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. F

    I Solution to PDEs via Fourier transform

    Suppose a PDE for a function of that depends on position, ##\mathbf{x}## and time, ##t##, for example the wave equation $$\nabla^{2}u(\mathbf{x},t)=\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)$$ If I wanted to solve such an equation via a Fourier transform, can I Fourier...
  2. D

    Fourier coefficients - dirichlet problem for annulus

    Hello 1. Homework Statement Find the Fourrier coefficients in the annulus problem of the text. uxx+uyy=0 in 0<a²<x²+y²<b² u=g(θ) for x²+y²=a² u=h(θ) for x²+y²=b² Homework Equations The solution is The Attempt at a Solution I have the solutions but when I solved it for...
  3. RJLiberator

    Fourier Analysis for |x| and Proving a Series Convergence

    Homework Statement Consider a 2pi-periodic function f(x) = |x| for -pi ≤ x ≤ pi a) Compute the Fourier series of the function f. b) Prove that (from n=1 to n=infinity)∑ 1/(2k-1)^2 = pi^2/8.**note all "sums" from here on out will be defined from n = 1 to n=infinity Homework EquationsThe Attempt...
  4. henry wang

    I Proving the Continuity of Fourier Transform in the Limit as L Tends to Infinity

    Quote: "The Fourier transform is a generalization of the complexFourier series in the limit as http://mathworld.wolfram.com/images/equations/FourierTransform/Inline1.gif. Replace the discrete http://mathworld.wolfram.com/images/equations/FourierTransform/Inline2.gif with the continuous while...
  5. Zarmina Zaman Babar

    A Fourier Transform of Piecewise linear spline wavelet

    Fourier Transform of Piecewise linear spline wavelet is defined by 1-|t|, 0<t<1; 0, otherwise, is (sinc(w/2))^2. Can anyone please show me the steps. Thanks
  6. henry wang

    I How is the constant pi/L deduced in Fourier series?

    How is pi/L part deduced in (n*pi*x)/L?
  7. P

    Generate the Fourier transform of the function

    Homework Statement a(x)=f-Nd(x) + f-(N-1)d(x) +...+ f(N-1)d(x) + fNdHomework Equations fd(x) = (1/a for |x-d| < a and 0 otherwise) Fourier transform of function g(x) is g~(p) = 1/root(2pi) ∫ dx e-ipx g(x) The Attempt at a Solution [/B] I have found the general Fourier transform for the...
  8. M

    Complex Fourier Series into a Cosine Series

    Homework Statement a. Represent f(x)=|x| in -2<x<2 with a complex Fourier series b. Show that the complex Fourier Series can be rearranged into a cosine series c. Take the derivative of that cosine series. What function does the resulting series represent? [/B]Homework Equations...
  9. L

    Pointwise, uniform convergence of fourier series

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  10. Ben Wilson

    Efficient 3D Inverse Fourier Transform on FORTRAN Code with Chi Array

    Hi, I have a FORTRAN code with an array called Chi that I want to run an inverse FT on. I have defined two spaces X and K which each consist of 3 vectors running across my physical verse and inverse space. My code (If it works??) is extremely slow and inefficient (see below). What is the best...
  11. R

    Sawtooth function Fourier transform

    Homework Statement For a periodic sawtooth function ##f_p (t) = t## of period ##T## defined over the interval ##[0, T]##, calculate the Fourier transform of a function made up of only a single period of ##f_p (t),## i.e. $$f(t)=\left\{\begin{matrix}f_p (t) \ \ 0<t<T\\0 \ \ elsewhere...
  12. R

    MATLAB How can I fix the dimensions mismatch in my split-step Fourier method code?

    I am trying to write a very basic Matlab code to preform the split-step Fourier method on the nonlinear Schrodinger equation: $$\frac{\partial A(z,T)}{\partial z} = -i \frac{\beta_2}{2} \frac{\partial^2A}{\partial T^2} + i \gamma |A|^2 A \ \ (1)$$ I want the program to make 3D plots of...
  13. Enigma322

    I Q about Fourier Coefficient Derivation

    For the calculation of cn u have to multiply the equation ∑ cn * ejnx by e-jmx what is the reason for this? in my textbook it says nothing about it and on some sites it just said "without justification" i guess what I am asking is why does this do what we want? ps: how do u properly make...
  14. I

    How Signals Are Sampled and Stored As A Fourier Transform?

