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. J

    Question about Fourier Series/Transform

    Hi guys, I'm now studying Fourier series/transform for representing signals in the frequency domain. I'm having a bit of a hard time getting the gist of it. Right now I'm using the book "signals and systems" (oppenheim) because that's the one my teacher uses. My problem is this: both the book...
  2. M

    Applying Convolution to a PDE with a Fourier Transform

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  3. evinda

    MHB How can we extend the solution of an initial value problem using Fourier series?

    Hello! (Wave) The following problem shall show the way with which the Fourier series can be used for the solution of initial value problems.Find the solution of the initial value problem $$y''+ \omega^2 y=\sin{nt}, y(0)=0, y'(0)=0$$ where $n$ is a natural number and $\omega^2 \neq n^2$. What...
  4. 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 +...
  5. dumbdumNotSmart

    Heat equation integral - Fourier Series coefficient is zero

    Homework Statement WE have a thermally insulated metallic bar (from enviroment/surroundings) . It has a temperature of 0 ºC. At t=0 two thermal sources are applied at either end, the first being -10 ºC and the second being 10 ºC. Find the equation for the temperature along the bar T(x,t), in...
  6. Vitani11

    How do you find the fourier expansion coefficients?

    Homework Statement I need to expand this piecewise function f(x) = h for a<x<L and f(x) = 0 for 0<x<a. I am told that this is a square wave so ao and an in the expansion are 0 (odd function). Therefore I only need to worry about bn. The limits on the integral are from a to L, but what about the...
  7. entropy1

    I Why is momentum the Fourier transform of position?

    Apart from the fact that it is, what is the physical significance of the fact that you can get the momentum distribution of a particle by taking the Fourier transform of its position distribution?
  8. 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...
  9. TheBigDig

    Need Help with Fourier Coefficients for 1-t Function?

    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 it...
  10. Vitani11

    Proving inverse Fourier transform of 1/(1+x^2) = 1/(1+x^2)

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  11. R

    Fourier Series of Sawtooth Wave from Inverse FT

    Homework Statement I want to find the Fourier series of the sawtooth function in terms of real sine and cosine functions by using the formula: $$f_p (t)=\sum^\infty_{k=-\infty} c_k \exp \left(j2\pi \frac{k}{T}t \right) \tag{1}$$ This gives the Fourier series of a periodic function, with the...
  12. M

    I Truncated Fourier transform and power spectral density

    Hello, I am trying to find an expression for the signal-to-noise ratio of an oscillating signal on top some white noise. In particular I would like to know how the SNR scales with the integration time. It is well known that during some integration time ##T##, the SNR increases as ##T^{1/2}##...
  13. V

    Fourier transform of periodic potential in crystal lattice

    Homework Statement Homework Equations I'm not sure. The Attempt at a Solution I started on (i) -- this is where I've gotten so far. I am asked to compute the Fourier transform of a periodic potential, ##V(x)=\beta \cos(\frac{2\pi x}{a})## such that...
  14. chikou24i

    B Fourier series of a step function

    Hello, can we make a Fourier series expansion of a (increasing or decreasing) step function ? like the one that I attached here. I just want to know the idea of that if it is possible.
  15. Marcus95

    Fourier Series Coefficient Symmetries

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  16. T

    Fourier Series of a Piecewise Function

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  17. D

    Fourier/heat problem involving hyperbolic sine

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  18. L

    I Wave equation solution using Fourier Transform

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  19. G

    Evaluate Fourier series coefficients and power of a signal

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  20. M

    Calculating Magnetic Field from FFT Amplitude

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  21. needved

    Help with found Fourier complex series of e^t

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  22. P

    Find equation obeyed following Fourier transform

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  23. D

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  24. D

    How Does Time Affect the Displacement of a Plucked Violin String?

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  25. D

    Fourier Transform of Polarization

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  26. bradzyc

    Fourier Series: Stamping Machine Positioning Function

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  27. N

    Change of variables in Heat Equation (and Fourier Series)

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  28. R

    Mathematica How to plot several terms in a Fourier series

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  29. J

    Fourier series expansion. Find value of a term in expansion

    Homework Statement Fourier series expansion of a signal f(t) is given as f(t) = summation (n = -inf to n = +inf) [3/(4+(3n pi)2) ) * e j pi n t A term in expansion is A0cos(6 pi ) find the value of A0 Repeat above question for A0 sin (6 pi t) Homework Equations Fourier expansion is summation...
  30. C

    Fourier Analysis: Inspect Waveforms in FIGURE 1

    Homework Statement b) state by inspection (i.e. without performing any formal analysis) all you can about each of the periodic waveforms shown in FIGURE 1 in terms of their Fourier series when analysed about t = 0 Homework Equations 3. The Attempt at a Solution Hi could someone please be...
  31. E

    Finding Fourier Series of f(x)=√(x2) -pi/2<x<pi/2

    Homework Statement Find the Fourier series of the function f(x) =√(x2) -pi/2<x<pi/2 , with period pi Homework EquationsThe Attempt at a Solution I have tried attempting the question, but couldn't get the answer. uploaded my...
  32. mnb96

    A What Conditions Determine a Zero Measure Set in Fourier Transforms?

