Fourier transform Definition and 950 Threads

Homework Statement My book uses the following equation to derive the diffraction condition for electromagnetic waves scattering in a crystal lattice: F= \int dV n(\mathbf{r}) \exp \left[i\Delta\mathbf{k}\cdot \mathbf{r} \right] F is the scattering amplitude and n is the electron density. I...

Homework Statement In this problem I'm trying to derive an explicit solution for Langmuir waves in a plasma. In part (a) of the problem I derived the wave equation (\partial_t_t+\omega_e^2-3v_e^2\partial_x_x) E(x,t) = 0 This matches the solution in the book so I believe it's correct...

I need to find the Fourier transform of f(x) which is given by the equation: -\frac{d^2f(x)}{dx^2}+\frac{1}{a^3}\int_{-\infty}^{\infty}dx'exp(-\lambda|x-x'|)f(x')=\frac{b}{a^2}exp(-\lambda|x|) ofcourse Iv'e taken the Fourier tarnsform of both sides, but I don't see how to calcualte the...

I need to find the fourie transform of f(x)=N*exp(-ax^2/2). (N and a are constants). well ofcourse iv'e put into the next integral: \int_{-\infty}^{\infty}f(x)exp(-ikx)dx Iv'e changed variables, that i will get instead of exp(-ax^2/2)exp(-ikx), exp(-z^2)exp(-ikz*sqrt(2/a)), but that didn't...

Are there any "real" examples of a Fourier transform being applied? When we see that something accelerates and then moves we can say its acceleration is being "integrated" to get a velocity, but what meaning does a Fourier transform have? I understand it's used in spectroscopy but I mean...

Homework Statement Find the Fourier Transform of \frac {1}{t} Homework Equations Euler's equations I think... The Attempt at a Solution I tried splitting up the integral into two. One from -\inf to 0 and the other from 0 to \inf . Not much help there. I tried using...

X(w) = 1/(j*(w*hbar-Ek)+(hbar/T2)) - 1/(j*(w*hbar+Ek)+(hbar/T2)) The inverse Fourier transform of the above equation using MATLAB will obtain the following: x(t) = 2*j/hbar*heaviside(t)*sin(t/hbar*Ek)*exp(-t/T2) We can see that the values of x(t) are all imaginary values, however this...

an equation involves an integration. After an inverse Fourier transform of the equation, will the integration limits change? (maybe you can take a look at the attached file) Thanks!

Does anyone know what the eigenfunctions of the spherical Fourier transform are? I want to expand a spherically symmetric function in these eigenfunctions. Are they Bessel functions? Legendre functions?

Homework Statement By the convolution theorem one would expect that if you convolved your image with a kernel in the spatial domain, you would get the exact same result if you multiplied the FFT of the image array by the FFT of the kernel. My problem is that I don't get the same results...

Input: sine wave at 10Hz, amplitude 1. After the transform the plot has a spike at 10Hz with amplitude 0.5. If I vary the amplitude of the sine wave I get: sine amp. - FT spike amp. 1 - 0.5 2 - 2 4 - 8 So it seems A' = A^2/2 Is this because power is proportional to A^2 and it is...

Homework Statement Hi, I have this problem and I don't know how to finish it: Using the Cauchy Theorem, prove that the Fourier tranform of \frac{1}{(1+t^2)} is \pi.e^{-2.\pi.|f|} .( you must show the intergration contour) Stetch the power spectrum. I applied the Fourier transform...

I have been recently reading papers on Generalized Pencil of Functions and Prony Method (parameteric modeling). It turns out that GPOF/Prony are very good in extracting resonances from a given data and don't suffer from the so called 'windowing effects' associated with FT. My question is...

Hi. I have a question regarding the continuous time Fourier Transform of an input signal: x(t) \rightarrow X(j\omega) then \int_{-\infty}^{t}x(\tau)d\tau \rightarrow \frac{X(j\omega)}{j\omega} + \pi X(0)\delta(\omega) but if I want to write it in terms of f = \frac{\omega}{2\pi}...

Homework Statement Let f(x) be an integrable complex-valued function on \mathbb{R}. We define the Fourier transform \phi=\mathcal{F}f by \[\phi(t)=\int_{\infty}^{\infty} e^{ixt} f(x) dx.\] Show that if f is continuous and if $\phi$ is integrable, then \[f(x)=\frac{1}{2\pi}...

