\right) = \int \frac{1}{\left(-x^2 + i \epsilon...
  33. B

    Hi,a question regarding Fourier transform and energy of a signals

    Somehow I have really hard time wrapping my head around the concept.I mean,I get it,but I can't seem to solve any questions regarding it. Here are some examples ,and I just get stuck.Its a part of test,so I think it shouldn't be that hard to solve,and if it looks hard,I know there are some...
  34. F

    Fourier Analysis of Sawtooth Signal with To = 1

    Homework Statement Sawtooth signal with To = 1, at T=0, x = 0, at T=1, x =1 verify: a_{k} = \left\{\begin{matrix} \frac{1}{2}, for k=0; & \\\frac{j}{2\pi k}, for k \neq 0; & \end{matrix}\right. Homework Equations \frac{1}{T_{0}} \int_{0}^{T_{0}} te^{-j(2\pi/T_{0}))kt}dt The Attempt...
  35. B

    Saw-tooth Wave and Fourier Series amplitude of oscillation

    Homework Statement An oscillator with free period \tau is critically damped and subjected to a force with the saw-tooth form \F(t)=c(t-n\tau) for (n-0.5)\tau<t<(n+0.5)\tau for each integer n. Find the amplitudes a_n of oscillation at the angular frequencies 2\pi n/\tau if c is a...
  36. jbrussell93

    Calculating Fourier Coefficients

    I'm having trouble finding a definite answer to this question: When finding the Fourier series of a function is it always possible to find ##a_0## by first finding ##a_n## and just plugging in ##n=0##?
  37. E

    Justify an equality involving hyperbolic cosine and Fourier series

    Homework Statement The problem: Justify the following equalities: \cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2} I am trying to figure out how to start this. When I insert the Euler identity of \coth (using...
  38. B

    MATLAB Need help plotting Fourier series in matlab

    Hi all my first post as I need to seek help! I have just learned some simple Fourier series stuff and would like to be able to plot my answers in matlab. Assuming this is correct I was wondering if someone would be able to walk me through plotting this equation in Matlab...
  39. U

    Fourier Transform Scaling Property help

    Hi, I'm following the proof of the "Scaling Property of the Fourier Transform" from here: http://www.thefouriertransform.com/transform/properties.php ...but don't understand how they went from the integral to the right hand term here: The definition of the Fourier Trasform they...
  40. E

    Solving a simple second order PDE, do I need the Fourier?

    Solving a "simple" second order PDE, do I need the Fourier? Homework Statement The problem as given: y'' + 2y' + 5y = 10\cos t We want to find the general solution and the steady-state solution. We're using \mu y'' + c y' + k y = F(t) as our general form. OK, so I first want the general...
  41. D

    MHB Fourier Transform of a function squared.

    Consider \(u_t = -u_{nxxx} - 3(u^2)_{nx}\). The Fourier Transform is linear so taking the Inverse Fourier transform of the Fourier Transform on the RHS we have \begin{align} -\mathcal{F}^{-1}\left[\mathcal{F}\left[u_{nxxx} - 3(u^2)_{nx}\right]\right] &= -\mathcal{F}^{-1}...
  42. J

    MHB Find Fourier Sin Series for f(x)=1, 0 < x < c

    3. Fourier sin series for f(x) = 1, 0 < x < pi is given by 1 = 4/n E 1/ (2n-1) times sin (2n-1) x, (0 < x < n). Using this, find the Fourier sinc series for f(x)= 1, on 0 < x < c where c > 0. Then find the Fourier series for g(x), x > 0 where g(x) = 1, 0 < x < c, -1, c < x < 2c, g (x + 2c)...
  43. J

    MHB Fourier Cosine Series: Equivalence for {x}

    2. Fourier cosine series correspondence for f(x)= x, o < x < pi given by x ~ pi / 2 - 4/n, E infinity on top and n=1 on bottom. cos (an-1)/x / (2n-1)squared, (0 < x < pi). Explain why this correspondence is actually an equality for 0 is less than or equal to x and x is less than or equal to...
  44. D

    MHB Fast Fourier Transform and its inverse

    Does every FFT have \(i\) in it? Given \(u_t = -(u_{xxx} + 6uu_x)\). \(f'''(x) = \mathcal{F}^{-1}\left[(ik)^3\mathcal{F}(f(x))\right]\) \(f'(x) = \mathcal{F}^{-1}\left[(ik)\mathcal{F}(f(x))\right]\) The only equation I have used the pseudo-spectral method on was the NLS which is \(u_t =...
  45. D

    MHB Fourier series--showing converges to pi/16

    When a Fourier series contains only sine and cosine terms, evaluating the series isn't too difficult. However, I want to show a Fourier series with sine and sinh converges to \(\frac{\pi}{16}\). \[ T(50, 50) = \sum_{n = 1}^{\infty}...
  46. A

    What does the fourier transform do

    Today I found a program, which does Fourier transforms on pictures and tried it on some basic patterns. One of those was a lattice of dots and I have attached this and its Fourier transform to the thread. I would very much like if someone in basic details could explain what is going on. Why...
  47. A

    Finding the Fourier Series of a periodic rectangular wave

    Homework Statement The problem/question is attached in the file called "homework". In the third signal (the peridic rectangular wave), I am requested (sub-question b) to find the Fourier series of the wave. Homework Equations The file called "solution" presents a detailed solution to the...
  48. B

    Finite Fourier Transform on a 3d wave

    Finite Fourier Transform on a 2d wave How does the finite Fourier transform work exactly? The transform of f(x) is \widetilde{f}(\lambda_{n}) =\int^{L}_{0} f(x) X_{n} dx If I had a 3d wave equation pde and I applied Finite Fourier transform on the pde for z(x,y,t)=X(x)Y(y)T(t)...
  49. S

    What is Fourier Analysis and its Applications?

    Is this college or graduate math? Is it pure or applied math? Is it useful for physics and electrical engineering?
  50. E

    Odd & Even Functions (was thread Fourier Series )

    Odd & Even Functions (was thread "Fourier Series") Homework Statement determine if the functions below are odd even or neither: a) f(x)=x^2+2 b) f(x)=(x^2+2)tan(x^2) c) f(x) = (x^2+2)sin(x)tan(x^2)Homework Equations even - f(x) = f(-x) odd - f(-x)=-f(x) The Attempt at a Solution I've managed...
Back
Top