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

    Understanding the Fourier Sine Transform: Valid Inputs and Applications

    I am having a hard time understanding how the Fourier sine transform works. I understand that you input a function and you get a function as an output, but I have no idea how certain inputs are even valid. Here is what the book gives for the transform: F(\omega...
  2. H

    How to Choose Values for N, N1, and N2 in MATLAB for Fourier Analysis?

    Hi I'm trying to use the fft function in MATLAB to compute the discrete Fourier transform of a box signal. I'm told to assume that the signal x[n] is periodic with period N and the vector contains one period. x[n]= box [n] I'm am going to use these commands to make my vector...
  3. K

    Finding the fourier series coefficients for cos(pi)x for unit periods

    Hi all, How do I compute the Fourier series coefficients for unit periods for cos(pi)x, the interval is from -1/2 to 1/2. I know the formula but I am getting a wrong answer ?
  4. B

    Fourier Transfrom and expectation value of momemtum operator

    Homework Statement Using <\hat{p}n> = ∫dxψ*(x)(\hat{p})nψ(x) and \hat{p} = -ihbar∂x and the definition of the Fourier transform show that <\hat{p}> = ∫dk|\tilde{ψ}(k)|2hbar*k 2. The attempt at a solution Let n = 1 and substitute the expression for the momentum operator. Transform the...
  5. I

    Fourier integral representation of function

    Homework Statement How can I get to that answer? Homework EquationsThe Attempt at a Solution I'm stuck don't know how to get to the answer that I got from wolframalpha. there is no solution there unfortunately. I know how to get to that denominator w^2-1 but I don't know how to get to...
  6. I

    Fourier Series Convergence for Square Wave Function

    Homework Statement what values does the Fourier series for f(t) converge to if t = 0 and t = 2? Homework Equations The Attempt at a Solution My answers the red rectangles for the even function t=0 >> 1 and t=2 -->1.5 and odd function t=0 >> 0 and t=2 -->1.5 because at t=0 is continuity...
  7. M

    How do I integrate sec(x) sin(nx) over a specific interval?

    So I know that sec(x) has period 2Pi, and it's even so I don't need to figure out coefficients for bn. Let's take the limits of the integral to go from -3/2 Pi to 1/2 Pi. How do I integrate sec(x) sin(nx) dx?! Am I on the right path? PS: I know that this doesn't satisfy the Dirichlet...
  8. J

    Using Fourier to solve a problem.

    I've been kicking this around for a few days. I know I'm overlooking something simple it has been a long time since I've had to do this. I'm trying to sketch the output in the time domain of a 1 kHz square wave passing through a communication channel whose bandwidth is 0 to 10 kHz. I'm trying...
  9. S

    What's the Fourier transform of these functions?

    Homework Statement How can I figure out the Fourier transform of the following: I'd prefer to use tables if at all possible. 1. d(z)=d_{eff}sign[\cos[2\pi z]/\Lambda]) (note this is one function inside another one.) 2. d(z)=d_{eff}(1/2)(sign[\cos[2\pi z]/\Lambda]+1) 3...
  10. I

    Where Does \(\frac{1}{2}x\) Come From at \(k=1\)?

    Homework Statement How did \frac{1}{2}x come from at k=1? Homework EquationsThe Attempt at a Solution because k=1 will make the first term at denominator 2(k-1) = \frac{0}{0}
  11. B

    Fourier Transform for unevenly sampled date

    Dear people, I am trying to analyze data from test bench which consists of a magnetically levitated spindle. We have a rotor/spindle which rotates and moves vertically up and down as it rotates. I measure the angle of rotation and the verticle displacement at a steady rate of 10,000 samples...
  12. I

    Fourier series (2 same functions different inequality signs)

    Homework Statement these two functions will give the same Fourier series? because when I write the graph they look the same? Homework Equations The Attempt at a Solution in the picture thank you
  13. S

    Analysis of vector fields, fourier and harmonics

    Hi I am working on a optimization problem involving vector fields. In order to define a objective function I need a measure (scalar quantity) of some properties of the vector field. The vector field comes from a finite element analysis, that is the vector field is calculated on a discretized...
  14. K

    How many terms are needed for Fourier Isometry to be under 5%?