Alright. So... Conceptually I completely understand what I'm doing. I'm just a bit confused about how, mechanically, to solve this. I basically have the following: W(w)=(1/2)[p2(w+6)+p2(w-6)] Where p2(w) is a pulse of width 2. Of course, w is omega, and the +/-6 is the shift. This is a...

Question: Is the Fourier Transform of a Hermitian operator also Hermitian? In the case of the density operator it would seem that it is not the case: \rho(\mathbf{r}) = \sum_{i=1}^N \delta(\mathbf{r}-\mathbf{r}_i) \rho_k = \sum_{i=1}^N e^{-i\mathbf{k} \cdot \mathbf{r}} I have a hard...

Homework Statement Richard Robinett defined the Fourier transform with an exp(-ikx) and the inverse Fourier transform with an exp(ikx). I have always seen the opposite convention and I thought it was not even a convention but a necessity to do it the other in order to apply it to some Gaussian...

Im kind of stuck in one of my signals problems. A triangular function defined as: V(t)= (-A/T)t + A when 0< t< T; V(t)= (A/T)t + A when -T< t< 0; otherwise, the function is 0. I have to find the Fourier transform of this function. Could anyone help me??

Can someone help me and tell me the steps to solve the inverse Fourier transform of the following function (10*sin(3*omega)) / (omega+Pi) Thanks!

I have seen two formulations of the dirac delta function with the Fourier transform. The one on wikipedia is \int_{-\infty}^\infty 1 \cdot e^{-i 2\pi f t}\,dt = \delta(f) and the one in my textbook (Robinett) is 1/2\pi \int_{-\infty}^\infty 1 \cdot e^{-i f t}\,dt = \delta(f) I...

I need to find the Fourier Transform (FT) of: x(t)=\sum^{\infty}_{n=-\infty}((-1)^{n}\delta(t-nT)) Not really sure how to solve this problem, so any help will be appreciated. Also, if you guys know a good reference for non-uniform sampling and reconstruction, please post it.

I hope this is OK to post here. I thought it would be better here than in the math questions forum, since you are EEs, and probably have more experience dealing with things related to the delta function. Problem Let \hat{x}(t) = \sum_{k=-\infty}^{\infty}\delta(t-2k). Now let x(t) =...

Is there any way to calculate the Fourier transform of the functions \frac{d\pi}{dx}-1/log(x) and \frac{d\Psi}{dx}-1 (both are understood in the sense of distributions) i believe that these integrals (even with singularities) exist either in Cauchy P.V or Hadamard finite part...

Hi all, I just know the Fourier series can be applied in differential equation solving, and that's all. Can anyone tell me the physical meaning of the Fourier transform, and fast Fourier transform too. Thank you very much.

I don't know if this question should be posted here, but I'll give it a shot anyways. I am trying to find f(x,y), which can be obtain by doing the backward Fourier integral to F(\omega_x, \omega_y). I have 2 questions. 1. Is there any Fortran code that could evaluate the (numerical)...

I have two signals one continuous oscillating at a high frequency and another one instantaneous at a lower frequency. How can I use a Fourier transform to single out the low frequency one? See at attached picture for what I am trying to do. Edit: Yeah by the way, data is collected in a...

Prove: FT^2(f(x))=f(-x) where FT is the Fourier transform. I tried to change x into -x' but with no success. Do I need to separate cases for even f and odd f?

Hi, What is meant when they say photons are Fourier transform limited? Thanks

Homework Statement I'd like to prove a F/T pair and to confim if they are correct. s(t) = A Sin[w0 t] * rect[t/T - T/2] ... (1) it's Fourier transform is S(f) = exp(-j w T)*T/2*A* {Sinc[(w+w0)T/2/Pi] + Sinc[(w-w0)T/2/Pi]} ...(2) where rect is rectangular function Homework...

I am solving laplaces equation in the half plane and I have the following boundary condition of which I need to find the Fourier transform in the x-direction S_\epsilon(x) = sgn(x)exp(\epsilon|x|), \epsilon >0 sgn(x)=\left\{\begin{array}{cc}1,&\mbox{ if } x<0\\-1, & \mbox{ if } x<0\\...

How can i solve this Fourier Transform question? Homework Statement f(t)= a/((a^2)+(t^2)) if a>0 find the Fourier transform Homework Equations Just give me a hint to solve or first step for solving. Then i will solve. The Attempt at a Solution Thanks for help.

Homework Statement f(t)=N(e^(-a(t^2))) N and a are constants Find the Fourier transform of this problem? Homework Equations http://mathworld.wolfram.com/FourierTransformGaussian.html The Attempt at a Solution http://mathworld.wolfram.com/FourierTransformGaussian.html...

