Fourier Definition and 1000 Threads

The FT decomposes images into its individual frequency components In its absolute crudest form, would the sum of these two images (R) give the L image?

Suppose a PDE for a function of that depends on position, ##\mathbf{x}## and time, ##t##, for example the wave equation $$\nabla^{2}u(\mathbf{x},t)=\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)$$ If I wanted to solve such an equation via a Fourier transform, can I Fourier...

Hello 1. Homework Statement Find the Fourrier coefficients in the annulus problem of the text. uxx+uyy=0 in 0<a²<x²+y²<b² u=g(θ) for x²+y²=a² u=h(θ) for x²+y²=b² Homework Equations The solution is The Attempt at a Solution I have the solutions but when I solved it for...

Homework Statement Consider a 2pi-periodic function f(x) = |x| for -pi ≤ x ≤ pi a) Compute the Fourier series of the function f. b) Prove that (from n=1 to n=infinity)∑ 1/(2k-1)^2 = pi^2/8.**note all "sums" from here on out will be defined from n = 1 to n=infinity Homework EquationsThe Attempt...

Quote: "The Fourier transform is a generalization of the complexFourier series in the limit as http://mathworld.wolfram.com/images/equations/FourierTransform/Inline1.gif. Replace the discrete http://mathworld.wolfram.com/images/equations/FourierTransform/Inline2.gif with the continuous while...

Fourier Transform of Piecewise linear spline wavelet is defined by 1-|t|, 0<t<1; 0, otherwise, is (sinc(w/2))^2. Can anyone please show me the steps. Thanks

How is pi/L part deduced in (n*pi*x)/L?

Homework Statement a(x)=f-Nd(x) + f-(N-1)d(x) +...+ f(N-1)d(x) + fNdHomework Equations fd(x) = (1/a for |x-d| < a and 0 otherwise) Fourier transform of function g(x) is g~(p) = 1/root(2pi) ∫ dx e-ipx g(x) The Attempt at a Solution [/B] I have found the general Fourier transform for the...

Homework Statement a. Represent f(x)=|x| in -2<x<2 with a complex Fourier series b. Show that the complex Fourier Series can be rearranged into a cosine series c. Take the derivative of that cosine series. What function does the resulting series represent? [/B]Homework Equations...

Hello; I'm struggling with pointwise and uniform convergence, I think that examples are going to help me understand Homework Statement Consider the Fourier sine series of each of the following functions. In this exercise de not compute the coefficients but use the general convergence theorems...

Hi, I have a FORTRAN code with an array called Chi that I want to run an inverse FT on. I have defined two spaces X and K which each consist of 3 vectors running across my physical verse and inverse space. My code (If it works??) is extremely slow and inefficient (see below). What is the best...

Homework Statement For a periodic sawtooth function ##f_p (t) = t## of period ##T## defined over the interval ##[0, T]##, calculate the Fourier transform of a function made up of only a single period of ##f_p (t),## i.e. $$f(t)=\left\{\begin{matrix}f_p (t) \ \ 0<t<T\\0 \ \ elsewhere...

I am trying to write a very basic Matlab code to preform the split-step Fourier method on the nonlinear Schrodinger equation: $$\frac{\partial A(z,T)}{\partial z} = -i \frac{\beta_2}{2} \frac{\partial^2A}{\partial T^2} + i \gamma |A|^2 A \ \ (1)$$ I want the program to make 3D plots of...

For the calculation of cn u have to multiply the equation ∑ cn * ejnx by e-jmx what is the reason for this? in my textbook it says nothing about it and on some sites it just said "without justification" i guess what I am asking is why does this do what we want? ps: how do u properly make...

This question is a little basic but.. how are signals stored in a Fourier Transform function f(t)? In my PDE class we were always given a base function to put in terms of sin and cos. But when taking a bunch of samples, all I end up with is a table/array over some time T. How might I use this...

Homework Statement Find the Fourier transform of H(x-a)e^{-bx}, where H(x) is the Heaviside function. Homework Equations \mathcal{F}[f(t)]=\frac{1}{2 \pi} \int_{- \infty}^{\infty} f(t) \cdot e^{-i \omega t} dt Convolution theory equations that might be relevant: \mathcal{F}[f(t) \cdot...

Homework Statement A string of length L =8 is fixed at both ends. It is given a small triangular displacement and released from rest at t=0. Find out Fourier coefficient Bn. Homework Equations what should i use for U0(x) ? The Attempt at a Solution

I have a function f(x,y) which i have defined in this way: a vector x and a vector y meshgrid[x,y] z= f(meshgrid[x,y]). how do i do a 2-d Fourier transform of f(x,y)? the transform must be done without using operations like fft, and must be done using summations written in the code.

I have a function of 2 variables [f(x,y)] where if there was an ellipse in the x-y plane, all values of the function are 1 inside the ellipse and 0 outside. I can plot this function as a surface in 3d where it looks like an elevated ellipse hovering over an elliptical hole in a sheet. My...

