Fourier Definition and 1000 Threads

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.

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  1. binbagsss

    Fourier transform integration using well-known result

    Problem F denotes a forward Fourier transform, the variables I'm transforming between are x and k - See attachment Relevant equations So first of all I note I am given a result for a forward Fourier transform and need to use it for the inverse one. The result I am given to use, written out...
  2. redtree

    A Conjugate variables in the Fourier and Legendre transforms

    In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...
  3. LLT71

    I Can Fourier Analysis Represent Any Function Using Sin and Cos?

    has Fourier used sin(x) and cos(x) in his series because "there must be such interval [a,b] where integral of "some function"*sin(x) on that interval will be zero?" so based on that he concluded that any function can be represented by infinite sum of sin(x) and cos(x) cause they are "orthogonal"...
  4. J

    MHB Need help on Fourier Series (badly)

    Need help on Fourier series! Been stuck on this questions, it is too tough for me!
  5. M

    I Understanding Fourier Transforms

    Hello all, First time poster here so please excuse any mistakes as I'm unfamiliar with the conventions of this forum. Also before I get started I'd like to say I wasn't sure exactly where a question like this would go; I debated in the Math Programs and Latex section but figured general physics...
  6. L

    A I've tried to read the reason for using Fourier transform

    I've tried to read the reason for using Fourier transform in wave packets, I don't understand why. Please help me with this.
  7. D

    I Inverse Laplace to Fourier series

    I have the following laplace function F(s) = (A/(s + C)) * (1/s - exp(-sα)/s)/(1 - exp(-sT)) I think that the inverse laplace will be- f(t) = ((A/C)*u(t) - (A/C)*exp(-Ct)*u(t)) - ((A/C)*u(t-α) - (A/C)*exp(-C(t-α))*u(t-α)) and f(t+T)=f(t) Now I want to find the Fourier series expansion of f(t)...
  8. Captain1024

    Fourier Series Coefficients of an Even Square Wave

    Homework Statement Link: http://i.imgur.com/klFmtTH.png Homework Equations a_0=\frac{1}{T_0}\int ^{T_0}_{0}x(t)dt a_n=\frac{2}{T_0}\int ^{\frac{T_0}{2}}_{\frac{-T_0}{2}}x(t)cos(n\omega t)dt \omega =2\pi f=\frac{2\pi}{T_0} The Attempt at a Solution Firstly, x(t) is an even function because...
  9. MexChemE

    I Motivation for Fourier series/transform

    Hello, PF! I am currently learning Fourier series (and then we'll move on to the Fourier transform) in one of my courses, and I'm having a hard time finding motivation for its uses. Or, in other words, I can't seem to find its usefulness yet. I know one of its uses is to solve the heat...
  10. D

    Smearing an audio recording using Fourier transform

    Hi! I'd like to smear an audio recording, where the frequency content audibly changes, into an audio recording where it does not. Here's a recording of a sampled piano playing a melody, which will serve as an example: https://dl.dropboxusercontent.com/u/9355745/oldmcdonald.wav The frequency...
  11. J

    A Fourier transform of hyperbolic tangent

    Hello I am trying to determine the Fourier transform of the hyperbolic tangent function. I don't have a lot of experience with Fourier transforms and after searching for a bit I've come up empty handed on this specific issue. So what I want to calculate is: ##\int\limits_{-\infty}^\infty...
  12. Jezza

    Domain of a discrete fourier transform

    Homework Statement The (computing) task at hand is to take a function f(x) defined at 2N discrete points, and use the Discrete Fourier Transform (DFT) to produce F(u), a plot of the amplitudes of the frequencies required to produce f(x). I have an array for each function holding the value of...
  13. G

    I Understanding the Intuition Behind Fourier Series?

    I'm wondering if anyone could give me the intuition behind Fourier series. In class we have approximated functions over the interval ##[-\pi,\pi]## using either ##1, sin(nx), cos(nx)## or ##e^{inx}##. An example of an even function approximated could be: ## f(x) = \frac {(1,f(x))}{||1||^{2}}*1...
  14. Jezza

    How Does Slit Height Affect the Discrete Fourier Transform?

