Fourier Definition and 1000 Threads

  1. DavideGenoa

    Eigenvectors of Fourier transform operator #F:L^2\to L^2#

    Hi, friends! In order to find an orthogonal basis of eigenvectors of the Fourier transform operator ##F : L_2(\mathbb{R})\to L_2(\mathbb{R}),f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x## for Euclidean separable space ##L_2(\mathbb{R})##, so that ##F## would be represented by...
  2. T

    Fourier Series for f(x) = sin(3x/2) and Evaluating Series for (1/(4n^2-9))^2

    Homework Statement Evaluate following series: \sum_{n=1}^\infty \frac{1}{(4n^2-9)^2} by finding the Fourier series for the 2\pi-periodic function f(x) = \begin{cases} sin(3x/2) & 0<x<\pi \\ 0 & otherwise \end{cases} Homework Equations a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}...
  3. S

    Why didn't I see a peak at half the frequency in my FFT analysis of two waves?

    I am carrying out FFT analysis to compare two waves. One looks very much like a sine wave the other has an extra dip occurring at half the frequency of the main wave. I have been thinking around how I might expect this to show up in the FFT analysis. At first i was expecting to see a smaller...
  4. I

    Find the following fourier series in trigonometric form

    Homework Statement Find the following Fourier series in trigonometric form. Homework Equations $$y(t)=a_0+\sum\limits_{n=1}^{\infty} a_n cos(n\omega_{0}t)+b_n sin(n\omega_{0}t)$$ The Attempt at a Solution The graph above is represented by the function: $$ x(t) = \left\{ \begin{array}{ll}...
  5. D

    Fourier Transform of product of heaviside step function and another function

    Homework Statement A damped harmonic oscillator is driven by a force of the form f(t)=h(t) t^2 Exp(-t), where h(t) is a Heaviside step function. The Oscillator satisfies the equation x''+2x'+4x=f(t). Use pencil-and-paper methods involving Fourier transforms and inverse transforms to find the...
  6. grandpa2390

    Calculate the Fourier Transform using theorems

    Homework Statement Using a theorem (state which theorem you are using and give the formula), Calculate the Fourier Transform of 1. rect(x)triangle(x) 2.cos(pi*x)sinc(x) 3.rect(x)exp(-pi*x^2) 4.sinc(x)sin(pi*x) 5. exp(-pi*x^2)cos(pi*x) Homework Equations not sure what theorem to use for the...
  7. G

    Fourier Series (Half-range expansion)

    Homework Statement Homework Equations The Attempt at a Solution I don't really understand why my solution is wrong as I think I have substituted everything in correctly.. Is it okay if anyone can help me take a look at my solution? Thank you. :) My solution: (Only bn) My...
  8. R

    Inverse Fourier transform of decaying function

    Homework Statement Find the inverse Fourier transform of X(ejw = 1/(1-ae-jw)2 using the convolution theorem. Homework EquationsThe Attempt at a Solution I tried finding the partial fraction coefficients but without success.
  9. Math Amateur

    MHB Proof of the Irrationality of e - attributed to Joseph Fourier

    (NOTE: Maybe this post belongs in the Number Theory Forum? Apologies if it is wrongly located!) I am reading Julian Havil's book, "The Irrationals: The Story of the Numbers You Can't Count On" In Chapter 4: Irrationals, Old and New, Havil gives a proof of the irrationality of e which was...
  10. DavideGenoa

    Identical Fourier coefficients of continuous ##f,\varphi\Rightarrow f=\varphi##

    Hi, friends! Let ##f:[a,b]\to\mathbb{C}## be an http://librarum.org/book/10022/173 periodic function and let its derivative be Lebesgue square-integrable ##f'\in L^2[a,b]##. I have read a proof (p. 413 here) by Kolmogorov and Fomin of the fact that its Fourier series uniformly converges to a...
  11. Hanyu Ye

    How to compute multidimensional inverse Fourier transform

    Hello, everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field: \tilde{p}=-\frac{i}{{{k}^{2}}}\mathbf{F}\centerdot \mathbf{k} where i is the imaginary unit...
  12. moriheru

    Books on Fourier Transforms & Other Transformations

    Any books on Fourier transforms and other transformations? (Thanks, for any help)
  13. P

    Find the fourier sine series of cosine.

