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. A

    A Uncertainty principle, removing infinity in the Fourier Transform

    I have come across a paper where it is stated that if the infinity assumption in the FT is removed, the uncertainty doesn't hold. Is this a sensible argument? Thank you.
  2. A

    I Invert a 3D Fourier transform when dealing with 4-vectors

    I am having trouble following a step in a book. So we are given that $$\varphi (x) = \int \frac {d^3k}{(2\pi)^3 2\omega} [a(\textbf{k})e^{ikx} + a^*(\textbf{k})e^{-ikx}] $$ where the k in the measure is the spatial (vector) part of the four-momentum k=(##\omega##,##\textbf{k}##) and the k in the...
  3. J

    Fourier transform of a power signal or a voltage signal

    Homework Statement By using Fourier transform, I want to calculate power of signal. I confuse that f(x) in attached equation represents voltage or power. Is that possible when f(x) means power to use Fourier transform. Homework Equations The Attempt at a Solution
  4. M

    A Which Transform to Use for Solving Thermoelastic PDEs?

    I've a system of partial diff. eqs. in thermo-elasticity, I can solve it using normal mode analysis method but I need to solve it using laplace or Fourier
  5. dRic2

    Fourier Transform of 1/(1+x^4)

    Homework Statement Calculate ##F(\frac 1 {1+x^4})##. Homework Equations ##\hat f (ξ) = \int_ℝ \frac 1 {1+x^4} e^{-2\pi i ξ x} dx## and Residue Theorem The Attempt at a Solution I know the function has to be real and even because ##\frac 1 {1+x^4}## is real and even, but I can't work out the...
  6. dRic2

    I Can the Fourier Transform of an L^1 Function be Bounded by its L^1 Norm?

    Hi, I have to show that if ##f \in L^1(ℝ^n)## then: $$ ||\hat f||_{C^0(ℝ^n)} \le ||f||_{L^1(ℝ^n)}$$ Since ##|f(y)e^{-2 \pi i ξ ⋅y}| \le |f(y)|##, using the dominated convergence theorem, it is possible to show that ##\hat f \in C^0(ℝ^n)## but now I don't know how to go on. Thanks is advance.
  7. S

    B What Is the Fourier Transform of a Constant?

    It is often reported that the Fourier transform of a constant is δ(f) : that δ denotes the dirac delta function. ƒ{c} = δ(f) : c ∈ R & f => Fourier transform however i cannot prove this Here is my attempt:(assume integrals are limits to [-∞,∞]) ƒ{c} = ∫ce-2πftdt = c∫e-2πftdt = c∫ƒ{δ(f)}e-2πftdt...
  8. S

    I Fourier transform for cosine function

    Fourier Transform problem with f(t)=cos(at) for |t|<1 and same f(t)=0 for |t|>1. I have an answer with me as F(w)=[sin(w-a)/(w-a)]+[sin(w+a)/(w+a)]. But I can't show it.
  9. C

    Solving a 2D PDE using the Fourier Transform

    Homework Statement Solve the following partial differential equation , using Fourier Transform: Given the following: And a initial condition: Homework EquationsThe Attempt at a Solution First , i associate spectral variables to the x and t variables: ## k ## is the spectral variable...
  10. 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...
  11. N

    Fourier Series: How to interpret the function?

    This is a rather simple question, but am I understanding the following correctly? 1. Homework Statement The Attempt at a Solution This isn't really the problem, but I have a feeling my problem the assignments, is me misunderstanding the function description. I don't see how this 2 pi...
  12. G

    Help finding ths Fourier transform

    Homework Statement find the Fourier transform of the following function in two ways , once using direct computation , and second using the convolution therom . Homework Equations Acos(w0t)/(d2+t2) The Attempt at a Solution I tried first to solve directly . used Euler's identity and got...
  13. M

    How Should Exponential Terms Be Integrated in Fourier Transforms?

