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. K

    Discrete Fourier series derivation

    Hello,*please refer to the table above. I started from x(n)=x(n*Ts)=x(t)*delta(t-nTs), how can we have finite terms for discrete time F.S can anyone provide me a derivation or proof for Discrete F.S.?
  2. A

    MHB Fourier Transforms: Theory & Applications

  3. E

    Using the Fourier Transform on Partitioned Images

    If I cut my image into several portions and use the Fast Fourier Transform on each portioned image, will I achieve the same result as if I used Fast Fourier Transform on the whole image? I have this concern because I need to process a large image using the Fast Fourier Transform, the problem is...
  4. lucasLima

    Help with DC in fourier transform please

    Hello everyone, So, i have a big test tomorrow and my professor said i should study the DC level in Fourier transform , in the frequency domain. So, i did a little research and found out that the dc level is the percentage of the time a signal is active, and that's all. Can't see how that's...
  5. C

    Where can I start learning about Fourier Transforms/Series?

    I'm trying to learn about Fourier Transforms, specifically how they relate to equalizers, but I can't seem to find any academic guidance. I've asked my maths teacher for help, and I've looked through my school library, but I can't find a single source to start learning about Fourier Transforms...
  6. M

    Fourier series of periodic function

    Homework Statement Periodic function P=3 f(t) = 0 if 0<t<1 1 if 1<t<2 0 if 2<t<3 a) Draw the graph of the function in the interval of [-3,6] b) Calculate the Fourier series of f(x) by calculating the coefficient. Homework EquationsThe Attempt at a Solution a) in attached...
  7. Aristotle

    Can somebody check my work on this Fourier Series problem?

    Homework Statement Homework Equations The Attempt at a Solution Since P=2L, L=1 ? a_o = 1/2 [ ∫(from -1 to 0) -dx + ∫(from 0 to 1) dx ] = 1/2 [ (0-1) + (1-0) ] = 1/2(0) = 0 a_n = - ∫ (from -1 to 0) cosnπx dx + ∫ (from 0 to 1) cosnπx dx = 0 b_n = - ∫ (from -1 to 0) sinnπx dx...
  8. kostoglotov

    Fourier, square sign wave, f(x)sin(kx) integration

    I'm not sure whether to put this here or in Linear Algebra, if any Mod feels it should go in Linear Algebra I won't mind. I've just been introduced to Fourier Series decompositions in my Linear Algebra text, and I understand all the core concepts so far from the Linear Algebra side of it (a...
  9. kostoglotov

    Verifying the Fourier Series is in Hilbert Space

    The text does it thusly: imgur link: http://i.imgur.com/Xj2z1Cr.jpg But, before I got to here, I attempted it in a different way and want to know if it is still valid. Check that f^{*}f is finite, by checking that it converges. f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x +...
  10. B

    Dirac Delta Function - Fourier Series

    1. Homework Statement Find the Fourier series of ##f(x) = \delta (x) - \delta (x - \frac{1}{2})## , ## - \frac{1}{4} < x < \frac{3}{4}## periodic outside. Homework Equations [/B] ##\int dx \delta (x) f(x) = f(0)## ##\int dx \delta (x - x_0) f(x) = f(x_0)##The Attempt at a Solution...
  11. E

    Inverse Fourier transform of ## \frac{1}{a+jw} ##

    Fourier transform is defined as $$F(jw)=\int_{-\infty}^{\infty}f(t)e^{-jwt}dt.$$ Inverse Fourier transform is defined as $$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(jw)e^{jwt}dw.$$ Let ##f(t)=e^{-at}h(t),a>0##, where ##h(t)## is heaviside function and ##a## is real constant. Fourier...
  12. T

    Electrodynamics Fourier Analysis (Fouriers Trick)

    Homework Statement Two infinitely grounded metal plates at y=0 and y=a are connected at x=b and x=-b by metal strips maintained at a constant potential V. Find the potential inside the rectangular pipe.Homework Equations Laplaces EquationThe Attempt at a Solution I posted a photo of what I've...
  13. O

    Continuous Time Fourier Series of cosine equation

    Homework Statement Using the CTFS table of transforms and the CTFS properties, find the CTFS harmonic function of the signal 2*cos(100*pi(t - 0.005)) T = 1/50 Homework Equations To = fundamental period T = mTo cos(2*pi*k/To) ----F.S./mTo---- (1/2)(delta[k-m] + delta[k+m]) The Attempt at...
  14. H