I am stumped on this... Given a discrete function, and transform pair: x(n) \leftrightarrow \hat x (e^{j\omega}) What is the transform of: x_3(n) = (n-1)^2 x(n) I really don't know how to do this. I have a table proprety for nx(n) [/tex], but nothing with n^2 x(n) . The only...

how can I find Fourier transform of 1/(1+4t^2)? hmmm =/

I have a practice exam I'm going through, and I am stumped on one of the basic problems. How is this a transform pair? 10 X(jt) <-----> 20 \pi x (-\omega) I don't see how one can make this relation. What is the 10 X (jt) . thanks in advance

Homework Statement I am trying to show given f(x)=(sinax)/x, a>0 that the transform is 0, |k|>a (pi/2)^1/2, |k|<a Homework Equations The Attempt at a Solution so far i have f transform =1/(2pi)^1/2.[integral from -inf to +inf]exp[-ikx](sinax)/x.dk...

Homework Statement I need to have the Fourier transform of a Gaussian Homework Equations The Attempt at a Solution ∫(exp[-ax^2])(exp[-ikπx]) dx I tried by braking the last exponential into sine and cosine terms.The sine term is odd and it cancels.Then,I cannot evaluate the...

Hello, does anyone know how a converging lens forms the Fourier transform of an aperture when the obs. screen is at distance=f? If each point emits a spherical wave, the lens should make it then parallel and the FT should be the interference resulting from that. However, if we decompose...

Homework Statement Fourier transform of a constant Homework Equations The Attempt at a Solution I am trying to prove that Fourier Transform of a constant is a Dirac delta function.I have fed f(x)=1 in the formula of Forward Fourier transform and got F(k)=int{exp[-ik*pi*x]}dx I...

hi every one! i want to know the Fourier transform of x(t) x(t)=exp(-t/a)*sin(a*t), where a ,b is constant and can it be work out by matlab? another question is : how to proof the Fourier transform of x(t) who follows normal distribution n(u, sigm^2 ) is also normal...

Why is it that we don't use contour integration when we take the integral of a complex function to find the Fourier transform: X(j\omega) = \int_{-\infty}^\infty x(t) e^{- j\omega t} dt

We have \int \frac{d^3 \texbf{q}}{(2 \pi)^3} \frac {e^{i \texbf{q} \dot \texbf{r}}} {q^2 + K^2} = \frac {e^{-Kr}} {4 \pi r} How do we get from the left hand side to the right hand side? I've tried regular Fourier transform of the function under the complex exponential I think that gives...

When we study physics at the faculty we are told that any non-sinusoidal wave can be regarded as a combination of sinusoidal waves of different frecuencies, with the ‘weight’ of the different frecuencies given by the Fourier transform. On the other hand, if we have an electromagnetic wave, we...

Hi, I got a problem in which I have to find the Fourier Transform of a function f(t) defined: f(t) = { 1 - |t|, |t| < 1 0, |t| > 1 } Well , I found the Fourier transform by working out the integral f(t)e^(-iwt) with the limits being -inf to +inf (and I...

Hi all, I'm working in an exercise of advanced optics related to diffraction, in Fraunhoffer's aproximation. I need to calculate the FT of a gaussian multiplied by a rectangle function, i.e, FT(exp(-x^2)*rect(x/a)), and I can't obtain a result expressed using analytical common functions. I...

I am reading something about Electromagnetic wave and Antenna, and come across some equations that the author says are "Azimuth Fourier Transform" and "Inverse Azimuth Fourier Transform". While I am somewhat familiar with Fourier Transform in the time/frequency domains, "Azimuth Fourier...

My notes (from a physics course) justifies the following equality by invoking the convolution thm: \int_{-\infty}^{+\infty} \chi(\omega)\vec{E}_0(\omega)e^{-i\omega t}d\omega=\int_{-\infty}^{+\infty} \chi(\tau)\vec{E}_0(\tau-t)d\tau From a mathematical standpoint (i.e. without reference to...

ok, i have a wave packet which is defined between (-pi/(2b)) and (pi/(2b)) as cos(bx), and it's zero everywhere else. here's what I've done so far: i normalized and solved for b, getting pi/2. so now I'm thinking i should calculate the a0, an, and bn series, and add, right? and here is...

how would i calculate Fourier transform of functions such as 1/(1+t^2)? because if you try to integrate the product of the above function and e^(-jxt), you would realize it's nonintegrable or something at least my ti-89 does not calculate it for me. any other way?