Hello! (Wave) I want to calculate the Fourier transform of $g(x)=|x|$. I got so far that $\hat{g}(\omega)=2 \left[ \frac{x \sin{(x \omega)}}{\omega}\right]_{x=0}^{+\infty}-2 \int_0^{+\infty} \frac{\sin{(x \omega)}}{\omega} dx$ Is it right so far? How can we calculate $\lim_{x \to +\infty}...

Homework Statement A certain function ##v(x)## has Fourier transform ##V(\nu)##. The plot of the function is shown in the figure attached below. For each of these functions give their Fourier transform in terms of ##V(\nu)##. And also state if the FT is Hermitian/anti-Hermitian, even/odd...

Homework Statement So well, in class we were shown this equation for the Fourier transform: http://puu.sh/nHsWo/042d1d01ba.png First equation turns a function of time into frequency(notice there's no - in the exponent of e) Second one does the opposite(notice there is a - in the exponent of...

Homework Statement Hello everyone, I'm new to the great field that is Fourier analysis, and have a question about the way in which to determine if the function is a odd or even function. Given the function, of one period f(x) = { x; 0 <= x < =1, 1; 1 < x < 2, (3 -x); 2 <= x <= 3: Is...

Homework Statement Find the Fourier series for the following function (0 ≤ x ≤ L): y(x) = Ax(L-x) Homework EquationsThe Attempt at a Solution 1. We start with the sum from n to infinity of A_n*sin(n*pi*x/L) where An = B_n*Ax(l-x) 2. We have the integral from 0 to L of f(x)*sin(m*pi*x/L) dx...

Homework Statement A free particle moving in one dimension is initially bound by a very narrow potential well at the origin. At time ##t = 0## the potential is switched off and the particle is released; its wave function is: ##\psi (x,0) = N e^{-\frac{|x|}{\lambda}}## where λ is a positive...

Homework Statement From the derivation of v(x,t) and i(x,t) I am stuck on how the inverse Fourier transform of e^(-jwx/u) was calculated. I am trying to understand how the PDE was fully solved here: http://fourier.eng.hmc.edu/e84/lectures/transmission_line/node1.htmlHomework Equations Not...

Homework Statement Find the Fourier transform of x(t) = 4 / (4 - i*t)^2 where i is imaginary Homework Equations Duality Property F(t) ↔ 2πf(-ω) when f(t) ↔ F(ω) The Attempt at a Solution I am not sure if duality property is the way to solve this. I look at a list of properties and this...

Homework Statement Assume ## \phi(k_x ) = \sqrt2 {\pi}## for ## \bar{k}_x - \delta \le k_x \le \bar{k}_x + \delta##, and ##= 0## for all other values of ##k_x##. Calculate ##\psi(x, 0)##, and show that ## \Delta x \Delta k_x \approx 1 ## holds if ## \Delta x## is taken as the width at half...

Homework Statement I am doing #9. Homework EquationsThe Attempt at a Solution I've been looking at a lot of similar problems on the internet. The main difference between this one and them is that this one has an interval of [0,4] while they often have intervals of [0,pi] or [-pi,pi] In my...

Homework Statement Homework EquationsThe Attempt at a Solution So I am tasked with answer #3 and #4. I have supplied the indicated parenthesis of 8 also with the image. Here is my thinking: Take the Fourier series for |sin(θ)|. Let θ = 0 and we see a perfect relationship. sin(0) = 0 and...

I found this formula in a paper: \int exp( \frac{x1 + i x2}{ \sqrt 2} \eta^* - \frac{x1 - i x2}{ \sqrt 2} \eta) D(\eta)/ \pi d^2 \eta the author calls it the Fourier transform of D. It is the first time thar i see this formula. How common is this notation? Can we use it without problem?

Homework Statement In my PDE course we have a homework question stating the following: Let ϑ(x) = x in the interval [-pi, pi ]. Find its Fourier Coefficients. Homework Equations From my notes on this type of question: a_o = 2c_o = 1/pi * integral from -pi to pi [f(x) dx] a_n = c_n + c_(-n)...

Find the Fourier sin series expansion of dirac delta function $\delta(x-a)$ in the half-interval (0,L), (0 < a < L): Now $b_n = \frac{1}{L} \int_0^L f(x)sin \frac{n \pi x}{L}dx $ - but L should be $\frac{L}{2}$ for this exercise... So I would get $ \frac{2}{L} \int_0^L f(x)sin \frac{n \pi...

I don't know if it is the right section to post in. I have a problem with a "simple" Fourier transform. This is the function to transform: f(t)=\frac{\sin\left({2\pi t}\right)}{t}. My first idea was to write that as \sin\left({2\pi t}\right)\cdot\frac{1}{t} but then my fantasy crashed against a...

Hi - I've got myself mixed up here, please see what I am missing below ... Show $ \int e^{ik \cdot (r - r')} \frac{d^3 k}{(2 \pi)^3 k^2} = \frac{1}{4 \pi}|r-r'| $ Let R = r-r', then $k \cdot (r - r') = kR cos \theta$ Next I would translate into spherical polar coords, using $\int d^3 k =...

