Fourier transform Definition and 1000 Threads

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...

For some reason I can't post everything at once... gives me an error 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(...

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...

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...

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?

I want to write the dielectric profile of the following system, so I can then write its Fourier transform as an integral... The plates are semi-infinite. So far I have: Epsilon, for: -infinity < z <= 0 1, for: 0 < z < ? Epsilon, for: ? <= z < +infinity I need to find the...

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 Can someone please help me with this problem. I am wondering how I would be able to calculate the integral of (sinx)^n/x^n using the Fourier Transform? I am given these formulas for the Fourier Transform of spaces of square integrable functions. SO I know that the...

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...

Homework Statement I am supposed to find the Fourier Transform of the following: suppose t(subu) is a translation of the function f by u, so that f*t(subu) = f(t-u). suppose also that 1 means denotes a characteristic function so that the characteristic function has the value 1 from -T to...

Homework Statement Find the Fourier Transform of the following function: y(t) = \left( \begin{array}{cc} 0,& \ \ t<1 \\1-e^{-(t-1)},& \ \ 1 < t < 5 \\e^{-(t-5)}-e^{-(t-1)},& \ \ t \geq 5 \end{array} Homework Equations I employed the following transforms in my attempt...

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...