Fourier Definition and 64 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.

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  1. M

    Lorentzian line profile of emitted radiation

    First of all i tried to follow the textbook. Here they start of by modelling the atom as an harmonic oscilator: Then they find the solution as: They neglect the second term as omega_0 >> gamma which also makes good sense so they end up with: So far so good. After this they state the...
  2. A

    Fourier sine and cosine transforms of Heaviside function

    Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. I tried each part of the problem, but I'm only left in terms of limits as x -> infinity of sin or cos...
  3. QuarkDecay

    Fourier transformation for circular apertures

    My notes say that the Resolution of the Aperture(in the Electric field of the wave) is the Fourier transformation of the aperture. Then gives us the equation of the aperture: and says that for the circular aperture in particular also: My attempt at solving this: We know that the Fourier...
  4. P99

    How to obtain the frequency response and the spectrum graph of this function?

    I think that is with the Fourier transform.
  5. D

    B Fourier Analysis on musical chords in different instruments

    I wanted to do an investigation about how the same musical chord can have the same pitch but sound different on different musical instruments. Like how chord C major would sound higher played in the electric guitar than a C major played on piano. How should I approach this investigation?
  6. C

    I What's the definition of "periodic extension of a function"?

    I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...
  7. A

    I How can one see the noise in the Fourier Domain (Nyquist Frequency)?

    Say we have a transform of a line profile that extends out to the Nyquist frequency such that you cannot see the noise level, what could you change in your spectrograph arrangement that would allow you to see the noise level in the Fourier domain? My thought is that we can apply a filter, P(s)...
  8. entropy1

    Randomizing phases of harmonics

    Suppose I decompose a discrete audio signal in a set of frequency components. Now, if I would add the harmonics I got, I would get the original discrete signal. My question is: if I would randomize the phases of the harmonics first, and then add them, I would get a different signal, but would it...
  9. thereddy

    Discrete Fourier transform question

    Summary:: Discrete Fourier transform exam question Hi there, I'm not really sure how to do this question at all. Any help would be appreciated.
  10. C

    A Partial differential equation containing the Inverse Laplacian Operator

    I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...
  11. M

    Engineering Reconstruct a signal by determining the N Fourier Coefficients

    %My code: %Type of signal: square T = 40; %Period of the signal [s] F=1/T; % fr D = 23; % length of signal(duration) dt=(D/T)*100; N = 50; %Number of coefficients w0 = 2*pi/T; %signal pulse t1= 0:0.002:T; % original signal sampling x1 = square((2*pi*F)*(t1),dt);%initial square signal...
  12. N

    I Separation of variables - Getting the Fourier coefficients

    Hey there! I am current taking an introductory course on PDE's, and our professor hasn't really emphasized last part of solutions from separation of variables. Now its not strictly going to be on the exam, however I remember doing this with ease a few years back, but for some reason now I...
  13. Ineedhelp0

    I Parseval's theorem and Fourier Transform proof

    Given a function F(t) $$ F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega $$ I am looking for a proof of the following: $$ \int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d...
  14. E

    Deduce the formula of D'Alembert with Fourier transform

    Well what I did was first use the inverse Fourier transform: $$u(x,t)=\frac{1}{2\pi }\int_{-\infty }^{\infty }\tilde{u}(\xi ,t)e^{-i\xi x}d\xi$$ I substitute the equation that was given to me by obtaining: $$u(x,t)=\frac{1}{2\pi }\left \{ \int_{-\infty }^{\infty}\tilde{f}(\xi)cos(c\xi...
  15. Behrouz

    A Finding a specific amplitude-frequency in the time domain

    Hello, I have a signal and got the FFT result of that. I have shown them both below along with the MATLAB code. May I ask if there is any method to find the time zone(s) in the signal that a specific frequency has(have) happened? The reason I'm asking this is that I want to specify the time...
  16. I

    I Fourier's Trick and calculation of Cn

    I understand that the solutions to the time-independent Schrodinger equation are complete, so a linear combination of the wavefunctions can describe any function (i.e. ##f(x) = \sum_{n = 1}^{\infty}c_n\psi_n(x) = \sqrt{\frac{2}{a}} \sum_{n=1}^{\infty} c_n\sin\left(\frac{n\pi}{a}x\right)## for...
  17. Cassius1n