    Integral arising from the inverse Fourier Transform

    Homework Statement [/B] I was using the Fourier transform to solve the following IVP: \frac{\partial^2 u}{\partial t \partial x} = \frac{\partial^3u}{\partial x^3} \\ u(x,0)=e^{-|x|} Homework Equations [/B] f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{f}(\omega)e^{i\omega...
  15. D

    Find Fourier Series of g(t): Simplification & Formula Analysis

    1. Find the Fourier series of : $$g(t)=\frac{t+4}{(t^2+8t+25)^2}$$ 2. I have been trying to write the function to match the formula $$\mathcal{F} [\frac{1}{1+t^2}] = \pi e^{-\mid(\omega)\mid}$$ 3. I have simplified the function to $$(t+4)(\frac{1}{9}(\frac{1}{1+\frac{(t+4)^2}{9}})^2)$$...
  16. W

    C_0 coefficient of Complex Fourier transforms

    Mod note: Moved from technical math section, so no template was used. Hey! So the complex Fourier transform of the square wave $$ f(x) = \begin{cases} 2 & x \in [0,2] \\ -1 & x \in [2,3] \\ \end{cases}, \space \space f(x+3) = f(x)$$ is ##C_k = \frac{3j}{2 \pi k}( e^{-j \frac{4 \pi k}{3}}...
  17. A

    Relationship between Fourier transform and Fourier series?

    What is the relationship between the Fourier transform of a periodic function and the coefficients of its Fourier series? I was thinking Fourier series a special version of Fourier transform, as in it can only be used for periodic function and only produces discrete waves. By this logic, aren't...
  18. M

    MHB Calculating Fourier Co-Efficent An of an Even Square Function

    I've been trying to answer this question for several days now with no results. Here is the question Imgur: The most awesome images on the Internet Now, I know the answer is -4/npi, but after integrating the function piece-wise (broke it into 3 separate integrals) I got 4sin(npi/2)/npi...
  19. grandpa2390

    Fourier Transform deduce the following transform pair

    Homework Statement I'm supposed to be using the similarity theorem and the shift theorem to solve: cos(πx) / π(x-.5) has transform e^(-iπs)*Π(s) Homework Equations similarity theorem f(ax) has transform (1/a)F(s/a) shift theorem f(x-a) has transform e^(-i2πas)F(s) The Attempt at a Solution...
  20. Amith2006

    Fourier transform of vector potential

    Homework Statement I have question on doing the following indefinite integral: $$\int{d^3x(\nabla^2A^{\mu}(x))e^{iq.x}}$$ Homework Equations This is part of derivation for calculating the Rutherford scattering cross section from Quarks and Leptons by Halzen and Martin. This books gives the...
  21. M

    Bloch Function Recursion Relation of Fourier Components

    Homework Statement This is just a problem to help me understand. Determine the dispersion relations for the three lowest electron bands for a 1-D potential of the form ##U(x) = 2A\cos(\frac{2\pi}{a} x)## Homework Equations I will notate ##G, \,G'## as reciprocal lattice vectors. $$\psi_{nk}(x)...
  22. F

    MRI and Fourier transform to form an image

    I read about how MRI works briefly, by flipping the water molecules using a magnetic field to the correct state then send the radio wave to these atoms and have it bounces back to be received by receiver coils and apply Fourier Transform to figure out the imaging. My question is, how does...
  23. L

    Fourier transform of function which has only radial dependence

    3d Fourier transform of function which has only radial dependence ##f(r)##. Many authors in that case define \vec{k} \cdot \vec{r}=|\vec{k}||\vec{r}|\cos\theta where ##\theta## is angle in spherical polar coordinates. So \frac{1}{(2\pi)^3}\int\int_{V}\int e^{-i \vec{k} \cdot...
  24. H

    Can I use the Fourier Transform to analyze the Sun's Spectrum?

    I'm learning digital signal processing in my engineer class, but I'm more interested in apply these things into Astrophysics, so i know a little bit about for what is useful the Fourier Transform, so i thought why not use this in Analyzing the sun spectra! But what do you think!? Is it useful...
  25. A

    Finding a Fourier representation of a signal

    Given the following signal, find the Fourier representation, ##V(jf)= \mathfrak{F}\left \{ v(t) \right \}##: ## v(t)=\left\{\begin{matrix} A, & 0\leqslant t\leqslant \frac{T}{3}\\ 2A, & \frac{T}{3}\leqslant t\leqslant T\\ 0, & Else \end{matrix}\right. ## Then sketch ##V(jf)##. Homework...
  26. C

    Complex Degree of Coherence (Cittert-Zernike)

