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. Qiao

    Fourier optics with concave (diverging) lenses

    Hey, I was wondering, since for a convex lens the Fourier transform of a fields is in their real focus plane. Is it for a concave lens that the Fourier transform of a field is in the virtual focus plane? I can't find any book or paper that talks about how concave lenses work in terms of...
  2. R

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    I am engineering student and studying signal processing. The term Fourier transform comes in the discussion several times. There are many transforms like Laplace transform,Z transform,Wavelet transform.But as per my view ,Fourier transform is mostly used compared to others in general. My...
  3. T

    Fourier series, is this valid?

    Hi, I have a Fourier problem that i do not know if it is valid to do the calculations like this. The Fourier transform looks like this ## \hat{v}(x,\omega) = \frac{\hat{F}(\omega)}{4(EI)^{\frac{1}{4}}i \omega^{\frac{3}{2}}(\rho A)^{\frac{3}{4}}}\left[ e^{-i\left[\omega^2 \frac{\rho A}{EI}...
  4. D

    Fourier Transform of a full rectified sine wave

    Homework Statement Derive the FT for a full-wave rectified sine wave, i.e., |sin(wt)| Homework Equations $$1/(√2π)\int_{a}^{b} |Sin[wt]| {e}^{-i w t}dt$$ The Attempt at a Solution I'm not entirely sure how to start doing this problem. What I tried doing was noticing that both of these...
  5. T

    Calculating Transverse Wave Propagation in a Semi-Infinite Beam

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  6. F

    Fourier transform in curvilinear coordinates

    Hello, can you suggest a good book reference to find this: I have a 3D coordinate system where the axis are: 1) locally tangential to a spiral in the equatorial plane; 2) perpendicular to 1 in the equatorial plane; 3) colatitude. The direction of axes 1 and 2 changes with position. I need to...
  7. R

    What does the exponential term mean in Fourier transform

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  8. J

    Quantum Fourier Transform circuit

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  9. M

    Fourier transform - why we need it?

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  10. Qiao

    Spatial Light Modulators and Fourier Optics

    Hi, I'm working with a Digital Micro-mirror Device type SLM and my goal is to convert my laser from a gaussian to flat-head intensity profile. And then the tricky part is to make the beam oscillate up and down on the camera using just the SLM. Apparently I was to naive to think that moving my...
  11. T

    Find the actual sum of a fourier series at a given point

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  12. I

    How Did the Author Determine the Fourier Sine Series for x^2?

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  13. I

    Deriving the fourier transform

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  14. I

    Fourier Derivation: Constants in Integrals Explained

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  15. R

    Triangle wave Fourier transform

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  16. I

    PDE: How to use Fourier Series to express a real function?

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  17. \Theta

    Finding the Fourier Coefficients for mechanics homework

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  18. M

    Steady wave eq and fourier transform

    Homework Statement $$u_{xx} + u_{yy} = 0 : x < 0, -\infty < y < \infty$$ Homework Equations We can use Fourier Transform, which is defined over some function ##f(x)## as ##F(f(x)) = 1/ 2\pi \int_{-\infty}^{\infty} f(x) \exp (i \omega x) dx##. The Attempt at a Solution Using the Fourier...
  19. Coffee_

    Fourier series, Hermitian operators

    (First of all I never saw Hilbert spaces in a mathematical class, only used it in intro QM so far, so please don't assume I know that much when answering.) Let's consider the Hilbert space on the interval [a,b] and the operator ##\textbf{L} = \frac{d^{2}}{dx^{2}} ##. Then ##\textbf{L}## is...
  20. matqkks

    MHB Why Use Half Range Fourier Series for Functions Like x and x^2?