    Hi All! I've been looking at this Fourier Transform integral and I've realized that I'm not sure how to integrate the exponential term to infinity. I would expect the result to be infinity but that wouldn't give me a very useful function. So I've taken it to be zero but I have no idea if you can...
  14. D

    Normalization of the Fourier transform

    Homework Statement The Fourier transfrom of the wave function is given by $$\Phi(p) = \frac{N}{(1+\frac{a_0^2p^2}{\hbar^2})^2}$$ where ##p:=|\vec{p}|## in 3 dimensions. Find N, choosing N to be a positive real number. Homework Equations $$\int d^3\vec{p}|\Phi(p)|^2=1$$ , over all p in the 3...
  15. R

    I Fourier smoothing and Savitzky-Golay filtering

    I am trying to recover a laser beam profile using numerical differentiation of the data obtained from a "knife-edge scan". I am trying to select between two different methods to smooth out the numerical noise. Here is my raw data and the derivative: Here, I arbitrarily chose the 13 points...
  16. A

    A Fourier transform of outgoing spherical waves

    Please, can anyone explain how formula (5) is obtained in J.J. Barton article ''Approximate translation of screened spherical waves" . Phys.Rew. A ,Vol.32,N2, 1985. ? https://doi.org/10.1103/PhysRevA.32.1019 The same formula are given in the book Pendry J.B. "Low enrgy electron diffraction. The...
  17. T

    I Informational content in 2D discrete Fourier transform

    When you do a discrete Fourier transform (DFT) of a one-dimensional signal, I understand that the second half of the result is the complex conjugate of the first half. If you threw out the second half of the result, you're not actually losing any data and you would be able to recreate the entire...
  18. thecastlingking

    Complex exponential Fourier series coefficients?

    Above is a question I need some help with. I can get to a certain point, but I can't get beyond. Can somebody with knowledge about this help?
  19. W

    B Fourier Transform: Geometric Interpretation?

    Hi, outside the mathematical proof that shows that sines of different frequency are orthogonal... is there geometric interpretation/picture of this phenomena?
  20. C

    Fourier Analysis and the Significance of Odd and Even Functions

    Homework Statement Q1. a) In relation to Fourier analysis state the meaning and significance of 4 i) odd and even functions ii) half-wave symmetry {i.e. f(t+π)= −f(t)}. Illustrate each answer with a suitable waveform sketch. b) State by inspection (i.e. without performing any formal analysis)...
  21. Matt Chu

    How Do You Prove the Fourier Transform Definition Using Integral Evaluation?

    Homework Statement Given a continuous non-periodic function, its Fourier transform is defined as: $$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(x) e^{-ikx} dx$$ The problem is proving this is true by evaluating the...
  22. H

    I Fourier transform -- what physical variables am I allowed to transform between?

    A common use of the Fourier transform in physics is to transform between momentum-space and position-space. But what physical variables am I allowed to transform between? For instance can I use the Fourier transform to go from momentum space to frequency space or whatever?
  23. D

    Finding the fourier spectrum of a function

    Homework Statement Find the Fourier spectrum ##C_k## of the following function and draw it's graph: Homework Equations 3. The Attempt at a Solution [/B] I know that the complex Fourier coefficient of a rectangular impulse ##U## on an interval ##[-\frac{\tau}{2}, \frac{\tau}{2}]## is ##C_k =...
  24. Cassius1n

    Heat loss in a conductor based on Fourier's law

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  25. R

    I 2D Fourier transform orientation angle

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  26. T

    A What Is a Fourier Series?