    Hello, for a function f∈L2(ℝ), are there known necessary and sufficient conditions for its Fourier transform to be zero only on a set of Lebesgue measure zero?
  33. Gopal Mailpalli

    Fourier Series for Periodic Functions - Self Study Problem

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  34. arpon

    I Understanding the Complex Conjugate Property in Fourier Transform

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  35. arpon

    I Dirac Delta using Fourier Transformation

    We know, $$\delta(x) = \begin{cases} \infty & \text{if } x = 0 \\ 0 & \text{if } x \neq 0 \end{cases}$$ And, also, $$\int_{-\infty}^{\infty}\delta(x)\,dx=1$$ Using Fourier Transformation, it can be shown that, $$\delta(x)=\lim_{\Omega \rightarrow \infty}\frac{\sin{(\Omega x)}}{\pi x}$$ Let's...
  36. Pouyan

    Fourier series and differential equations

    Homework Statement Find the values of the constant a for which the problem y''(t)+ay(t)=y(t+π), t∈ℝ, has a solution with period 2π which is not identically zero. Also determine all such solutions Homework Equations With help of Fourier series I know that : Cn(y''(t))= -n2*Cn(y(t)) Cn(y(t+π)) =...
  37. binbagsss

    Simple Fourier coefficient exponential

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  38. F

    I Question about Fourier transformation

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  39. binbagsss

    Relationships between Fourier coefficients

    Homework Statement I have ## f(t) = \sum\limits^{\infty}_0 a_{n} e^{2 \pi i n t} ## [1] and ## g(t) = \sum\limits^{\infty}_0 b_{n} e^{ \pi i n t} ## [2] I want to show that ##b_n = a _{2n} ## Homework Equations see above. The Attempt at a Solution [/B] So obviously you want to use the...
  40. F

    MATLAB Question about Fourier transformation in Matlab

    Hello everybody. I am triying to calculate a band-pass filter using the Fourier transform. I have a vector with 660 compomponents; one for each month. I am looking for a phenomenon which has a periodicity between 3 and 7 years (it's el niño, on the souhtern pacific ocean). I want to make zero...
  41. Svein

    Insights Using the Fourier Series To Find Some Interesting Sums - Comments

    Svein submitted a new PF Insights post Using the Fourier Series To Find Some Interesting Sums Continue reading the Original PF Insights Post.
  42. redtree

    I Fourier transform of the components of a vector

    Given the Fourier conjugates ##\vec{r}## and ##\vec{k}## where ##\vec{r} = [r_1,r_2,r_3]## and ##\vec{k} = [k_1,k_2,k_3]## , are ##r_1## /##k_1##, ##r_2##/##k_2##, ##r_3##/##k_3## also Fourier conjugates, such that: ##\begin{equation} \begin{split} f(\vec{r})&=[f_1(r_1),f_2(r_2),f_3(r_3)] \\...
  43. CricK0es

    Find the Fourier series for the periodic function

    < Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown > Hi all. I'm completely new to these forums so sorry if I'm doing anything wrong. Anyway, I have this question... Find the Fourier series for the periodic function f(x) = x^2 (-pi < x < pi)...
  44. MAGNIBORO

    Complex Fourier Series Problem

    Hi, I'm starting to studying Fourier series and I have troubles with one exercises of complex Fourier series with f(t) = t: $$t=\sum_{n=-\infty }^{\infty } \frac{e^{itn}}{2\pi }\int_{-\pi}^{\pi}t\: e^{-itn} dt$$ $$t=\sum_{n=-\infty }^{\infty } \frac{cos(tn)+i\, sin(tn)}{2\pi...
  45. Conservation

    What is the inverse Fourier transform of e^3iωt for solving ut+3ux=0?

    Homework Statement Solve ut+3ux=0, where -infinity < x < infinity, t>0, and u(x,0)=f(x).Homework Equations Fourier Transform where (U=fourier transform of u) Convolution Theorem The Attempt at a Solution I've used Fourier transform to get that Ut-3iwU=0 and that U=F(w)e3iwt. However, I'm...
  46. Captain1024

    Find Fourier transform and plot spectrum by hand & MATLAB

    Homework Statement Link: http://i.imgur.com/JSm3Tqt.png Homework Equations ##\omega=2\pi t## Fourier: ## Y(f)=\int ^{\infty}_{-\infty}y(t)\mathrm{exp}(-j\omega t)dt## Linearity Property: ##ay_1(t)+by_2(t)=aY_1(f)+bY_2(f)##, where a and b are constants Scaling Property...
  47. R

    Using the Fourier transform to interpret oscilloscope data

    We have a waveform that is composed of several waves, maybe something like this: If we Fourier transform the graph we get something like this: My question is, does the value of the largest column represent the peak to peak voltage of the waveform pictured above?
  48. N

    Understanding the Variables of the FT and DTFT: Intuition and Differences

    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...
  49. J

    Fourier transform: signal with filter

    Hi Guys, I'm having trouble with the following: A finite-time signal is the result of a filter G(t) applied to a signal. The filter is simply “on” (1) for t ∈ [0,T] and off (“0”) otherwise. If x(t) is the signal, and x(ω),its Fourier transform, compute the Fourier transform of the filtered signal...
  50. ShayanJ

    A Fourier transform and translational invariance

    Can anyone explain what does the author mean by the statement below? page 27 of this paperI don't understand the relation between the Fourier transform and translational invariance. Thanks
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