Fourier transform Definition and 950 Threads

Can anyone explain me how to prove the following identity? \frac{\partial \hat{f}}{\partial x}(0,0) = \int \int x^2f(x,y)dxdy where \hat{f} denotes the Fourier Transform of f(x,y) ?

Homework Statement Take the Fourier Transform of f(x)=rect(x/2)*comb(x) where rect is the rectangle function and comb is the Dirac comb. Sketch the results. Homework Equations The FT of a convolution is the product of the individual FTs. The Attempt at a Solution Taking the FT is...

I couldn't understand that why the Fourier integral is meaningless for f(x)=A*cos(ax) ? Any comments will be appreciated.

I am an undergrad student of physics so recommend me some good/classic books on the LT & FT . THANX

When I see a graph of a Fourier Transform, or something in the frequency domain, say band-limited from -\Omega to \Omega, I'm confused to what the interpretation is of the negative frequencies. Physically it would seem as though something considered in cycles/second for example, should be...

Am just playing around, and following examples of Fourier transform solutions of the heat equation, tried the same thing for the electrostatics Poisson equation \nabla^2 \phi &= -\rho/\epsilon_0 \\ With Fourier transform pairs \begin{align*} \hat{f}(\mathbf{k}) &= \frac{1}{(\sqrt{2\pi})^3}...

Homework Statement This is from Griffiths Introduction to Quantum Mechanics, Problem 2.21. Suppose a free particle, which is initially localized in the range -a<x<a, is released at time t=0: \Psi(x,0) = \begin{cases} \frac{1}{\sqrt{2a}}, & \text{if } -a<x<a,\\ 0, &...

Homework Statement (a) Solve \frac{\partial u}{\partial t}=k\frac{\partial ^{2} u}{\partial x^{2}} - Gu where -inf < x < inf and u(x,0) = f(x) (b) Does your solution suggest a simplifying transformation? Homework Equations I used the Fourier transform as: F[f(x)] = F(w) =...

Hello, can anyone help me with the following problem: The discrete Fourier transform (DFT) in matrix form can be done as follows F=M*f where f are the space domain samples, F are the spatial frequency domain samples and M is the DFT matrix containing the exp(j*...) terms. To compute the...

hello I wonder if the diffraction pattern after Laserlight going thru a grating pattern (say 10 slots) has anything to do with the Fourier transform of the grating pattern. I am not a physicist, but have some knowledge of Fourier math. I think the spatial frequencies of the grating pattern...

Hi everyone! I'm not sure if I'm posting this question in the right section. Please don't be mad at me if I'm mistaken. Can you please help me solve this problem? Calculate the value of the signal x(t), given its spectrum (see figure in attachment), at the time t=2/W. Attempted...

Hi everybody! I kindly request your help in optimizing the numerical integration of the following expression: \xi (r)=\frac{1}{2\pi ^2}\int_{-\infty}^{\infty}f(k)\cdot \sin(k\cdot r)\cdot dk f(k) vanishes outside the boundaries k=0 and k=2; I have got k and f(k) as float arrays, so we...

I keep doing questions on Fourier transforms where the 1/2pi isn't there. Example: F\left[\frac{\partial^2u\left(x,y\right)}{\partial x^2}\right] for which I thought the next step would be \[...

Homework Statement If F(k)=TF\{f(x)\},k\neq 0 where TF is the Fourier transform ,and F(0)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(u)du\neq 0 , show that TF\{\int_{-\infty}^{x}f(u)du\}=-i \frac{F(k)}{k} +\pi F(0)\delta(k) Homework Equations The Attempt at a...

"Sliding DFT" discrete Fourier transform... I was wondering if any of you had had experience with the sliding DFT algorithm. It is somewhat similar to the Goertzel algorithm. I am having some trouble understanding the mathematics of the algorithm, and I also cannot seem to identify a useful...

Find the Fourier transform \hat{u}(w,t) = \frac{1}{\sqrt{2 \pi}} \int^{\infty}_{- \infty}u(x,t)e^{(-ixw)}dx of the general solution u(x,t) of the PDE u_{t}= u_{xx} - u Should I start by solving the PDE, or is there another way to do it?

When would one consider to use the Laplace over the Fourier Transform and vice versa?

Homework Statement I need to find the Fourier Transform of this integro-differential equation: \begin{subequations} \begin{eqnarray} \nonumber \dot{\hat{{\cal E}}}(t) &=& -\kappa \hat{{\cal E}}(t) + i g\int_{-\infty}^{\infty} d \Delta\; \hat{{\cal \rho}}(\Delta)\,(...

