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. P

    Fast Fourier Transform (FFT) power spectrum angle

    Dear Physics Buddies, How are well all, okay I hope. I was wondering if I might browse all your infinite intellects and ask you a very simple question. I am working with some medical images in MATLAB and my collaborators would like to know the orientation of the fibres that it contains...
  2. Choisai

    Focused diffraction and Fourier transform

    After searching on the web and reading a bit, I found that lenses can perform Fourier transform. All you need to do is put a transparant object in front of it, like a transparant sheet with black stripes on it and a screen behind the lens(so basically a 4f setup). The lens will then perform a...
  3. A

    How to Approach This Challenging Fourier Transform Problem?

    Homework Statement Homework Equations here is list of Fourier transforms: http://uspas.fnal.gov/materials/11ODU/FourierTransformPairs.pdfThe Attempt at a Solution so I know the solution but I don't know how to get it. Here is what I think to do: the ramp function r(t) and the rect pτ(t). I...
  4. J

    Representing a real periodic valued function with Fourier series

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  5. B

    Fourier transform vs Inner product

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  6. Choisai

    What are Fourier transforms of optics?

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  7. D

    MHB Magnitude Fourier transform lowpass, highpass, or bandpass

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  8. J

    An analytic solution for a fourier transform

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  9. L

    Fourier transform. Impulse representation.

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  10. G

    The Maxwell equation and the Fourier Spectrum of an Electric Field

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  11. Y

    Fourier Series/Transformations and Convolution

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  12. D

    Fourier expansion of boolean functions

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  13. R

    Is there any symmetry I can use to find this Fourier sine series?

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  14. R

    Using a Fourier Cosine Series to evaluate a sum

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  15. E

    How Does the Raman Effect Impact SSFM in Nonlinear Fiber Optics?

    Hello everyone I am doing my own split step Fourier method (SSFM)code on Matlab to solve the Nonlinear schrodinger equation in nonlinear fiber optics My problem is that in the Nonlinear operator we just multiply it with the initial pulse during SSFM without doing any Fourier transform not...
  16. U

    Solving Fourier Inverse: Integrals and Techniques for Exam Revision

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  17. I

    Fourier transform of the linear function

    Hello, I was wondering if one can give meaning to the Fourier transform of the linear function: \int_{-\infty}^{+\infty} x e^{ikx}\, dx I found that it is \frac{\delta(k)}{ik} , does this make sense?
  18. binbagsss

    Fourier Transforms - The Convolution Theorem.

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  19. S

    MHB Calculating Fourier Cosine Series of cos(x) from 0 to \pi

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  20. T

    Extracting a fourier series of pure tones from a signal

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  21. A

    Fourier transform of sinusoidal functions

    Homework Statement Homework Equations sinc(x) = \frac{sin(x)}{x} The Attempt at a Solution bit unsure how to get started?? i know transform of rectangular pulse pτ(t)=τ*sinc(τω/2∏) also that sin(ωt)= ejωt-e-jωt / (2) I could also probably sketch sinc(t/2∏), if that helps.
  22. binbagsss

    Fourier Transformation - Convolution quick question

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  23. B

    Fourier transform of multivalued functions

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  24. N

    Is This Fourier Series an Odd or Even Function with a Period of 4s?

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  25. C

    Hilbert, Banach and Fourier theory

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  26. B

    Fourier transform and the frequency domain

    I understand that the Fourier transform maps one function onto another. So it is a mapping from one function space onto another. My question is, why is it often referred to as a mapping from time domain to the frequency domain? I don't understand why the image of the Fourier transform...
  27. T

    How to Approach the Fourier Transform of an Annulus?

    Hi guys, I've been using this site for a while now, but this is going to be my first post. I want to pick your brains to get some insight on this problem I'm tackling. I'm trying to take a Fourier Transform of a function. My function is a function of (r,phi) and it is a piecewise function...
  28. N

    Fourier Transform of x(t) and g(t) with Product Calculation

    Homework Statement x(t) = 5cos(2*pi*1000*t) and g(t) = ∑ from n=-infinity to infinity delta(t-n/10000) find Fourier transform of x(t) and g(t) and the product of the two The Attempt at a Solution x(w) = 5*sqrt(pi/2) [delta(w-2000pi)+delta(w+2000pi)] g(w) = 1 so would the...
  29. N