    Homework Statement Hi, so I am doing some past exam papers and there was this question; Homework EquationsThe Attempt at a Solution a0 and an both are equal to zero, this leaves only bn. Since you can only use the sine series for an odd function, and cos(t) is even, does this mean i have to...
  14. S

    Did I set this Fourier series up correctly?

    If you take the Fourier series of a function $f(x)$ where $0 < x < \pi$, then would $a_{0}$, $a_{n}$, and $b_{n}$ be defined as, $a_{0} = \displaystyle\frac{1}{\pi}\int_{0}^{\pi}f(x)dx$ $a_{n} = \displaystyle\frac{2}{\pi}\int_{0}^{\pi}f(x)\cos(nx)dx$ $b_{n} =...
  15. V

    Fourier Transform Real Function Wave Equation

    Hello, I hope somebody can help me with this. 1. Homework Statement I am supposed to show that if there is a function \phi(x,t) which is real, satisfies a linear wave equation and which satisfies \phi(x,0)=0 for x<0 then the Fourier Transform \tilde{\phi}(k) of \phi(x,0) is in the lower...
  16. T

    Fourier transform of function of a complex variable

    Can anyone point me to some material on applying the Fourier transform to the case of an analytic function of one complex variable? I've tried to generalize it myself, but I want to see if I'm overlooking some important things. I've started by writing the analytic function with u + iv where u...
  17. M

    Fourier Series For Function Not Centred at Zero

    Homework Statement I was working on a problem where I had been given a differential equation to be solved using separation of variables. Two coordinates: a time coordinate and a single spatial coordinate (1-D problem). Homework Equations The domain for the spatial part was [0, L]. Given...
  18. L

    Find fourier series of wave function

    Homework Statement Find Fourier series of f(x) = Acos(\pix/L) I know how to do this, I just don't know the value of L. If it's equal to \lambda/2, then I know the solution. But the question does not specify the value of L. L is just the length of the entire wave that I'm working with, right? If...
  19. S

    How Do Fourier Integral Operators Work in Mathematical Analysis?

    Hi everybody! I'm studying the Fourier integral operators but I can't resolve a pass. I'm considering the following operator: $$Au(x)=\frac{1}{{(2\pi h)}^{n'}}\int_{\mathbb{R}_y^m\times\mathbb{R}_\theta^{n'}} e^{i\Psi(x,y,\theta)/h}a(x,y,\theta,h)u(y)\, dy\, d\theta$$ where $$Au\in C^0...
  20. R

    Calculating Expectation Value of Momentum with Fourier Transform

    we have a wavefunction \psi (x) the question asks for \psi (p) and says to use this to calculate the expectation value of momentum. The problem is the expectation value of momentum is integrated over dx so after transforming how do you get the integral to be over dp? thanks for any help with...
  21. J

    Understanding the Significance of Fourier Analysis in Signal Processing

    Ok so this isn't a homework question per se, but I'm currently writing a report on Fourier Analysis but a bit stuck as to what the results can actually help with. I realized that I don't grasp how a Fourier Transform can be used. In the experiment we recorded the signal created by a remote...
  22. P

    Quarter period symmetry in Fourier series

    Suppose we have some function f(x) with period L. My book states that if it is even around the point x=L/4, it satisfies f(L/4-x)=-f(x-L/4), whilst if it is odd it satisfies f(L/4-x)=f(x-L/4). Then we define s=x-L/4 so we have for the function to be odd or even about L/4 that f(s)=±f(-s)...
  23. A

    Fourier Series without complex

    Homework Statement The problem is finding the Fourier series of f(t) = e^(-t) from [0,2] where T=2 and without using complex solution. [/B]Homework Equations f(t) = a0/2 + ∑ (anCos(nωt) +bnsin(nωt) NOT using f(t) = ∑dne^(inωt)The Attempt at a Solution I tried once but got completely wrong...
  24. C

    Fourier Series for f on the Interval [-π, π) | Homework Statement

    Homework Statement Define ##f : [−π, π) → \mathbb R ## by ##f(x)## = ##−1## if ##− π ≤ x < 0##, ##1## if ##0 ≤ x < π.## Show that the Fourier series of f is given by ##\frac{4}{π} \sum_{n=0}^\infty \frac{1}{(2k+1)} . sin(2k+1)x##Homework Equations The Fourier series for ##f## on the interval...
  25. K

    Fourier transformation on discrete function

    Hi there, I am reading a material on the application of Fourier transformation in physics. One application is to transform the position-dependent function to k-dependent function, i.e.## F(k) = FFT[f(x)]## We know that the in physics, the wavenumber could be written in momentum as...
  26. B

    MHB Fourier Transform to find sidebands with 2 different frequencies

    Hi Folks, I need to evaluate the following function f(t)=A[1+B \cos(\omega_1 t+ \phi)] \cos(\omega_2 t+ \phi) to find f(\omega) using the Fourier transform. Ie, the Fourier transform I use is f(\omega)=\displaystyle \frac{1}{\sqrt {2 \pi}} \int^{\infty}_{-\infty} f(t) (\cos \omega t+ j \sin...
  27. T

    What is the Fourier transform of cos(2πf0x - π/4)?