    Homework Statement [/B] This is a computing coursework problem. (There is a reasonably long theory preamble). Create a single slit centred on the origin (the centre of your array) width 10 and height 1. The array containing the imaginary parts will be zero and the array containing the real...
  15. Captain1024

    Fourier Transform in the Form of Dirac-Delta Function

    Homework Statement Given x(t)=8cos(70\pi t)+4sin(132\pi t)+8cos(24\pi t), find the Fourier transform X(f) in the form of \delta function. Homework Equations X(f)=\int ^{\infty}_{-\infty}x(t)e^{-j\omega _0t}dt cos(\omega t)=\frac{e^{j\omega t}+e^{-j\omega t}}{2} sin(\omega t)=\frac{e^{j\omega...
  16. Captain1024

    Evaluate the Fourier Transform of a Damped Sinusoidal Wave

    Homework Statement Evaluate the Fourier Transform of the damped sinusoidal wave g(t)=e^{-t}sin(2\pi f_ct)u(t) where u(t) is the unit step function. Homework Equations \omega =2\pi f G(f)=\int ^{\infty}_{-\infty} g(t)e^{-j2\pi ft}dt sin(\omega _ct)=\frac{e^{j\omega _ct}-e^{-j\omega _ct}}{2j}...
  17. J

    Reconstruction of the Fourier transform from its parts

    I am using ROOT to calculate the Fourier transform of a digital signal. I can extract the individual parts of the transform, the magnitude and phase in the form of a 1D histogram. I am attempting to reconstruct the transforms from the phase and magnitude but cannot seem to figure it out. Any...
  18. K

    I Fourier transform of a sum of shifted Gaussians

    My first thought was simply that the Fourier transform of a sum of Gaussians functions that are displaced from the origin by different amounts would just be another sum of Gaussians: F{G1(x) + G2(x)} = F{G1(x)} + F{G1(x)} where a generalized shifted Gaussian is: G(x) = G0exp[-(x - x0)2 / 2σ2]...
  19. K

    I Generalized version of the Fourier Transform

    Hello everyone, I was trying to develop a sort of generalized version of the Fourier Transform. My question in particular is: Given a function f(x,u), is there a function g(x,u) with \int_{-\infty}^\infty f(x,u)g(x,u')\mathrm{d}x=\delta(u-u') For f(x,u)=e^{2\pi ixu} the solution would be...
  20. FeDeX_LaTeX

    I Discrete Convolution of Continuous Fourier Coefficients

    Suppose that we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{R}, whose continuous Fourier coefficients \hat{f} are known. The convolution theorem tells us that: $$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$ where \ast denotes the...
  21. C

    Sum of sinosoids that can be a Fourier Series expansion

    Homework Statement I was given a problem with a list of sums of sinusoidal signals, such as Example that I made up: x(t)=cos(t)+5sin(5*t). The problem asks if a given expression could be a Fourier expansion. Homework Equations [/B]The Attempt at a Solution My guess is that it has something to...
  22. Joppy

    MHB Fourier Transform of Periodic Functions

    A tad embarrassed to ask, but I've been going in circles for a while! Maybe i'll rubber duck myself out of it. If f(t) = f(t+T) then we can find the Fourier transform of f(t) through a sequence of delta functions located at the harmonics of the fundamental frequency modulated by the Fourier...
  23. W

    MHB I am trying to figure out the right fast fourier transform size.

    I am using a Tascam recorder to record an environmental nuisance noise that is occurring in my home. I then use Virtins Multi Instrument Software, which includes an oscilloscope, band pass filter, and a spectrum analyser. Noise source is probably machinery at a legal marijuana grow op. That...
  24. entropy1

    B Time-evolving Fourier transform

    I am a little familiar with Fourier Analysis, but I don't know where to get tools to get the answer to this question: Consider a discrete signal A[0..N-1], consisting of N samples. Suppose we Fourier transform it and get a series of harmonics. Now, consider the discrete signal A[1..N], that is...
  25. dreens

    I Orthogonal 3D Basis Functions in Spherical Coordinates

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  26. T

    Calculating Coefficients of Fourier Series Homework

    Homework Statement I'm calculating the coefficients for the Fourier series and I got to part where I can't simplify an any further but I know I have to. a_n = \frac{1}{2π}\Big[\frac{cos(n-1)π}{n-1}-\frac{cos(n+1)π}{n+1}-\frac{1}{n-1}+\frac{1}{n+1}\Big]Homework EquationsThe Attempt at a...
  27. I