    Heat loss in a conductor based on Fourier's law

    Homework Statement Find the admissible current density Jadm for a wire that has no insulation and also for a wire that has two layers of insulation and compare it to Jadm for the case when the wire has only one layer of insulation. 2. The attempt at a solution and equations In the image...
  18. J6204

    Extending function to determine Fourier series

    In the following question I need to find the Fourier cosine series of the triangular wave formed by extending the function f(x) as a periodic function of period 2 $$f(x) = \begin{cases} 1+x,& -1\leq x \leq 0\\ 1-x, & 0\leq x \leq 1\\\end{cases}$$ I just have a few questions then I will be able...
  19. M

    E&M separation of variables and Fourier

    Homework Statement Boundary conditions are i) V=0 when y=0 ii) V=0 when y=a iii) V=V0(y) when x=0 iv) V=0 when x app infinity. I understand and follow this problem (separating vars and eliminated constants) until the potential is found to be V(x,y) = Ce^(-kx)*sin(ky) Condition ii...
  20. S

    A Convolution operator spectrum

    Hi everyone, I have some knowledge of Hilbert spaces and Functional Analysis and I have the following question. I ofter have read that "Fourier transform diagonalize the convolution operator". So, we can say that for LTI systems (that can always be described with a convolution and "live" in...
  21. A

    Solving the heat equation using FFCT (Finite Fourier Cosine Trans)

    Homework Statement Solve the following heat Eq. using FFCT: A metal bar of length L is at constant temperature of Uo, at t=0 the end x=L is suddenly given the constant temperature U1, and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature at any...
  22. A

    Fourier Transform

    Homework Statement Hello everyone, am trying to solve this Fourier Trans. problem, here is the original solution >> Q/ How did he come up with this result and where is my mistake? Homework Equations All equation are in the above attached picture The Attempt at...
  23. A

    Supressed harmonics with certain initial conditions

    I'm currently reading class notes from an introductory waves course, written by the professor himself. I'm stuck in the Fourier analysis part, because he gives the formulas for the nth mode amplitude of a standing wave with fixed ends and then states some properties which I can't really make...
  24. R

    Matching Discrete Fourier Transform (DFT) Pairs

    Homework Statement [/B] I am trying to match each of the following 28-point discrete-time signals with its DFT: Set #1: Set #2: Homework Equations The Attempt at a Solution Set #1 We have already established (here) that: ##Signal 1 \leftrightarrow DFT3## ##Signal 4 \leftrightarrow...
  25. G

    Find Fourier coefficients - M. Chester text

    Homework Statement I am self studying an introductory quantum physics text by Marvin Chester Primer of Quantum Mechanics. I am stumped at a problem (1.10) on page 11. We are given f(x) = \sqrt{ \frac{8}{3L} } cos^2 \left ( \frac {\pi}{L} x \right ) and asked to find its Fourier...
  26. M

    Power signal calculation using Parseval's Theorem

    Homework Statement Hi guys, I have the following transmitted power signal: $$x(t)=\alpha_m \ cos[2\pi(f_c+f_m)t+\phi_m],$$ where: ##\alpha_m=constant, \ \ f_c,f_m: frequencies, \ \ \theta_m: initial \ phase.## The multipath channel is: $$h(t)=\sum_{l=1}^L \sqrt{g_l} \ \delta(t-\tau_l).$$...
  27. DeathbyGreen

    A Fourier Transforming a HgTe 2D Hamiltonian

    Hi! I am currently trying to derive the Fourier transform of a 2D HgTe Hamiltonian, with k_x PBC and vanishing boundary conditions in the y direction at 0 and L. Here is the Hamiltonian: H = \sum_{k}\tilde{c_k}^{\dagger}[A\sin{k_x}\sigma_x + A\sin{k_y}\sigma_y + (M-4B+2B[\cos{k_x} +...
  28. M