Show that the 3-D FT of a radially symmetric function may be rewritten as a Fourier sin transform i.e. $ \frac{1}{({2\pi})^{{3}_{2}}} \int_{-\infty}^{\infty}f(r)e^{ik \cdot r} \,d^3x = \frac{1}{k} \sqrt{\frac{2}{\pi}} \int_{-\infty}^{\infty} \left[ rf(r) \right] sin(kr) \,dr $ The example...

Homework Statement The odd 2π-periodic function f(x) is defined by f(x) = x2 π > x > 0 -x2 −π<x<0 Find the coefficient bn in the Fourier series f(x) = a0/2 + ∑(an cos(nx) + bn sin(nx)). What are the values of the coefficients a0 and an and why? Homework Equations bn = 1/π ∫...

Hello, My name is Thibaut. I am looking to improve my code in python in order to have a better look a my Fourier transform. as you can see on the image, we barely see any detail of the peaks on the image. Also it's not centred. the zero order peak in on the corner, not in the centre. Any idea...

I was reading about this Fourier transforming property of lens,when I came by the experimental setup for Fourier optics(with laser and a 4f correlator system).Part of the setup was that of Fraunhofer diffraction and we get the Fourier transform of the aperture at the focal point of first lens...

Homework Statement What is the Fourier transform of the function graphed below? According to some textbooks the Fourier transform for this function must be: $$ab \left( \frac{sin(\omega b/2)}{\omega b /2} \right)^2$$ Homework EquationsThe Attempt at a Solution I believe this triangular...

How does it work? (The derivative rules of FT) We look at $$F[x(t)]=\hat{x}(f)$$ $$\mathcal{l} \text{ is a distribution, with}\tilde{x}=tx(t)$$ $$\mathcal{F}[Dl(x)]=\mathcal{F}l'(x)=2\pi il(\mathcal{F}\tilde{x})=2\pi i \mathcal{F}l(\tilde{x})$$ Till here I fully understand. But next step...

The equation of motion of ##\phi^4## theory is ##(\partial^{2}+m^{2})\phi = -\frac{\lambda}{3!}\phi^{3}##. Why can't this equation be solved using Fourier analysis? Can't we simply write the equation in Fourier space and take it from there?

Please help me find my mistake - "find the Sine F/series of f(x)=x over the half-interval (0,L)" I get $ b_n=\frac 2L \int_{0}^{L}x Sin \frac{2n\pi x}{L} \,dx $ $ = \frac 2L \left[ x(-Cos \frac{2n\pi x}{L}. \frac{L}{2n\pi x}\right] + \frac {1}{n\pi} \int_{0}^{L} Cos \frac{2n\pi x}{L} \,dx$...

Find the Fourier Transform of $ e^{-a|t|}Cosbt $ I'd like to simplify this using $Cosbt = Re\left\{e^{ibt}\right\}$ $\therefore \hat{f}(\omega) = Re\left\{ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\left(-a+ib+iw\right)|t|} \,dt \right\} = Re\left\{ \frac{1}{\sqrt{2\pi}}...

Hello, I hope I am posting this in the correct forum topic. It really is more of a "mathy" type of question, but I am posting it here because it deals with radar, and this type of math is used a lot in radar. To the mods, feel free to move it to a more suitable location if desired. I have...

Hi - firstly should I be concerned that the dirac function is NOT periodic? Either way the problem says expand $\delta(x-t)$ as a Fourier series... I tried $\delta(x-t) = 1, x=t; \delta(x-t) =0, x \ne t , -\pi \le t \le \pi$ ... ('1' still delivers the value of a multiplied function at t)...

Hi - frustratingly I get some problems right 1st time, others just defy me (Headbang) $f(x) = -x, [-\pi,0]; = x, [0,\pi]$ I get $a_0 = \pi$ and $a_n = \frac{-4}{\pi \left(2n-1\right)^2}$ which agrees with the book - but I thought I'd check $b_n$ for practice, it should = 0 according to the...

Hi, appreciate some help with this FS problem - $f(t)= 0$ on $[-\pi, 0]$ and $f(t)=sin\omega t$ on $[0,\pi]$ I get $a_0=\frac{2}{\pi}$ and $b_1 = \frac{1}{2}$, which agree with the book; all other $b_n = 0$ because Sin(mx)Sin(nx) orthogonal for $m \ne n$ But $a_n...

Hi, in a section on FS, if I were given $\sum_{n=1}^{\infty} \frac{Sin nx}{n} $ I can recognize that as the Sin component of a Fourier Series, with $b_n = \frac{1}{n} = \frac{1}{\pi} \int_{0}^{2 \pi}f(x) Sin nx \,dx$ Can I find the original f(x) from this? Differentiating both sides doesn't...

Hi - an example in my book shows that FS coefficiants can be arrived at by minimizing the integrated square of the deviation, i.e. $ \Delta_p = \int_0^{2\pi}\left[ f(x) - \frac{a_0}{2}-\sum_{n=1}^{p}\left( a_nCosnx + b_nSinnx \right) \right]^2dx $ So we're looking for $ \pd{\Delta_p}{a_n}...