Homework Statement a light source a(x) is defined by a(x) = Acos(pi*x/a)[theta(x+(a/2)) -theta(x-(a/2))] calculate the diffraction pattern I(X) Homework Equations I(X)=2pi|a~((2pi/(LAMBDA*d))*X)|2 this is the equation for a diffraction pattern on a screen at distance d from...

Homework Statement Find the Fourier transform of the function f(x) = 1 if -1<x<1, f(x) = 0 otherwise 2. The attempt at a solution \hat{f}(w) = \frac{1}{\sqrt{2 \pi}} \int ^{1}_{-1}e^{-iwx}dx = \frac{1}{\sqrt{2 \pi}} [\frac{e^{-iwx}}{-iw}]^{1}_{-1} = \frac{1}{-iw \sqrt{2 \pi}}(e^{-iw} -...

Homework Statement Calculate Fourier transform of cos(x^2)Homework Equations The Attempt at a Solution I want, if it possible, a clue to solve the integral. I don't know how to proceed. I tried integration by parts, but i can't solve it. Sorry for my english. How can i use latex? Can...

So I have a power spectrum of a given function, which is supposed to be a superposition of four sinusoidal terms with frequencies that range from 1xomega to 4xomega. My spectrum looks something like this: http://upload.wikimedia.org/wikipedia/commons/4/4f/Triangle-td_and_fd.png What exactly...

Homework Statement I have been trying to solve the Fourier transform of exp(-|x|) Homework Equations Do I need to split the function into two parts with different limits,i.e. the first has a limit from -infinity to zero and the secod from zero to +infinity. Please advise? The...

Hi, I have a question about the FFT. I'm starting to learn the concepts behind it, but I'm struggling at this one particular thing... Ok, let's say you have this diagram. http://www.ece.uvic.ca/499/2004a/group05/image/radix2.jpg Can someone explain to me exactly what "N-point" means? Also...

It is a well known fact that \int dk \tilde{F}(k) = F(0) where the tilde denotes the Fourier transform. (take or leave some \pis) Is it possible to show this 1) without assuming that we know \int dx e^{ikx} = \delta(k) and 2) without saying: "well we know what the inverse Fourier transform...

a Fourier transform of an aperature results the pattern of the fraunhofer diffraction fringes at infinity of light passing that aperature. How can we understand that point physically? I tried much to think about it! but no use. Everyone, Please give your thought about this, so I can have...

Hello. I have a homework question here that I just can't seem to answer, and I am hoping if I can get pointed in the correct direction. The question asks "What is the difference between Frequency Responce of a system H(jw) and the Fourier Transform of a signal X(jw)?" It would be...

Hello guys, I have got a homework for Advanced Quantum Mechanics, actually I've tried to solve it in many ways my own, but I'm always forced to use computer at the end (For infinite series or improper integrals), I want to solve it my self, so I can do it also in the class! The problem is...

Homework Statement Just wanted to check if I did the Fourier transform of a somewhat long function correctly Homework Equations f(x) = (1+cos(\frac{2pix}{w}))rect2(\frac{x}{w}) they're not convolutions, just a modulation equation used in imaging studies 'rect' is rectangle function...

I don’t know if the question belongs to engineering or math but here it goes. I was taught that a sufficient (not necessary) condition for existence of Fourier transform of f(t) is f(t) is absolutely integratble. I was wondering what are the “necessary and sufficient conditions” for FT of f(t)...

For some reason I can't post everything at once... gives me a "Database error" so I will post in parts... A vector function can be decomposed to form a curl free and divergence free parts: \vec{f}(\vec{r})=\vec{f_{\parallel}}(\vec{r'})+\vec{f_{\perp}}(\vec{r'}) where...

A vector function can be decomposed to form a curl free and divergence free parts: \vec{f}(\vec{r})=\vec{f_{\parallel}}(\vec{r'})+\vec{f_{\perp}}(\vec{r'}) where \vec{f_{\parallel}}(\vec{r'}) = - \vec{\nabla} \left( \frac{1}{4 \pi} \int d^3 r' \frac{\vec{\nabla'} \cdot...

Hi all. I am revisiting Fourier transform now and am wondering why we need Fourier transform? I mean, what's so special of representing a function in another way (in terms of sine waves)? Actually, I am now working on a problem. I was just told that someone worked out something in Fourier...

Hi all! I am asking about a question about Fourier transform. I can only roughly remember things about Fourier transform. I am told that Fourier transform gives the steady state solution, is it? I can hardly relate these two concepts. Can someone try to explain? Many thanks.