    Working on some microwave stuff, read about this but can't understand the explanations online.
  27. P

    MHB Erin's question via email about a Fourier Transform

    $\displaystyle \begin{align*} F \left( \omega \right) &= \mathcal{F} \left\{ f \left( t \right) \right\} \\ &= \int_{-\infty}^{\infty}{ f\left( t \right) \mathrm{e}^{-\mathrm{j}\,\omega \, t}\,\mathrm{d}t } \\ &= \int_{-\infty}^{-2}{ 0\,\mathrm{d}t } + \int_{-2}^0{ \left( 1 + \frac{t}{2}...
  28. binbagsss

    Hecke Operators and Eigenfunctions, Fourier coefficients

    Homework Statement Consider the action of ##T_2## acting on ##M_k(\Gamma_{0}(N)) ##, and show that ##\theta^4(n)+16F ## and ##F(t)## are both eigenfunctions. Functions are given by: Homework Equations For the Hecke Operators ##T_p## acting on ##M_k(\Gamma_{0}(N)) ##, the Hecke conguence...
  29. R

    Fourier transform of integral e^-a|x|

    Homework Statement I am supposed to compute the Fourier transform of f(x) = integral (e-a|x|) Homework Equations Fourier transformation: F(p) = 1/(2π) n/2 integral(f(x) e-ipx dx) from -infinity to +infinity The Attempt at a Solution My problem is, that I do not know how to handle that there...
  30. D

    Question regarding Fourier Transform duality

    Homework Statement Given the Fourier transformation pair ##f(t) \implies F(jw)## where ##f(t) = e^{-|t|}## and ##F(jw)=\frac{2}{w^2+1}## find and make a graph of the Fourier transform of the following functions: a) ##g(t)=\frac{2}{t^2+1}## b) ##h(t) = \frac{2}{t^2+1}\cos (w_ot)## Homework...
  31. C

    Arbitrary Circulation Calculation with Fourier Series

    Homework Statement Homework Equations The Attempt at a Solution I am stuck trying to figure out why there are three different alphas and why in the equation we are supposed to use has a theta and what that means. If I can set up the Fourier series I can properley I know how to solve it for...
  32. N

    I Why is the Signal from a Discrete Fourier Transform considered periodic?

    https://en.wikipedia.org/wiki/Discrete_Fourier_transform Why is the signal obtained from a DFT periodic? The time signal x[n] is finite and the number of sinusoids being correlated with it is finite, yet its said the frequency spectrum obtained after the DFT is periodic. I've also read the...
  33. C

    Solving Fourier Cosine Series Homework w/ Matlab & Excel

    Homework Statement Homework Equations All I know is the a's have something to do with the integrals. The Attempt at a Solution I used FFT analysis in Matlab but I do not know what I am looking for. How do the a0s relate to the f(t) in the question and how would I even do run that equation in...
  34. mertcan

    I What Causes Repetition in Fourier Transforms of Audio and Visual Data?

    I would like to express that when I am viewing the repetitive Fourier transform on Internet I encounter that for instance twice Fourier transform may lead the same value at the end of first Fourier transform. When does repetitive( twice or third... consecutively)fourier transform be same with...
  35. Tspirit

    Fourier transform in the complex plane

    Homework Statement I am reading the book of Gerry and Knight "Introductory Quantum Optics" (2004). In page 60, Chapter 3.7, there is two equation referring Fourier Transformation in the complex plane as follows: $$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$...
  36. Jeviah

    How is the following fraction split for inverse Fourier?

    Hi i’m having problems with the following equations: X(w)=2/(-1+iw)(-2+iw)(-3+iw) This then becomes the following equation according the the tutorial, although there is no explanation as to how: X(w)=1/-1+iw, -2/-2+iw, +1/-3+iw The commas indicated the end of each fraction to make it easier...
  37. J

    I Interpretation of the Fourier Transform of a Cauchy Distribution

    Hi, I'm struggling with a conceptual problem involving the Fourier transform of distributions. This could possibly have gone in Physics but I suspect what I'm not understanding is mathematical. The inverse Fourier transform of a Cauchy distribution, or Lorentian function, is an exponentially...
  38. K

    I The Relationship Between Angular and Cyclic Frequency in Fourier Transform

    Hi everybody. There has been a thread about this on physics forums, where the Fourier transform X(w) of x(t) volts (with time units in seconds) could be considered as volt second, or volt per Hz. So when we see tables of Fourier transform pairs, we might see Fourier transform plots associated...
  39. DrClaude

    What is the significance of Joseph Fourier's 250th birthday?