    Hi, I need help with some basic Fourier transform properties stuff - its fairly simple though I think I am doing something wrong. So we know from the shifting property if h(x) has the Fourier transform H(f) then h(x-a) has the Fourier transform H(f)ei*2*π*f*a so I have the function cos(2πf0x...
  28. G

    Fourier Series (Clarification of Concept)

    Hi everyone. I ran into a problem while attempting my Fourier Series tutorial. I don't really understand the "L" in the general formula for a Fourier Series (integration form). I shall post my question and doubts as images. Thank you for any assistance rendered. <I am solving Q3 in the image.>
  29. M

    Help in split step Fourier method with initial rectangular pulse

    when i use this MATLAB code but with rectangular pulse shape instead of chirped pulse i didn't get the predicted output , so is there a limitation with this method in case of rectangular pulse or the MATLAB code is wrong...
  30. M

    Discrete Fourier Transforms of Signals

    Homework Statement Homework EquationsThe Attempt at a Solution I'd like to see if I have the right line of thinking in my solutions: a. The sampling frequency should be such that no aliasing or folding occurs, so it should be twice the frequency of the original signal. $$x(t) = -17...
  31. D

    Calculating Fourier Transform in Circular Wells

    Hi everyone, do you know how to calculate the Fourier transform for the infinitely deep circular well (confined system)? The radial wave function is given by R=N_m J_m (k r). k=\alpha_{mn}/R. R is the radius of the circular well. R(k R)=0. Thanks. Another question is that The k in J_{m}(k r)...
  32. S

    How to Calculate Error in Fourier Series and its Approximation of Angles?

    Does anyone know how to calculate the error between a function and its Fourier series representation as a function of the partial sums of the series? So far I haven't been able to find anything in the literature that talks about this. I'm also interested in looking at how well a Fourier series...
  33. E

    Complex form of Fourier series

    Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ $$a_0=\frac{2}{b-a}\int_{a}^{b}f(t)dt,$$ $$a_n=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi t}{b-a}dt,$$...
  34. C

    MHB Pole shifting for Fourier transform

    Hi, I have a simple harmonic oscillation problem whose Green function is given by $$\Bigl[\frac {d^2}{dt^2}+ \omega_{0}^{2}\Bigl] G(t, t') = \delta(t-t')$$ Now I found out the Fourier transform of $G(t, t')$ to be $$G(\omega)= \frac{1}{2\pi} \frac{1}{\omega_{0}^{2}-\omega^2}$$ which has poles...
  35. D

    Electrostatics Fourier Decomposition (problem setting up boundaries)

    Homework Statement An #a*b*c box is given in x,y,z (so it's length #a along the x axis, etc.). Every face is kept at #V=0 except for the face at #x=a , which is kept at #V(a,y,z)=V_o*sin(pi*y/b)*sin(pi*z/c). We are to, "solve for all possible configurations of the box's potential" Homework...
  36. P

    Solving cos ax/sin pi*x: Fourier Series Approach

    I' m trying to solve something as apparently simple like this cos ax/sin pi*x which appears solved in https://archive.org/details/TheoryOfTheFunctionsOfAComplexVariable in the page 157, exercise 9. second part. I'm trying by Fourier series, but by the moment I can't achieve it. Thanks.
  37. ranju

    Why do Fourier series require specific limits for integration?

    Homework Statement The major problem I am facing while solving for Fourier series is about the limits to be taken while integrating..! In the general equation of Fourier series the upper & lower limits are t1 & t1+T respectively..while solving for even functions we take t1 =-T/2..! Why is it...
  38. mnb96

    Sufficient condition for bounded Fourier transform

    Hello, Let's suppose we are given a function f:\mathbb{R}\rightarrow \mathbb{R}, and we assume its Fourier transform F=\mathcal{F}(f) exists and has compact support. What sufficient condition could we impose on f, in order to be sure that F is also bounded?
  39. E

    Can a discontinuous function have a uniformly convergent Fourier series?