    I Fourier transform of Coulomb potential

    Dear all, In my quantum mechanics book it is stated that the Fourier transform of the Coulomb potential $$\frac{e^2}{4\pi\epsilon_0 r}$$ results in $$\frac{e^2}{\epsilon_0 q^2}$$ Where ##r## is the distance between the electrons and ##q## is the difference in wave vectors. What confuses me...
  28. P

    Why Does the Generalised Fourier Series Use a Weight Function in L2 Space?

    Homework Statement By applying the Gram–Schmidt procedure to the list of monomials 1, x, x2, ..., show that the first three elements of an orthonormal basis for the space L2 (−∞, ∞) with weight function ##w(x) = \frac{1}{\sqrt{\pi}} e^{-x^2} ## are ##e_0(x)=1## , ##e_1(x)= 2x## ,##e_2(x)=...
  29. C

    I Fourier Transform for Solving Parameter Perturbation Problem

    Suppose that a parameter y= 123. That parameter is somehow "perturbed" and its instantaneous value is: y(t)= 123 + sin(t - 50°) * 9 + sin(t * 3 + 10°) * 3 + sin(t * 20 + 60°) * 4 Suppose that I don't know the above formula, but I can calculate y(t) for any t. Hence I decide to use the...
  30. Houeto

    A Fourier Transform of a piecewise function

    Here is the Problem Statement : Find Fourier Transform of the piecewise function Can someone sheds some lights on how to start solving this? Thanks
  31. DoobleD

    B Why is momentum the fourier transform of the wavefunction ?

    I think this is probably a very basic question: why does the Fourier transform of a wavefunction describing position probabilities gives us a function describing momentum probabilities ? Is there a fairly simple explanation for this ? What leads us to this relation ?
  32. Vajhe

    Fourier transform of the Helmholtz equation

    Hi guys, I have been trying to solve the Helmholtz equation with no luck at all; I'm following the procedure found in "Engineering Optics with MATLAB" by Poon and Kim, it goes something like this: Homework Statement Homework Equations Let's start with Helmholtz eq. for the complex amplitude ##...
  33. L

    I Why is the Fourier transform of a sinusoid assumed as this?

    Hello everyone. I'm trying to better understand structured illumination microscopy and in the literature, I keep coming across bits of text like this. Source: http://www.optics.rochester.edu/workgroups/fienup/PUBLICATIONS/SAS_JOSAA09_PhShiftEstSupRes.pdf From Fourier analysis, if I take the...
  34. H

    Maple Maple question: defining functions as inverse Fourier transforms

    Hi, I have a a Fourier transformed variable \hat{\eta}(k) defined as the following: \hat{\eta}(k)=\frac{e^{-k^{2}}\tanh k}{kU^{2}+(-B+\Omega U+E_{b}|k|-k^{2})\tanh k} The parameters U,B,\Omega,E_{b} have all been defined previously. I have naively tried the following: \eta...
  35. V

    I Solving u_x=(sin(x))*(u) in Fourier space

    Does anyone know if it is possible to solve an equation of the type u_x=(sin(x))*(u) on a periodic domain using the fft. I have tried methods using convolutions but have had no success thanks in advance
  36. thegreengineer

    I Fourier Series: I don't understand where I am wrong --

    Good afternoon people. Recently I started taking a course at my college about Fourier series but I got extremely confused. Here's what's going on. In school we were asigned to use the symmetry formulas to find the Fourier series of the following: f\left ( t \right )=\begin{cases} 1 & \text{ if...
  37. M

    Fourier Transform of a 2D Anisotropic Gaussian Function

    In an image processing paper, it was explained that a 2D Gabor filter is constructed in the Fourier domain using the following formula: $$ H(u,v)=H_R(u,v) + i \cdot H_I(u,v)$$ where HR(u,v) and HI(u,v) are the real and imaginary components, respectively. It also mentions that the real and...
  38. baby_1

    Fourier Series in cylindrical coordinate

    Homework Statement Here is my question Homework Equations I don't know with what formula does the book find Fourier series? The Attempt at a Solution
  39. sa1988

    I Is this even possible? Question about Fourier Series....