    Fourier series / calculate power over resistor

    Homework Statement V(t) = 4 for 0<t< 1 and 0 for 1<t<3 and repeats itself for all t (negative and positive) Find the first 5 harmonics of the Fourier series in cosine form and find the power if this is the voltage over 100 ohm resistor The Attempt at a Solutionpower = d_dc ^2 / R + .5sum...
  30. J

    Other definition for fourier series

    Is correct to define Fourier series like: f(t)=\sum_{k=0}^{\infty}a_k \cos \left (\frac{2 \pi k t}{T} \right ) + b_k \sin \left (\frac{2 \pi k t}{T} \right ) Where ak and bk: a_k=\frac{1}{T} \int_{-T}^{+T} f(t) \cos \left (\frac{2 \pi k t}{T} \right ) dt b_k=\frac{1}{T}...
  31. M

    How do we find the Fourier coefficient for a cosine term in a Fourier series?

    hey pf! okay, so if you've studied PDEs you know the value of a Fourier series, and the difficulty of determining a Fourier coefficient. my question relates to finding this coefficient. briefly, i'll define a Fourier series as f(x)=\sum_{n=0}^{\infty} A_n\cos\frac{n\pi x}{L}+B_n\sin\frac{n\pi...
  32. N

    How Do You Calculate Fourier Series Coefficients for a Piecewise Function?

    Homework Statement x(t) = 4 for 0 <x<1 and 0 otherwise and this process repeats for all values including negative. find X_0 and X_n and find the first 6th harmonics of the Fourier series in cosine form Homework Equations The Attempt at a Solution x_0 = 4/3 x_n =...
  33. I

    Fourier transform of a functional

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  34. V

    Fourier Transform of Distribution

    Hi, I hope somebody can help me with this one. Homework Statement Compute the Fourier Transform of the distribution x-a Homework Equations The Fourier Transform of a distribution is just the distribution evaluated with the Fourier Transform of a test function.The Attempt at a Solution See...
  35. A

    Fourier Transform: Determining Constant in Convolution

    I have an exercise with a function of the form: h(t) = f(t)g(t) and f(t) and g(t) both have discrete Fourier series, which implies that h does too. I want to find the Fourier series of h, so my teacher said I should apply the convolution theorem which would turn the product above into a...
  36. U

    How Do You Solve Differential Equations Using Fourier Transform?

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  37. A

    How Do Delta Functions Simplify the Fourier Transform in Quantum Mechanics?

    Homework Statement The exercise is a) in the attached trial. I have attached my attempt at a solution, but there are some issues. First of all: Isn't the example result wrong? As I demonstrate you get a delta function which yields the sum I have written (as far as I can see), not the sum...
  38. M

    Exploring Alternating Solutions in Fourier Series

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  39. dexterdev

    Which of the signals is not the result of fourier series expansion?

    Homework Statement Which of the signals is not the result of Fourier series expansion? options : (a) 2cos(t) + 3 cos(3t) (b) 2cos(\pit) + 7cos(t) (c) cos(t) + 0.5 Homework Equations Dirichlet conditionsThe Attempt at a Solution From observation, I thought all are periodic and so must be...
  40. P

    Problem in Convolution integral by fourier transformation

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  41. D

    Solving Real Valued Fourier Coefficients

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  42. J

    Airy integral by Fourier transform?

    http://calclab.math.tamu.edu/~fulling/m412/f07/airywkb.pdf Can someone walk me through this derivation of the Airy integral by Fourier transform? I have tried it but failed
  43. E

    Is the Discrete Fourier Transform a Unitary Transformation?

    I'm trying to prove that the discrete form of the Fourier transform is a unitary transformation So I used the equation for the discrete Fourier transform: ##y_k=\frac{1}{\sqrt{N}}\sum^{N-1}_{j=0}{x_je^{i2\pi\frac{jk}{N}}}## and I put the Fourier transform into a N-1 by N-1 matrix form...
  44. K

    Fourier Series for a Square-wave Function

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  45. D

    Fourier Series from m=1 to infinity

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  46. S

    Fourier Transforms: Proving Proportionality

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  47. N

    Do Fourier transforms always converge to 0 at the extreme ends?

    From -infinity to infinity at the extreme ends do Fourier transforms always converge to 0? I know in the case of signals, you can never have an infinite signal so it does go to 0, but speaking in general if you are taking the Fourier transform of f(x) If you do integration by parts, you get a...
  48. P

    Find the Fourier series solution to the differential equation

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  49. N

    Fourier series of a periodic function not starting at x=-L

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  50. N

    Why Is the Coefficient Doubled in the Fourier Sine Series Calculation?

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