    This question is a little basic but.. how are signals stored in a Fourier Transform function f(t)? In my PDE class we were always given a base function to put in terms of sin and cos. But when taking a bunch of samples, all I end up with is a table/array over some time T. How might I use this...
  15. E

    Fourier Transform of Heaviside function

    Homework Statement Find the Fourier transform of H(x-a)e^{-bx}, where H(x) is the Heaviside function. Homework Equations \mathcal{F}[f(t)]=\frac{1}{2 \pi} \int_{- \infty}^{\infty} f(t) \cdot e^{-i \omega t} dt Convolution theory equations that might be relevant: \mathcal{F}[f(t) \cdot...
  16. nazmus sakib

    Fourier sine series for a triangular wave on a finite string

    Homework Statement A string of length L =8 is fixed at both ends. It is given a small triangular displacement and released from rest at t=0. Find out Fourier coefficient Bn. Homework Equations what should i use for U0(x) ? The Attempt at a Solution
  17. Ben Wilson

    MATLAB Matlab syntax for 2-d fourier transform

    I have a function f(x,y) which i have defined in this way: a vector x and a vector y meshgrid[x,y] z= f(meshgrid[x,y]). how do i do a 2-d Fourier transform of f(x,y)? the transform must be done without using operations like fft, and must be done using summations written in the code.
  18. Ben Wilson

    MATLAB Fourier transform of a 2D shape

    I have a function of 2 variables [f(x,y)] where if there was an ellipse in the x-y plane, all values of the function are 1 inside the ellipse and 0 outside. I can plot this function as a surface in 3d where it looks like an elevated ellipse hovering over an elliptical hole in a sheet. My...
  19. evinda

    MHB Fourier Transform: Calculate $\hat{g}(\omega)$

    Hello! (Wave) I want to calculate the Fourier transform of $g(x)=|x|$. I got so far that $\hat{g}(\omega)=2 \left[ \frac{x \sin{(x \omega)}}{\omega}\right]_{x=0}^{+\infty}-2 \int_0^{+\infty} \frac{\sin{(x \omega)}}{\omega} dx$ Is it right so far? How can we calculate $\lim_{x \to +\infty}...
  20. R

    What are the Fourier transform properties for various functions?

    Homework Statement A certain function ##v(x)## has Fourier transform ##V(\nu)##. The plot of the function is shown in the figure attached below. For each of these functions give their Fourier transform in terms of ##V(\nu)##. And also state if the FT is Hermitian/anti-Hermitian, even/odd...
  21. D

    Very Basic Fourier Transform Equation

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  22. C

    Odd or Even? - Arbritrary Period Fourier Series

    Homework Statement Hello everyone, I'm new to the great field that is Fourier analysis, and have a question about the way in which to determine if the function is a odd or even function. Given the function, of one period f(x) = { x; 0 <= x < =1, 1; 1 < x < 2, (3 -x); 2 <= x <= 3: Is...
  23. RJLiberator

    How to Find Fourier Series for a Given Function Using Sine Series?

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  24. B

    Particle Released From Narrow Potential - Fourier Transform

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  25. N

    Inverse Fourier Tranform of Transmission Lines Wave Equation

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  26. J

    Fourier Transform using duality property?

    Homework Statement Find the Fourier transform of x(t) = 4 / (4 - i*t)^2 where i is imaginary Homework Equations Duality Property F(t) ↔ 2πf(-ω) when f(t) ↔ F(ω) The Attempt at a Solution I am not sure if duality property is the way to solve this. I look at a list of properties and this...
  27. ognik

    Fourier Analysis of Wave Packet: Is This Calculation Correct?

    Homework Statement Assume ## \phi(k_x ) = \sqrt2 {\pi}## for ## \bar{k}_x - \delta \le k_x \le \bar{k}_x + \delta##, and ##= 0## for all other values of ##k_x##. Calculate ##\psi(x, 0)##, and show that ## \Delta x \Delta k_x \approx 1 ## holds if ## \Delta x## is taken as the width at half...
  28. RJLiberator

    Help solving fourier cosine series related problem

    Homework Statement I am doing #9. Homework EquationsThe Attempt at a Solution I've been looking at a lot of similar problems on the internet. The main difference between this one and them is that this one has an interval of [0,4] while they often have intervals of [0,pi] or [-pi,pi] In my...
  29. RJLiberator