    Integral of absolute value of a Fourier transform

    Homework Statement Hi guys, I am going to calculate the following integral: $$\int_0^{f_c+f_m} |Y(f)|^2\, df$$ where: $$Y(f)=\frac{\pi}{2} \alpha_m \sum_{l=1}^{L} \sqrt{g_l}\left [ e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0) + e^{-j(\omega \tau_l + \theta_m)} \delta(\omega +...
  29. M

    Calculating Magnetic Field Strength from FFT

    Hello All, Briefly on the exposition; I'm an undergraduate assistant to a professor. We contribute to the Muon g-2 experiment in Fermilab, designing and optimizing the magnetic-measurement equipment. As you might imagine I utilize the Fourier Transform often to analyze data. The data I'm...
  30. TheBigDig

    Finding Fourier Coefficient

    Homework Statement [/B] I am looking for help with part (d) of this question 2. Homework Equations The Attempt at a Solution I have attempted going through the integral taking L = 4 and t0 = -2. I was able to solve for a0 but I keep having the integrate by parts on this one. I've tried...
  31. Marcus95

    Fourier Series Coefficient Symmetries

    Homework Statement Let ## f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx) ## What can be said about the coefficients ##a_n## and ##b_n## in the following cases? a) f(x) = f(-x) b) f(x) = - f(-x) c) f(x) = f(π/2+x) d) f(x) = f(π/2-x) e) f(x) = f(2x) f) f(x) = f(-x) =...
  32. D

    Fourier/heat problem involving hyperbolic sine

    Homework Statement A rectangular box measuring a x b x c has all its walls at temperature T1 except for the one at z=c which is held at temperature T2. When the box comes to equilibrium, the temperature function T(x,y,z) satisfies ∂T/∂t =D∇2T with the time derivative on the left equal to zero...
  33. M

    Calculating Magnetic Field from FFT Amplitude

    So a little bit of background: I work in an undergraduate lab at UMass Amherst and am currently building/optimizing a faraday magnetometer for use in the Muon g-2 experiment at Fermilab. The magnetometer works as follows. A laser is shone through a crystal with a particular Verdet Constant at...
  34. D

    Fourier Series/Wave Problem

    Homework Statement A violin string is plucked to the shape of a triangle with initial displacement: y(x,0) = { 0.04x if 0 < x < L/4 (0.04/3)(L-x) if L/4 < x < L Find the displacement of the string at later times. Plot your result up to the n = 10...
  35. N

    Arguments of the FT and DTFT

    Could someone explain the intuition behind the variables of the FT and DTFT? Do I understand it correctly ? For FT being X(f), I understand that f is a possible argument the frequency, as in number of cycles per second. FT can be alternatively parameterized by \omega = 2 \pi f which...
  36. 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...
  37. 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...
  38. 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...
  39. dreens

    I Orthogonal 3D Basis Functions in Spherical Coordinates

    I'd like to expand a 3D scalar function I'm working with, ##f(r,\theta,\phi)##, in an orthogonal spherical 3D basis set. For the angular component I intend to use spherical harmonics, but what should I do for the radial direction? Close to zero, ##f(r)\propto r##, and above a fuzzy threshold...
  40. 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 ##...
  41. Ben Wilson

    3D Fourier Transforms?

    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...
  42. 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.
  43. 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...
  44. 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...
  45. 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...
  46. 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\...
  47. philton

    Fourier Transformation in a single dynamic gif BTW, how to insert a photo url to show an image here at Physics Forum?
  48. T

    What method does a receiver or transmitter use to approx....

    Hi, I'm just curious because I know wifi uses digital FFT to send and receive signals. (I can't really remember why) But when I imagine a signal being sent its like a squiggily wave, so what method does the reciever use to approximate the instantanious values of the signal into a mathematical...
  49. T

    Fourier Reminder

    It's been quite a few years but I recently watched a video about how every picture can be represented by a number of overlapping constructive and destructive peaks from a Fourier (transform or series? I don't remember which). I remember that Fourier series was for periodic and transform was for...
  50. M

    Exact fourier transformation

    Hello, I am having a bit of trouble with calculating the Fourier transformation of a harmonic load. I have the function f(t) = A * sin(ωt) in the time-domain. I would like to represent this function in the frequency domain. What would be its amplitude? Thank you