    Hey guys. I just started a class on Fourier Analysis and I'm having a difficult time understanding this question. Any help would be much appreciated! Homework Statement Verify that the Fourier Isometry holds on [−π, π] for f(t) = t. To do this, a) calculate the coefficients of the orthogonal...
  15. C

    What is the purpose of the exp[-(t^2)/2] term in Fourier transforms?

    I need more help understanding Fourier Transforms. I know that they transform a function from the time domain to the frequency domain and vice versa, but the short cuts to solve them just straight up confuse me. http://www.cse.unr.edu/~bebis/CS474/Handouts/FT_Pairs1.pdf This list of relations...
  16. M

    Discrete Fourier Transform and Hand-waving

    Hi all, I'm reading the following PDF about the DFT: http://www.analog.com/static/imported-files/tech_docs/dsp_book_Ch8.pdf Please see pages 152-153. So the inverse DFT (frequency to space, x[i] = ...) is given on page 152. Then it is claimed that the amplitudes for the space-domain...
  17. C

    Fourier Transform of a Gaussian Pulse

    Homework Statement Consider a Gaussian pulse exp[-(t/Δt)^2/2]exp(i*w*t), where Δt is its approximate pulse width in time. Use the Fourier transform to find its spectrum. Homework Equations The Fourier transform of a Gaussian is a Gaussian. If a Gaussian is given by f(t) = exp(-t^2/2)...
  18. J

    Can the Fourier Transform Be Defined Without the Minus Sign?

    Hi All, Usually the Fourier transform is defined as the one in the Wiki page here (http://en.wikipedia.org/wiki/Fourier_transform), see the definition. My question is can I define Fourier transform as \intf(x)e^{2\pi ix \varsigma}dx instead, i.e., with the minus sign removed, as the...
  19. K

    Resonant Frequencies from Fourier Analysis

    I did Fourier analysis on a set of force data from a vibrating string. In my graph of magnitute and frequency, I'm getting major peaks at 62.1 Hz and 249.0 Hz. There is a tiny blip in the data at 125 Hz and nothing at 186 Hz. I have two questions. Do the peaks at 62.1 and 249 mean that those...
  20. J

    Fourier Series = Re(Power Series)

    Somebody posted a question about Fourier series yesterday that got me thinking about an argument I heard some time before. If we have a (complex-valued) analytic function f, then any closed loop in the complex plane will be mapped by f to another closed loop. (If the loop doesn't enclose any...
  21. R

    Fourier Transform of Undefined Function

    Homework Statement I'm trying to derive the result on slide 1 of this link: http://www.physics.ucf.edu/~schellin/teaching/phz3113/lec13-3.pdf Unfortunately, I'm not sure how to integrate the Fourier transform when my u(x,t) function is undefined. Could someone help me get the...
  22. D

    How to Find the Inverse Fourier Transform of A(r', ω)?

    Homework Statement I have been given the following: A(r', ω) = μ/4∏*∫ J(r', ω)*exp(-j*k*R)/R dV' And am being asked to find the inverse FT of A(r', ω) Homework Equations Given that k = ω/c and R = |r - r'| The Attempt at a Solution I know what the inverse FT transform is, but...
  23. I

    Fourier Series Representation of Signals (Proof)

    Hi guys, I was studying the proof below and just can't figure out the the first highlighted step leads to the second and I was wondering if you guys can help me to fill that in. (: Thank you so much for your help in advance guys!
  24. M

    Discrete Fourier Transform on even function

    The DCT of an even function is comprised of just cosine coefficients, correct? I'm playing around in MATLAB and I came up with a simple even function 1.0000 0.7500 0.5000 0.2500 0 0.2500 0.5000 0.7500 1.0000 0.7500 0.5000 0.2500 0 0 0 0...
  25. N

    Grating Spacing With Fourier Optics

    Just ahead of time, no this is not related to homework or coursework in my case. I am a TA for an optics lab and need to know it so I can help the students in my class. I am trying to find an expression for the x-spacing for a grating imaged using the 4f method, which is a grating that is 1...
  26. stripes