    Fourier Series and deriving formulas for sums of numerical

    Homework Statement Homework EquationsThe Attempt at a Solution So I am tasked with answer #3 and #4. I have supplied the indicated parenthesis of 8 also with the image. Here is my thinking: Take the Fourier series for |sin(θ)|. Let θ = 0 and we see a perfect relationship. sin(0) = 0 and...
  30. naima

    Fourier transform with complex variables

    I found this formula in a paper: \int exp( \frac{x1 + i x2}{ \sqrt 2} \eta^* - \frac{x1 - i x2}{ \sqrt 2} \eta) D(\eta)/ \pi d^2 \eta the author calls it the Fourier transform of D. It is the first time thar i see this formula. How common is this notation? Can we use it without problem?
  31. RJLiberator

    Understanding Fourier Coefficients using PDE

    Homework Statement In my PDE course we have a homework question stating the following: Let ϑ(x) = x in the interval [-pi, pi ]. Find its Fourier Coefficients. Homework Equations From my notes on this type of question: a_o = 2c_o = 1/pi * integral from -pi to pi [f(x) dx] a_n = c_n + c_(-n)...
  32. ognik

    MHB Please help me find Fourier series mistake

    Find the Fourier sin series expansion of dirac delta function $\delta(x-a)$ in the half-interval (0,L), (0 < a < L): Now $b_n = \frac{1}{L} \int_0^L f(x)sin \frac{n \pi x}{L}dx $ - but L should be $\frac{L}{2}$ for this exercise... So I would get $ \frac{2}{L} \int_0^L f(x)sin \frac{n \pi...
  33. L

    MHB Solve Fourier Transform: f(t)=sin(2πt)/t

    I don't know if it is the right section to post in. I have a problem with a "simple" Fourier transform. This is the function to transform: f(t)=\frac{\sin\left({2\pi t}\right)}{t}. My first idea was to write that as \sin\left({2\pi t}\right)\cdot\frac{1}{t} but then my fantasy crashed against a...
  34. ognik

    MHB How Does Spherical Polar Coordinates Simplify the Coulomb Potential in K-Space?

    Hi - I've got myself mixed up here, please see what I am missing below ... Show $ \int e^{ik \cdot (r - r')} \frac{d^3 k}{(2 \pi)^3 k^2} = \frac{1}{4 \pi}|r-r'| $ Let R = r-r', then $k \cdot (r - r') = kR cos \theta$ Next I would translate into spherical polar coords, using $\int d^3 k =...
  35. ognik

    MHB Finding the Spherical Polar Fourier Transform with Variable Change

    Show that the 3-D FT of a radially symmetric function may be rewritten as a Fourier sin transform i.e. $ \frac{1}{({2\pi})^{{3}_{2}}} \int_{-\infty}^{\infty}f(r)e^{ik \cdot r} \,d^3x = \frac{1}{k} \sqrt{\frac{2}{\pi}} \int_{-\infty}^{\infty} \left[ rf(r) \right] sin(kr) \,dr $ The example...
  36. S

    Solving the Fourier Series of a 2π-Periodic Function

    Homework Statement The odd 2π-periodic function f(x) is defined by f(x) = x2 π > x > 0 -x2 −π<x<0 Find the coefficient bn in the Fourier series f(x) = a0/2 + ∑(an cos(nx) + bn sin(nx)). What are the values of the coefficients a0 and an and why? Homework Equations bn = 1/π ∫...
  37. T

    Python Improving Fourier Transform Visualization in Python

    Hello, My name is Thibaut. I am looking to improve my code in python in order to have a better look a my Fourier transform. as you can see on the image, we barely see any detail of the peaks on the image. Also it's not centred. the zero order peak in on the corner, not in the centre. Any idea...
  38. W

    Exploring the Fourier Transforming Property of Lenses

    I was reading about this Fourier transforming property of lens,when I came by the experimental setup for Fourier optics(with laser and a 4f correlator system).Part of the setup was that of Fraunhofer diffraction and we get the Fourier transform of the aperture at the focal point of first lens...
  39. R

    Fourier Transform (Triangular Pulse)

    Homework Statement What is the Fourier transform of the function graphed below? According to some textbooks the Fourier transform for this function must be: $$ab \left( \frac{sin(\omega b/2)}{\omega b /2} \right)^2$$ Homework EquationsThe Attempt at a Solution I believe this triangular...
  40. S