Hi everyone I am trying to prove that if a signal g(t) is its own Fourier Transform (so that G(f) = g(f), i.e. they have the same functional form), then g(t) must be a Gaussian. I know that the Fourier Transform of a Gaussian is a Gaussian, so that's not the point of the exercise. Simon...

Why is this so? \displaystyle F\left[ \frac{1}{1-e^{-\pi x}} \right] = i \frac{1+e^{-2k}}{1-e^{-2k}} Here, -\infty < x < \infty. It has to be done by contour integration, by the way. Unfortunately, I'm having difficulty with the whole thing.

Why shud one take the Fourier transform of a wavefunction and multiply the result with its conjugate to get the probability? Why can't it be Fourier transform of the probability directly? thank you

Hi, Just when I thought I'd grasped the Discrete Fourier Transform properly,something comes along and messes me up ... and my books don't seem to treat it. Say you have a square 2D image and you want to do an Ideal LowPass Filter. Well, in general, filters need to be odd-number-sized so...

Hey everybody. I was studying Fourier transforms today, and I thought, what if you took the transform of an ordinary sine or cosine? Well, since they only have one frequency, shouldn't the transform have only one value? That is, a delta function centered at the angular frequency of the wave...

Homework Statement calculate the inverse Fourier transform of \left( a^2 + \left( bk \right)^2 \right)^{-1} The Attempt at a Solution I know that FT[e^{-|x|)}](k) = ( \pi (k^2 + 1 ) )^{-1}. I've tried to to concatenate the shift FT or the strech FT, but the "+1" in the known FT is in the...

I would like to do an inverse Fourier transform using MATLAB's IFFT. I am confused by MATLAB'S single input of X for its IFFT function. Has anyone had experience using MATLAB for these tranforms? I would like to do an inversion of Fourier transform for my function y(iw) at some value real...

Hello All, As I understand it, the wavefunction Psi(x) can be written as a sum of all the particle's momentum basis states (which is the Fourier transform of Psi(x)). I was woundering if the wavefunction's complex conjugate Psi*(x) can be written out in terms of momentum basis states, similar...

If I use the following code in Mathematica f1[t_] := Cos[w t + d1]; f2[t_] := Cos[w t + d2]; data1 = Table[f1[t], {t,1,10000}]; data2 = Table[f2[t], {t,1,10000}]; ft1 = Fourier[data1]; ft2 = Fourier[data2]; To take the Fourier transform of two data sets, how can I use the resulting data...

Hi! I want to find the Fourier transform of \int_{-\infty}^t f(s-t)g(s) ds . The FT \int_{-\infty}^t h(s) ds \rightarrow H(\omega)/i\omega + \pi H(0) \delta(\omega) is found in lots of textbooks. So if I let h(s) = f(s-t)g(s), I need to find the FT of h(s) H(\omega) =...

OK< I've been trying to understands Fourier Transforms with no success. Does anybody know a tutorial or website that explains it completely? My math background is Calculus AB, and my Physics background is reg. physics, but I am into QM, and already know basic wave equations and can apply...

[SOLVED] Fourier transform of a function such that it gives a delta function. ok say, if you Fourier transform a delta function G(x- a), the transform will give you something like ∫[-∞ ∞]G(x-a) e^ikx dx a is a constant to calculate, which gives you e^ka (transformed into k space)...

The Schwartz space on \mathbb{R}^d is defined to be S(\mathbb{R}^d) := \{f\in C^{\infty}(\mathbb{R}^d,\mathbb{C})\;|\; \|f\|_{S,N}<\infty\;\forall N\in\{0,1,2,3,\ldots\}\} where \|f\|_{S,N} := \underset{|\alpha|,|\beta|\leq...

How would one obtain a Fourier Transform solution of a non homogeneous heat equation? I've arrived at a form that has \frac{\partial }{ \partial t }\hat u_c (\omega,t) + (\omega^2 + 1)\hat u_c (\omega,t) = -f(t) My professor gave us the hint to use an integrating factor, but I don't see...

Homework Statement I need to take the inverse Fourier transform of \frac{b}{\pi(x^2+b^2)}Homework Equations f(t)=\int_{-\infty}^{\infty}e^{itx}\frac{b}{\pi(x^2+b^2)}dx It might be useful that \frac{2b}{\pi(x^2+b^2)}=\frac{1}{b+ix}+\frac{1}{b-ix}The Attempt at a Solution I know the result...

If i have a signal S(t) (the plot would be voltage vs time) and I take its Fourier transform, what are the units of the vertical axis? The horizontal axis can either be frequency in hertz or in radians, but what about the other axis? I guess generally I plot the magnitude of the transform since...