    Fourier series coefficients and convergence

    Homework Statement Third question of the day because this assignment is driving me crazy: Suppose that \left\{ f_{k} \right\} ^{k=1}_{\infty} is a sequence of Riemann integrable functions on the interval [0, 1] such that \int ^{0}_{1} |f_{k}(x) - f(x)|dx \rightarrow 0 as k \rightarrow...
  27. H

    Computing the Hilbert transform via Fourier transform

    I know the result: \widehat{H(f)}=i\textrm{sgn}\hspace{1mm}(k)\hat{f} I thought I could use fft, and ifft to compute the transform easily, is there a MATLAB command for sgn? Mat
  28. W

    Where Did the 2's in the Denominator Go in the Fourier Analysis Proof?

    Hi. Just going through my notes from the last lecture I remember having some troubles understanding the proof the lecturer gave for the following theorem: Suppose that f is Riemann integrable and that all its Fourier coefficients are equal to 0, then f(x)=0 at all points of continuity. The...
  29. D

    Fourier series: relation of coefficients

    Hi, The Fourier series can (among others) expressed in terms of sines and cosines with coefficients a_n and b_n and solely by sines using amplitudes A_n and phase \phi_n. I want to express the latter using a_n and b_n. Using a_n = A_n \sin(\phi_n) \\ b_n = A_n \cos(\phi_n) I...
  30. jegues

    Fourier Transform and Modulation

    Homework Statement See figure attached. Homework Equations The Attempt at a Solution See pdf attached for my attempt at the solution. I'm a little confused as to how to draw the phase spectrum for y(t). Would it simply be a line equation of, -\frac{\pi}{6000}f \pm...
  31. Jalo

    Fourier transform of a even/odd function

    Homework Statement Is the Fourier transform of a even/odd function also even/odd ? Homework Equations The Attempt at a Solution So far this result seems to be true. I can't find a confirmation however... Thanks ahead. Daniel.
  32. M

    Fourier transform of the hyperbolic secant function

    Homework Statement Hi there! I'm just trying to figure out the Fourier transform of the hyperbolic secant function... I already know the outcome: 4\sum\ ((-1)^n*(1+2n))/(ω^2*(2n+1)^2) But sadly, I cannot figure out how to work round to it! :( maybe one of you could help me... Homework...
  33. L

    Discrete Fourier transform mirrored?

    Why does a discrete Fourier transform seems to produce two peaks for a single sine wave? It seems to be the case that the spectrum ends halfway through the transform and then reappears as a mirror image; why is that? And what is the use of this mirror image? If I want to recover the frequency...
  34. Jalo

    What is the Cosine Fourier Transform of an Exponential Function?

    Homework Statement Find the cosine Fourier transform of the function f(t)=e-at Homework Equations The Attempt at a Solution F(w)=(2/π)0.5∫dt e-atcos(wt) The integral is from 0 to +∞ Using euler's formula I got the result F(w)=(2/π)0.5( eit(w-a)/i(w-a) - e-it(w+a)/i(w+a)...
  35. TrickyDicky

    Fourier transform as (continuous) change of basis

    Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform. My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where...
  36. A

    Fourier series representation of delta train

    The Fourier series of a delta train is supposedly (1/T) + (2/T ) Ʃcos(nωt) ... where T is period and ω=2*Pi/T ...but when I plot this, it doesn't give me just a spike towards positive infinity, but towards negative infinity as well (see attached pic), so this does not seem to converge to the...
  37. D

    MHB Solve for A_n: Fourier Coefficient $$u_y(x,\pi) = 0$$

    $$ u_y(x,\pi) = \frac{x}{\pi} + \sum_{n = 1}^{\infty}nB_n\sin xn\cosh\pi n = 0. $$ How can I solve for $A_n$ here?
  38. maistral

    Crude Fourier Series approximation for PDEs.