    Today I had a maths exam with a question which was worded something like: Write ##sin(3x-x_0)## as its Fourier representation. By doing a suitable integral or otherwise, find the possible values of its Fourier coefficients. You may find the following useful: ##sin(\alpha-\beta) =...
  40. sa1988

    Manipulating Fourier transforms

    Homework Statement Homework EquationsThe Attempt at a Solution I think I'm ok with the first part. I start with: ##\widetilde{f}(p) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \! e^{-ipx}f(x) \, \mathrm{d}x## Then moving on to the transform for ##e^{ip_0 x}f(x)## I get...
  41. sa1988

    Fourier Transform and Partial Differential Equations

    Homework Statement Homework EquationsThe Attempt at a Solution First write ##\phi(x,t)## as its transform ##\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \! e^{ipx} \widetilde{\phi}(p,t) \, \mathrm{d}p## which I then plug into the PDE in the question to get...
  42. P

    How's Fourier series modified for function f(t)= f(2Pi t)?

    Homework Statement How are the coefficients of the Fourier series modified for a function with a period 2πT? Homework Equations a0 = 1/π ∫π-π f(x) dx an = 1/π ∫π-π f(x) cos(nx) dx bn = 1/π ∫π-π f(x) sin(nx) dx The Attempt at a Solution I tried letting x= t/T so dx = dt/T and the limits x = ±...
  43. P

    Fourier Series: Solving Homework Equations for f(x)

    Homework Statement The following function is periodic between -π and π: f(x) = |x| Find the Coefficients of the Fourier series and, by examining the Fourier series at x=π or otherwise, determine: 1 + 1/32 + 1/52 + 1/72 ... = Σ∞j=1 1/(2j - 1)2 Homework Equations f(x) = a0/2 + ∑∞n=1 ancos(nx) +...
  44. S

    Fourier transform of sin(3pix/L)

    Homework Statement Homework EquationsThe Attempt at a Solution So we want sine in terms of the exponentials when we take the Fourier transform F(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}dx where f(x)=\sin(3\pi x/L). Let a=3pi/L. Then \sin(ax)=\frac{e^{iax}-e^{-iax}}{2i}. (Is this correct?) Then we...
  45. SU403RUNFAST

    Sketching a periodic function and Fourier analysis

    Homework Statement So i have a function f(x)=x^2 that is periodic -a<x<a and need to sketch this function from -3a<x<a. I know how to find the Fourier coefficients though. Homework Equations f(x)=x^2 sketch it periodically The Attempt at a Solution I know that a function is only periodic...
  46. Jianphys17

    I Lebesgue measure and Fourier theory

    Hi everyone, in this days i was seeing a little of Fourier series and transform, and i wondered if it was necessary to better understand before the measure and Lebesgue integral before studying it. Or it's not necessary?
  47. Alana02011114

    Solving Coefficient not using Fourier Series coefficient

    Given the Laplace's equation with several boundary conditions. finally i got the general solution u(x,t). One of the condition is that: u(1,y)=y(1-y) After working on this I finally got: ∑An sin(π n y )sinh (π n) = y(1-y) However, i was asked to find An, by not using Fourier series...
  48. Nemo's

    Fourier series neither odd nor even

    Homework Statement I'm trying to calculate the Fourier Series for a periodic signal defined as: y = x 0<x<2Π y = 0 2Π≤x<3Π Homework Equations Fn = 1/T ∫T f(t)cos(kwοt + θk)[/B] cn/2 + ∑k=1k=∞(cn)cos(kwοt+θk) cn= 2|Fn| θk=∠Fn The Attempt at a Solution I got Cn =...
  49. S

    I Fourier transform of Dirac delta

    In lectures, I have learned that F(k)= \int_{-\infty}^{\infty} e^{-ikx}f(x)dx where F(k) is the Fourier transform of f(x) and the inverse Fourier transform is f(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx}f(k)dk . But on the same chapter in the lecture notes, there is an example solving...
  50. B

    I Fourier transform sum of two images

    The FT decomposes images into its individual frequency components In its absolute crudest form, would the sum of these two images (R) give the L image?
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