    Homework Statement A light source consists of two long thin parallel wires, separated by a distance, W. A current is passed through the wires so that they emit light thermally. A filter is placed in front of the wires to only allow a narrow spectral range, centred at λ to propagate to a...
  27. I

    Convolution (Possibly using Fourier transform)

    Homework Statement Find a function ##u## such that ##\int_{-\infty}^\infty u(x-y)e^{-|y|}dy=e^{-x^4}##. Homework Equations Not really sure how to approach this but here's a few of the formulas I tried to use. Fourier transform of convolution ##\mathscr{F} (f*g)(x) \to \hat f(\xi ) \hat g(\xi...
  28. R

    Exploring Fourier Series: An=An*sin() & bn=An*cos()

    Consider the following article: https://en.wikipedia.org/wiki/Fourier_series At definition, they say that an = An*sin() and bn = An*cos() So with these notations you can go from a sum having sin and cos to a sum having only sin but with initial phases. Why can I write an = An*sin() and bn =...
  29. R

    Fourier Transform: Nonperiodic vs Periodic Signals

    In a book the Fourier transform is defined like this. Let g(t) be a nonperiodic deterministic signal... and then the integrals are presented. So, I understand that the signal must be deterministic and not random. But why it has to be nonperiodic (aperiodic). The sin function is periodic and we...
  30. LunaFly

    Why is Fourier Transform of a Real Function Complex?

    Homework Statement Find the Fourier transform F(w) of the function f(x) = [e-2x (x>0), 0 (x ≤ 0)]. Plot approximate curves using CAS by replacing infinite limit with finite limit. Homework Equations F(w) = 1/√(2π)*∫ f(x)*e-iwxdx, with limits of integration (-∞,∞). The Attempt at a Solution I...
  31. I

    Calculate indefinite integral using Fourier transform

    Homework Statement Use the Fourier transform to compute \int_{-\infty}^\infty \frac{(x^2+2)^2}{(x^4+4)^2}dx Homework Equations The Plancherel Theorem ##||f||^2=\frac{1}{2\pi}||\hat f ||^2## for all ##f \in L^2##. We also have a table with the Fourier transform of some function, the ones of...
  32. ognik

    Fourier Transforms, Green's function, Helmholtz

    Homework Statement I've gotten myself mixed up here , appreciate some insights ... Using Fourier Transforms, shows that Greens function satisfying the nonhomogeneous Helmholtz eqtn $$ \left(\nabla ^2 +k_0^2 \right) G(\vec{r_1},\vec{r_2})= -\delta (\vec{r_1} -\vec{r_2}) \:is\...
  33. ognik

    MHB Greens Function for Hemmholtz using Fourier

    I've gotten myself mixed up here , appreciate some insights ... Using Fourier Transforms, shows that Greens function satisfying the nonhomogeneous Helmholtz eqtn $ \left(\nabla ^2 +k_0^2 \right) G(\vec{r_1},\vec{r_2})= -\delta (\vec{r_1} -\vec{r_2}) $ is $ G(\vec{r_1},\vec{r_2})=...
  34. B

    Optical Fourier Transform for Propagation

    Homework Statement The complex amplitudes of a monochromatic wave of wavelength ##\lambda## in the z=0 and z=d planes are f(x,y) and g(x,y), redprctively. Assume ##d=10^4 \lambda##, use harmonic analysis to determine g(x,y) in the following cases: (a) f(x,y)=1 ... (d) ##f(x,y)=cos^2(\pi y / 2...
  35. E

    Fourier Transform and Convolution

    Considering two functions of ##t##, ##f\left(t\right) = e^{3t}## and ##g\left(t\right) = e^{7t}##, which are to be convolved analytically will result to ##f\left(t\right) \ast g\left(t\right) = \frac{1}{4}\left(e^{7t} - e^{3t}\right)##. According to a Convolution Theorem, the convolution of two...
  36. N

    Bounds of a Fourier Transform

    If I have a wave function given to me in momentum space, bounded by constants, and I have to find the wave function in position space, when taking the Fourier transform, what will be my bounds in position space?
  37. ognik

    MHB Help with Fourier integral using contour

    Find $ F=\frac{\hbar}{2\pi i} \int_{-\infty}^{\infty} \frac{e^{-i \omega t}}{E_0-\frac{i\Gamma}{2} -\hbar \omega} \,d\omega $ using contour integration. I have a couple questions I'd like some help with please... Taking out the $\hbar$ in the denominator, which cancels with the $\hbar$ outside...
  38. ognik

    MHB How do we find A0 in Fourier series for f(x)=x?