    Is there a way to "crudely" approximate PDEs with Fourier series? By saying crudely, I meant this way: Assuming I want a crude value for a differential equation using Taylor series; y' = x + y, y(0) = 1 i'd take a = 0 (since initially x = 0), y(a) = 1, y'(x) = x + y; y'(a)...
  39. F

    First nonzero terms of Fourier sine series

    Homework Statement Consider the function φ(x) ≡ x on (0, l). Find the sum of the first three (nonzero) terms of its Fourier sine series. Reference: Strauss PDE exercise 5.1.3 Homework Equations The Attempt at a Solution I have found the coefficient without difficulty, it is...
  40. A

    Obtain a fourier series equation from a given graph

    The problem statement: Obtain a Fourier Series Expression Form from the above graph: I can't post the graph, so I will describe it. It's a periodic function with period 1 and magnitude 5. The equation is the following: f(x) = -x, -1/2<x<1/2 I'm really stuck at trying to obtain a series...
  41. Jalo

    Fourier transform of a function

    Homework Statement a) Find the Fourier transform of the function f(x) defined as: f(x) = 1-3|x| , |x|<2 and 0 for |x|>2 b) Find the values of the inverse Fourier transform of the function F(k) obtained in a) Homework Equations F(k) = \frac{1}{\sqrt{2π}}\int f(t) eikx dx f(x) =...
  42. J

    Fourier representation of a random function

    Consider continuous function x(t), which has zero time average: \lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2} x(t)\,dt = 0 and exponentially decaying autocorrelation function: \lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2} x(t)x(t-\tau)\,dt = C_0e^{-\gamma |\tau|}...
  43. J

    Maxwell Equations and Fourier Expansions

    Homework Statement The field E(r,t) can be written as a Fourier expansion of plane waves E(r,t)=∫E(k,w)e^{i(kr-wt)}d^{3}kdw with similar expansions for other fields. Need to show the derivation of kXE(k,w)=wB(k,w) from Faraday's law ∇XE(r,t)=-∂B(r,t)/∂t and also the derivation of...
  44. J

    Why complex discrete Fourier transform?

    I've been trying to figure out why it's standard to use complex discrete Fourier transforms instead of just the real version. It's discussed a bit here. http://dsp.stackexchange.com/questions/1406/real-discrete-fourier-transform As far as I can tell there's a hypothetical efficiency...
  45. F

    How can the Fourier Integral Theorem be used to evaluate improper integrals?

    Homework Statement Show that integral from 0 - > infinity (cos(alpha*x)/(alpha^2 + 1))dalpha = (pi/2)exp(-x) Homework Equations The Attempt at a Solution cos(alpha*x) = (1/2)(exp(i*alpha*x)+exp(-i*alpha*x)) Really don't know where to go from here.
  46. fluidistic

    Solving Fourier Coefficients for A_m & B_m

    Homework Statement I've reached a relation but then I need to obtain the coefficients ##A_m## and ##B_m##'s, those are the only unknowns. Here's the expression: ##\sum _{m=0}^\infty a^m [A_m \cos (m \theta ) + B_m \sin (m \theta )]=T_0\sin ^3 \theta##. Homework Equations Fourier...
  47. A

    Fourier Transform: Limit in Infinity of Exponential Function

    In calculating some basic Fourier transform I seem stumble on the proble that I don't know how to take the limit in infinity of an exponentialfunction with imaginary exponent. In the attached example it just seems to give zero but I don't know what asserts this property. I would have thought...
  48. fluidistic

    Infinite series, probably related to Fourier transform?

    Homework Statement A function f(x) has the following series expansion: ##f(x)=\sum _{n=0}^\infty \frac{c_n x^n}{n!}##. Write down the function ##g(y)=\sum _{n=0}^\infty c_n y^n## under a closed form in function of f(x). Homework Equations Not sure at all. The Attempt at a Solution...
  49. D

    PDE involving Fourier Sine Series

    Homework Statement Solve: PDE: 9Uxx=Utt BC: U(0, t) = U(∏, t) = 0 IC: U(x, 0) = sin4x + 7sin5x Ut = x 0 < x < ∏/2 = ∏ - x ∏/2 < x < ∏ Homework Equations Fourier Sine Series and Cosine Series Equations The Attempt at a Solution...
  50. M

    Exponential fourier series expansion

    Hey, thanks for taking the time to look ay my post (: I have attached a file which shows the question I am stuck on, and my attempt at working it out. My problem is the answer I get, is different to what my Lecturer gets (shown in the attachment). He worked it out a different way to me, he...
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