    If we have a standard function like x or x^2 defined between 0 and pi. Then why should we be interested in extending this function to give a Fourier series which resembles this function between 0 and pi? What is the whole purpose of this process? Does it have any real life application or is it...
  21. K

    Fourier Transform and Hilber transform, properties

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  22. M

    Uncovering the Mystery of Using Cosine Transform in Fourier Analysis

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  23. T

    Improving Frequency Resolution with Window Functions in FFT Calculations

    Okay I have a question involving calculating the FFT of a signal from a sensor. I have simulated many different scenarios in MATLAB of various noise characteristics involving the signal. I want to take the FFT of a noisy signal. As long as my expected input signal has a higher amplitude than...
  24. blue_leaf77

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  25. A

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  26. J

    Generating 1/f^b colored time series

    Please excuse (and ignore) this if this is not the right place to ask this. I am an ecologists and need to generate a time series with a specific color or frequency spectra. I never learned how Fourier transforms work in class and while I get the gist from reading there are so many subtleties...
  27. A

    Simple Fourier transform problem

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  28. matqkks

    MHB Fourier Series Animation Resources for Motivating Students

    Are there any resources which show Fourier series approximating a given waveform? I am looking for examples which have a real impact on students and provides motivation. I am trying to find something visual but it could be just audio based. Something to start the topic of Fourier series so that...
  29. matqkks

    Fourier Series Animation: Examples to Inspire Students

    Are there any resources which show Fourier series approximating a given waveform? I am looking for examples which have a real impact on students and provides motivation. I am trying to find something visual but it could be just audio based. Something to start the topic of Fourier series so that...
  30. M

    Understanding the Fourier Transform in Solving the Heat Equation

    Hi PF! I was wondering if you could clarify something for me. Specifically, I am solving the heat equation ##u_t = u_{xx}## subject to ##| u(\pm \infty , t ) | < \infty##. Now this implies a solution of sines and cosines times an exponential. Since we have a linear PDE, we may superimpose each...
  31. E

    How does split step Fourier method help four wave mixing?

    Just a question How does solving the nonlinear schrodinger equation using split step Fourier method makes us understand the four wave mixing process in optical fiber ? Any examples on how that happens Thank you
  32. ElijahRockers

    Nth derivative Fourier transform property

    Homework Statement I am given f(t) = e^-|t| and I found that F(w) = ##\sqrt{\frac{2}{\pi}}\frac{1}{w^2 + 1}## The question says to use the nth derivative property of the Fourier transform to find the Fourier transform of sgn(t)f(t), and gives a hint: "take the derivative of e^-|t|" I also...
  33. C

    Fourier Series Homework (Discontinuous Function)

    Homework Statement I have attached a screenshot of the question. I know how to use Fourier's theorem for one function but have no idea how to attempt it with a discontinuous function like this. I tried working out a0 by integrating both functions with the limits shown, adding them and...
  34. D

    Summing series through Fourier series

    Hi, I was trying to sum the Series S(a)=1+exp(-a^2)+exp(-4a^2)+exp(-9a^2)+... According to the notes where I found it it could be done through Fourier Series. I managed to find a relation between S(a) and S(pi/a), and it works, but I can't find S(a) alone. Can anybody help me find a way to do...
  35. matqkks

    MHB Uncovering the Hidden Significance of Fourier Series in Physics and Engineering

    If we have a simple periodic function (square wave) which can be easily written but the Fourier series is an infinite series of sines and cosines. Why bother with this format when we can quite easily deal with the given periodic function? What is the whole point of dealing this long calculation...
  36. matqkks

    Uncovering the Hidden Power of Fourier Series

    If we have a simple periodic function (square wave) which can be easily written but the Fourier series is an infinite series of sines and cosines. Why bother with this format when we can quite easily deal with the given periodic function? What is the whole point of dealing this long calculation?
  37. I

    A question regarding Fourier transform in electron microscop

    I have recorded a micrograph of a 2-D array at a magnification of 43,000x on my DE-20 digital camera, which has a 6.4 μm pixel size and a frame size of 5120 × 3840 pixels. This magnification is correct at the position of the camera. I then compute the Fourier transform of the image. What is the...
  38. A