    My book says the expansion of $f(x)=x, -\pi \lt x \lt \pi = \sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{n}$, I get double that so please tell me where this is wrong: f(x) is odd, so $a_n=0$ $ b_n=\frac{1}{\pi} \int_{-\pi}^{\pi}x Sin(nx) \,dx = \frac{1}{\pi} [\frac{1}{n^2}Sin(nx) - \frac{x}{n}...
  39. I

    Convolutions, Fourier coefficients

    Homework Statement When ##f## and ##g## are ##2\pi##-periodic Riemann integrable functions define their convolution by ##(f*g)(x) = \frac{1}{2\pi} \int_0^{2\pi} f(y)g(x-y)dy## Denoting Fourier coefficients by ##c_n(f)## show that ##c_n(f * g) = c_n(f)c_n(g)##. Homework Equations ##c_n =...
  40. davidbenari

    Is a function really equal to its Fourier series?

    Suppose all Dirichlet conditions are met and we have a function that has jump discontinuities. Dirichlet's theorem says that the series converges to the midpoint of the values at the jump discontinuity. What bothers me then is: Dirichlet's theorem is basically telling us the series isn't the...
  41. B

    Can You Confirm My Fourier Series Calculation for a Square Wave?

    Hello, I think that I have done this correctly, but this is the first problem I have done on my own and would appreciate confirmation. 1. Homework Statement Find the Fourier series corresponding to the following functions that are periodic over the interval (−π, π) with: (a) f(x) = 1 for...
  42. B

    Fourier Transform of a sin(2pi*x)

    I have been very briefly introduced to Fourier transformations but the topic was not explained especially well (or I just didn't understand it!) We were shown the graphs with equations below and then their Fourier transformation (RHS). I understand the one for cos(2pist) but NOT the sin(2pist)...
  43. I

    How Can Fourier Series Aid in Solving the Sinc Function Integral?

    Homework Statement Compute ##\int_0^\infty \frac{\sin x}{x}dx## using that ##\frac{\sin x}{x} = \frac{b_0}{2} +\sum_1^\infty b_n \cos nx \; \; , \; \; 0 < x < \pi## with ##b_n = \frac{1}{\pi} \int_{(n-1)\pi}^{(n+1)\pi} \frac{\sin x}{x}dx##. Homework Equations Perhaps the following...
  44. G

    How Fourier components of vector potential becomes operators

    Hello. I'm studying quantization of electromagnetic field (to see photon!) and on the way to reach harmonic oscillator Hamiltonian as a final stage, sudden transition that the Fourier components of vector potential A become quantum operators is observed. (See...
  45. K

    Different forms of the discrete Fourier Transform

    Hi I am trying to program excel to take the DFT of a signal, then bring it back to the time domain after a low pass filter. I have a code that can handle simple data for example t = [ 0, 1, 2, 3] y = [2, 3, -1, 4] So I think everything is great and so I plug in my real signal and things go off...
  46. T

    Can fourier sine series approximate even functions?

    I am learning Fourier series and have come across the sine, cosine, and imaginary exponential expressions. To my knowledge, these individual terms form a basis since they are all orthogonal to each other. I am just wondering: can a Fourier sine series be used to model a purely even function...
  47. S

    Definition clarification for Fourier transform

    I have been studying Fourier transforms lately. Specifically, I have been studying the form of the formula that uses the square root of 2π in the definition. Now here is the problem: In some sources, I see the forward and inverse transforms defined as such: F(k) = [1/(√2π)] ∫∞-∞ f(x)eikx dx...
  48. S

    One question on the sampling theorem in Fourier transform

    Hello everyone, The question that I have may not be fully relevant to the title, but I thought that could be the best point to start the main question! I'm working on 2-D data which are images. For some reason, I have converted my data to a 1-D vector, and then transformed them to the...
  49. I

    Value of Fourier Series from Two Equations

    Homework Statement Derive ##\sum_{n=1}^\infty \frac{1}{n^2+b^2} = \frac{\pi}{2b}\coth b\pi - \frac{1}{2b^2}## from either ##e^{b\theta} = \frac{e^{2\pi b}-1}{2\pi} \sum_{-\infty}^\infty \frac{e^{in\theta}}{b-in}## for ##0 < \theta < 2\pi##. or ##e^{b\theta} = \frac{\sinh...
  50. J

    Quantum Books on waves with Fourier Transforms

    There are many waves and oscillations books out there that also include Fourier analysis but very few give the subject a thorough treatment, they just pass it in a few pages. If anybody has any sources(particularly books) that have Fourier analysis and particularly Fourier Transforms, I would...
Back
Top