    Joseph Fourier turned 250 this week: https://www.nature.com/articles/d41586-018-03389-w Joyeux anniversaire, Mr Fourier !
  40. T

    Fourier Transformation of ODE

    Homework Statement I am to solve an ODE using the Fourier Transform, however I am quite inexperienced in using this method so I'd like some advice: Homework Equations a) The Fourier Transform b) The Inverse Fourier Transform The Attempt at a Solution I started by applying the Fourier...
  41. DoobleD

    I Fourier series of Dirac comb, complex VS real approaches

    Hello, I tried to compute the Fourier series coefficients for the Dirac comb function. I did it using both the "complex" formula and the "real" formula for the Fourier series, and I got : - complex formula : Cn = 1/T - real formula : a0 = 1/T, an = 2/T, bn = 0 This seems to be valid since it...
  42. Peter Alexander

    Fourier transform of exponential function

    1. The problem statement, all variables, and given/known data Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) =...
  43. J

    Testing my Discrete Fourier Transform program

    Homework Statement I've written a program that calculates the discrete Fourier transform of a set of data in FORTRAN 90. To test it, I need to "generate a perfect sine wave of given period, calculate the DFT and write both data and DFT out to file. Plot the result- does it look like what you...
  44. L

    Find the Fourier Series of the function

    Homework Statement Find the Fourier series of the function ##f## given by ##f(x) = 1##, ##|x| \geq \frac{\pi}{2}## and ##f(x) = 0##, ##|x| \leq \frac{\pi}{2}## over the interval ##[-\pi, \pi]##. Homework Equations From my lecture notes, the Fourier series is ##f(t) = \frac{a_0}{2}*1 +...
  45. A

    Trouble determining the Fourier Cosine series for a Function

    Homework Statement I am only interested in 9 (a) Determine the Fourier Cosine series of the function g(x) = x(L-x) for 0 < x < L Homework Equations The Answer for 9 a. g(x) = (L^2)/6 - ∑(L^2/(nπ)^2)cos(2nπx/L) This is the relevant equation given where ω=π/L f(t) = a0+∑ancos(nωt) a0=1/L...
  46. J

    Fourier transform (got right answer, but not matching graph)

    Homework Statement Homework Equations Scaling property and property of dual. I got the answer. The Attempt at a Solution I got the answer using scaling property and using property of dual. x1(t)---> X2(W)----(another Fourier transform)--->2(3.14) x1(-w) But I think the final answer should be...
  47. A

    I Complex Fourier Series: Even/Odd Half Range Expansion

    Does the complex form of Fourier series assume even or odd half range expansion?
  48. O

    I Using Fourier Transform to Solve ODE with Initial Conditions

    Hi, let's take this ode: y''(t) = f(t),y(0)=0, y'(0)=0. using the FT it becomes: -w^2 Y(w) = F(w) Y(w)=( -1/w^2 )F(w) so i can say that -1/w^2 is the Fourier transorm of the green's function(let's call it G(w)). then y(t) = g(t) * f(t) where g(t) = F^-1 (G(w)) (inverse Fourier transorm) how can...
  49. A

    Fourier transform between variables of different domains

    I'm doing a research project over the summer, and need some help understanding how to construct an inverse Fourier transform (I have v. little prior experience with them). 1. Homework Statement I know the explicit form of ##q(x)##, where $$ q(x) = \frac{M}{2 \pi} \int _{- \infty}^{\infty} dz...
  50. Duke Le

    I [Signal and system] Function with fourier series a[k] = 1

    We have: Period T = 4, so fundamental frequency w0 = pi/2. This question seems sooo easy. But when I use the integral: x(t) = Σa[k] * exp(i*k*pi/2*t). I get 1 + sum(cos(k*pi/2*t)), which does not converge. Where did I went wrong ? Thanks a lot for your help.
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