    Converge pointwise with full Fourier series

    I am working on a simple PDE problem on full Fourier series like this: Given this piecewise function, ##f(x) = \begin{cases} e^x, &-1 \leq x \leq 0 \\ mx + b, &0 \leq x \leq 1.\\ \end{cases}## Without computing any Fourier coefficients, find any values of ##m## and ##b##, if there is any...
  39. C

    Units of Fourier expanded field

    If I write the basic scalar field as $$\phi(x)=\int\frac{d^3k}{(2\pi)^3}\frac1{\sqrt{2E}}\left(ae^{-ik\cdot x}+a^\dagger e^{ik\cdot x}\right),$$ this would seem to imply that the creation and annihilation operators carry mass dimension -3/2. That's the only way I can get the total field...
  40. B

    Exercise proofreading about Fourier Series

    Homework Statement I have solved the following exercise, but I have obtained the half of the correct result! I can't understand where is the problem... ##f(x)=\begin {cases} 0, x \in[-\pi, 0]\\cos x, x \in[0, \pi]\end{cases}## 1) Find the Fourier Series (base: ##{\frac{1}{\sqrt{2 \pi}}...
  41. ElijahRockers

    Fourier series coefficients: proof by induction

    Homework Statement Given f = a0 + sum(ancos(nx) + bnsin(nx)) and f' = a0' + sum(an'cos(nx) + bn'sin(nx)) The sums are over all positive integers up to n. show that a0' = 0, an' = nbn, bn' = -nan Then prove a similar formula for the coefficients of f(k) using induction. Homework EquationsThe...
  42. J

    Fourier COSINE Transform (solving PDE - Laplace Equation)

    I'm trying to solve Laplace equation using Fourier COSINE Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't think so). NOTE: U(..) is the Fourier Transform of u(..) This are the equations (Laplace...
  43. S

    Does a diffraction grating with a shape form fourier image

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  44. C

    Solving for Φ(k) in Quantum Fourier Transform with ψ(x,0)=e^(-λ*absvalue(x))

    Homework Statement Assume ψ(x,0)=e^(-λ*absvalue(x)) for x ± infinity, find Φ(k) Homework Equations Φ(k)=1/√(2π)* ∫e(-λ*absvalue(x))e(-i*k*x)dx,-inf, inf[/B]The Attempt at a Solution , my thought was Convert the absolute value to ± x depending on what of the number line was being...
  45. moriheru

    Fourier coefficients in string theory

    If a function f'(u) has Fourier coefficients anμ and bnμ, by integration one can make new coefficients Anμ ,Bnμ which include constants of integration. My question how can I verify that : Anμcos nτ + Bnμ sin nτ= -i/2 ((Bnμ) -Anμi) einτ- (Bnμ-iAnμ) e-inτ I assume this is the complex form of...
  46. matqkks

    MHB Real-Life Applications of Fourier Series

    Why are Fourier series important? Are there any real life applications of Fourier series? Are there examples of Fourier series which have an impact on students learning this topic. I have found the normal suspects of examples in this field such as signal processing, electrical principles but...
  47. matqkks

    Why are Fourier series important?

    Are there any real life applications of Fourier series? Are there examples of Fourier series which have an impact on students learning this topic. I have found the normal suspects of examples in this field such as signal processing, electrical principles but there must be a vast range of...
  48. fisher garry

    Fourier transform for the wave vector dervation problem

    Below is my walkthrough of a Fourier transform. My problem is that I want to do all the similar steps for a Fourier transform between position x and the wave vector k. That is working on a solution of the maxwell equations. The maxwell equations has many possible solutions for example: $$...
  49. G

    Linking Fourier Transform, Vectors and Complex Numbers

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  50. H

    Help with Eigenvalue Equation and Fourier Transform

    Homework Statement Homework Equations The Attempt at a Solution I did Fourier transform directly to the eigenvalue equation and got Psi(p)=a*Psi(0)/(p^2/2m-E) But the rest, I don't even know where to start. Any opinion guys?
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