    Fourier transformation and test function -- Question in the derivation

    How does it work? (The derivative rules of FT) We look at $$F[x(t)]=\hat{x}(f)$$ $$\mathcal{l} \text{ is a distribution, with}\tilde{x}=tx(t)$$ $$\mathcal{F}[Dl(x)]=\mathcal{F}l'(x)=2\pi il(\mathcal{F}\tilde{x})=2\pi i \mathcal{F}l(\tilde{x})$$ Till here I fully understand. But next step...
  41. S

    Solving phi-fourth theory using Fourier analysis

    The equation of motion of ##\phi^4## theory is ##(\partial^{2}+m^{2})\phi = -\frac{\lambda}{3!}\phi^{3}##. Why can't this equation be solved using Fourier analysis? Can't we simply write the equation in Fourier space and take it from there?
  42. ognik

    MHB Is the half interval Fourier series for f(x)=x over (0,L) correct?

    Please help me find my mistake - "find the Sine F/series of f(x)=x over the half-interval (0,L)" I get $ b_n=\frac 2L \int_{0}^{L}x Sin \frac{2n\pi x}{L} \,dx $ $ = \frac 2L \left[ x(-Cos \frac{2n\pi x}{L}. \frac{L}{2n\pi x}\right] + \frac {1}{n\pi} \int_{0}^{L} Cos \frac{2n\pi x}{L} \,dx$...
  43. ognik

    MHB Fourier Transform limits problem

    Find the Fourier Transform of $ e^{-a|t|}Cosbt $ I'd like to simplify this using $Cosbt = Re\left\{e^{ibt}\right\}$ $\therefore \hat{f}(\omega) = Re\left\{ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\left(-a+ib+iw\right)|t|} \,dt \right\} = Re\left\{ \frac{1}{\sqrt{2\pi}}...
  44. X

    How can I represent this expression as a Fourier Transform?

    Hello, I hope I am posting this in the correct forum topic. It really is more of a "mathy" type of question, but I am posting it here because it deals with radar, and this type of math is used a lot in radar. To the mods, feel free to move it to a more suitable location if desired. I have...
  45. ognik

    MHB Find Fourier series of Dirac delta function

    Hi - firstly should I be concerned that the dirac function is NOT periodic? Either way the problem says expand $\delta(x-t)$ as a Fourier series... I tried $\delta(x-t) = 1, x=t; \delta(x-t) =0, x \ne t , -\pi \le t \le \pi$ ... ('1' still delivers the value of a multiplied function at t)...
  46. ognik

    MHB Help with Fourier series mistake

    Hi - frustratingly I get some problems right 1st time, others just defy me (Headbang) $f(x) = -x, [-\pi,0]; = x, [0,\pi]$ I get $a_0 = \pi$ and $a_n = \frac{-4}{\pi \left(2n-1\right)^2}$ which agrees with the book - but I thought I'd check $b_n$ for practice, it should = 0 according to the...
  47. ognik

    MHB Understanding Fourier Series: Solving a Problem with Sinusoidal Functions

    Hi, appreciate some help with this FS problem - $f(t)= 0$ on $[-\pi, 0]$ and $f(t)=sin\omega t$ on $[0,\pi]$ I get $a_0=\frac{2}{\pi}$ and $b_1 = \frac{1}{2}$, which agree with the book; all other $b_n = 0$ because Sin(mx)Sin(nx) orthogonal for $m \ne n$ But $a_n...
  48. ognik

    MHB How to decompose a fourier series

    Hi, in a section on FS, if I were given $\sum_{n=1}^{\infty} \frac{Sin nx}{n} $ I can recognize that as the Sin component of a Fourier Series, with $b_n = \frac{1}{n} = \frac{1}{\pi} \int_{0}^{2 \pi}f(x) Sin nx \,dx$ Can I find the original f(x) from this? Differentiating both sides doesn't...
  49. ognik

    MHB Fourier series coefficient problem

    Hi - an example in my book shows that FS coefficiants can be arrived at by minimizing the integrated square of the deviation, i.e. $ \Delta_p = \int_0^{2\pi}\left[ f(x) - \frac{a_0}{2}-\sum_{n=1}^{p}\left( a_nCosnx + b_nSinnx \right) \right]^2dx $ So we're looking for $ \pd{\Delta_p}{a_n}...
  50. J

    Fourier series of square wave on Matlab?

    Homework Statement How Can i do this on matlap the question in Attached files Homework Equations The Attempt at a Solution i try a lot but i failed
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