    Let's say I have Fourier series of some function, f(t), f(t)=\frac{a0}{2}+\sum_{n=1}^{\infty}(an\cos{\frac{2n\pi t}{b-a}}+bn\sin{\frac{2n\pi t}{b-a}}), where a and b are lower and upper boundary of function, a0=\frac{2}{b-a}\int_{a}^{b}f(t)dt, an=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi...
  40. B

    DTFT of x(n)*(-1)^n: Effect & Transform Explained

    Whais is the effect of a multiplication by (-1)^n in the DTFT ?? In other words, what's is this transform : x(n)* (-1)^n ??
  41. K

    Fourier transform - Other possible wave forms

    Hi, I am totally a non-math guy. I had to attend a training (on automobile noise signals) that had a session discussed about Fourier Transform (FT). Let me pl. write down what I understand: "The noise signal observed at any point in the transmission line can be formed using a sum of many sine...
  42. Xenosum

    Real Scalar Field Fourier Transform

    Homework Statement Silly question, but I can't seem to figure out why, in e.g. Peskin and Schroeder or Ryder's QFT, the Fourier transform of the (quantized) real scalar field \phi(x) is written as \phi (x) = \int \frac{d^3k}{(2\pi)^3 2k_0} \left( a(k)e^{-ik \cdot x} + a^{\dagger}(k)e^{ik...
  43. E

    MATLAB How can I optimize my MATLAB code for faster Fourier series plot?

    Hi! Here is my m-file for Fourier series plot: clear clc syms n a0=input('Enter coefficient a0: '); an=input('Enter coefficient an: '); bn=input('Enter coefficient bn: '); a=input('Enter lower boundary: '); b=input('Enter upper boundary: '); t=linspace(a,b,10000); sum=0; for n=1:10 %%n could...
  44. homer

    Principal Values & Fourier Transforms in Quantum Physics

    E.g., if I have a time independent wavefunction \psi(x) with Fourier transform \tilde{\psi}(k), in computing the expectation of momentum are we calculating the principal value \lim_{R \to \infty} \int_{-R}^{R} dk\,\lvert \tilde{\psi}(k)\lvert^2\, \hbar k instead of the improper integral...
  45. U

    Can I Calculate the Fourier Transform of an Image by Hand?

    Hi all, I have a somewhat qualitative understanding of image Fourier transforms and what they represent which for the most part is sufficient for me. However i am interested to know how when i use an image analysis program to produce the Fourier transform of a real image, what is actually...
  46. R

    Integration test of dirac delta function as a Fourier integral

    Homework Statement Problem: a) Find the Fourier transform of the Dirac delta function: δ(x) b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves). c) test by integration, that the delta function represented by a Fourier integral integrates...
  47. E

    Representing Signals with Fourier Series in Multisim

    Is it possible to represent some signal in terms of Fourier series in Multisim? For example, Fourier series of sawtooth voltage with period T=2pi is $$\sum_{n=1}^{\infty }\frac{2}{n}(-1)^{n+1}sin{(nt)}=2sin{(t)}-sin{(2t)}+\frac{2}{3}sin{(3t)}-\frac{1}{2}sin{(4t)}+...$$. These terms on right side...
  48. I

    Fourier Transform of a sinc like equation

    I have been given this y(t)=\frac{sin(200πt)}{πt} All I want is to find, is how the rectangular pulse will look like if I take the transformation of the above. That "200" kind of confusing me, because it isn't a simple sinc(t)=\frac{sin(πt)}{πt} I need somehow to find the height of the...
  49. ZetaOfThree

    Volume of Brillouin zone is the same as Fourier primitive parallelepip

    In Kittel's solid state text, problem 2.3, he says that the volume of the Brillouin zone is the same as a primitive parallelepiped in Fourier space. Somehow I can't see why this is true. Can someone help me see why this is true? Also, is the same relationship true between Wigner-Seitz cells and...
  50. S

    Applying Integration by Parts and Eikonal Equation to Fourier Integral Operators

    Hi! I have a question for you. At the end of the post there's a link. There's the homework which I have to do for an exam. I have to study the Fourier Integral Operator that there is at the begin of the paper. I did almost all the homework but I can't do a couple of things. First: at the point...
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