Fourier Definition and 1000 Threads

Can anyone explain what does the author mean by the statement below? page 27 of this paperI don't understand the relation between the Fourier transform and translational invariance. Thanks

Problem F denotes a forward Fourier transform, the variables I'm transforming between are x and k - See attachment Relevant equations So first of all I note I am given a result for a forward Fourier transform and need to use it for the inverse one. The result I am given to use, written out...

In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...

has Fourier used sin(x) and cos(x) in his series because "there must be such interval [a,b] where integral of "some function"*sin(x) on that interval will be zero?" so based on that he concluded that any function can be represented by infinite sum of sin(x) and cos(x) cause they are "orthogonal"...

Need help on Fourier series! Been stuck on this questions, it is too tough for me!

Hello all, First time poster here so please excuse any mistakes as I'm unfamiliar with the conventions of this forum. Also before I get started I'd like to say I wasn't sure exactly where a question like this would go; I debated in the Math Programs and Latex section but figured general physics...

I've tried to read the reason for using Fourier transform in wave packets, I don't understand why. Please help me with this.

I have the following laplace function F(s) = (A/(s + C)) * (1/s - exp(-sα)/s)/(1 - exp(-sT)) I think that the inverse laplace will be- f(t) = ((A/C)*u(t) - (A/C)*exp(-Ct)*u(t)) - ((A/C)*u(t-α) - (A/C)*exp(-C(t-α))*u(t-α)) and f(t+T)=f(t) Now I want to find the Fourier series expansion of f(t)...

Homework Statement Link: http://i.imgur.com/klFmtTH.png Homework Equations a_0=\frac{1}{T_0}\int ^{T_0}_{0}x(t)dt a_n=\frac{2}{T_0}\int ^{\frac{T_0}{2}}_{\frac{-T_0}{2}}x(t)cos(n\omega t)dt \omega =2\pi f=\frac{2\pi}{T_0} The Attempt at a Solution Firstly, x(t) is an even function because...

Hello, PF! I am currently learning Fourier series (and then we'll move on to the Fourier transform) in one of my courses, and I'm having a hard time finding motivation for its uses. Or, in other words, I can't seem to find its usefulness yet. I know one of its uses is to solve the heat...

Hi! I'd like to smear an audio recording, where the frequency content audibly changes, into an audio recording where it does not. Here's a recording of a sampled piano playing a melody, which will serve as an example: https://dl.dropboxusercontent.com/u/9355745/oldmcdonald.wav The frequency...

Hello I am trying to determine the Fourier transform of the hyperbolic tangent function. I don't have a lot of experience with Fourier transforms and after searching for a bit I've come up empty handed on this specific issue. So what I want to calculate is: ##\int\limits_{-\infty}^\infty...

Homework Statement The (computing) task at hand is to take a function f(x) defined at 2N discrete points, and use the Discrete Fourier Transform (DFT) to produce F(u), a plot of the amplitudes of the frequencies required to produce f(x). I have an array for each function holding the value of...

I'm wondering if anyone could give me the intuition behind Fourier series. In class we have approximated functions over the interval ##[-\pi,\pi]## using either ##1, sin(nx), cos(nx)## or ##e^{inx}##. An example of an even function approximated could be: ## f(x) = \frac {(1,f(x))}{||1||^{2}}*1...

Homework Statement [/B] This is a computing coursework problem. (There is a reasonably long theory preamble). Create a single slit centred on the origin (the centre of your array) width 10 and height 1. The array containing the imaginary parts will be zero and the array containing the real...

Homework Statement Given x(t)=8cos(70\pi t)+4sin(132\pi t)+8cos(24\pi t), find the Fourier transform X(f) in the form of \delta function. Homework Equations X(f)=\int ^{\infty}_{-\infty}x(t)e^{-j\omega _0t}dt cos(\omega t)=\frac{e^{j\omega t}+e^{-j\omega t}}{2} sin(\omega t)=\frac{e^{j\omega...

Homework Statement Evaluate the Fourier Transform of the damped sinusoidal wave g(t)=e^{-t}sin(2\pi f_ct)u(t) where u(t) is the unit step function. Homework Equations \omega =2\pi f G(f)=\int ^{\infty}_{-\infty} g(t)e^{-j2\pi ft}dt sin(\omega _ct)=\frac{e^{j\omega _ct}-e^{-j\omega _ct}}{2j}...

I am using ROOT to calculate the Fourier transform of a digital signal. I can extract the individual parts of the transform, the magnitude and phase in the form of a 1D histogram. I am attempting to reconstruct the transforms from the phase and magnitude but cannot seem to figure it out. Any...

My first thought was simply that the Fourier transform of a sum of Gaussians functions that are displaced from the origin by different amounts would just be another sum of Gaussians: F{G1(x) + G2(x)} = F{G1(x)} + F{G1(x)} where a generalized shifted Gaussian is: G(x) = G0exp[-(x - x0)2 / 2σ2]...

Hello everyone, I was trying to develop a sort of generalized version of the Fourier Transform. My question in particular is: Given a function f(x,u), is there a function g(x,u) with \int_{-\infty}^\infty f(x,u)g(x,u')\mathrm{d}x=\delta(u-u') For f(x,u)=e^{2\pi ixu} the solution would be...

Suppose that we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{R}, whose continuous Fourier coefficients \hat{f} are known. The convolution theorem tells us that: $$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$ where \ast denotes the...

Homework Statement I was given a problem with a list of sums of sinusoidal signals, such as Example that I made up: x(t)=cos(t)+5sin(5*t). The problem asks if a given expression could be a Fourier expansion. Homework Equations [/B]The Attempt at a Solution My guess is that it has something to...

A tad embarrassed to ask, but I've been going in circles for a while! Maybe i'll rubber duck myself out of it. If f(t) = f(t+T) then we can find the Fourier transform of f(t) through a sequence of delta functions located at the harmonics of the fundamental frequency modulated by the Fourier...

I am using a Tascam recorder to record an environmental nuisance noise that is occurring in my home. I then use Virtins Multi Instrument Software, which includes an oscilloscope, band pass filter, and a spectrum analyser. Noise source is probably machinery at a legal marijuana grow op. That...

I am a little familiar with Fourier Analysis, but I don't know where to get tools to get the answer to this question: Consider a discrete signal A[0..N-1], consisting of N samples. Suppose we Fourier transform it and get a series of harmonics. Now, consider the discrete signal A[1..N], that is...

I'd like to expand a 3D scalar function I'm working with, ##f(r,\theta,\phi)##, in an orthogonal spherical 3D basis set. For the angular component I intend to use spherical harmonics, but what should I do for the radial direction? Close to zero, ##f(r)\propto r##, and above a fuzzy threshold...

Homework Statement I'm calculating the coefficients for the Fourier series and I got to part where I can't simplify an any further but I know I have to. a_n = \frac{1}{2π}\Big[\frac{cos(n-1)π}{n-1}-\frac{cos(n+1)π}{n+1}-\frac{1}{n-1}+\frac{1}{n+1}\Big]Homework EquationsThe Attempt at a...

Dear all, In my quantum mechanics book it is stated that the Fourier transform of the Coulomb potential $$\frac{e^2}{4\pi\epsilon_0 r}$$ results in $$\frac{e^2}{\epsilon_0 q^2}$$ Where ##r## is the distance between the electrons and ##q## is the difference in wave vectors. What confuses me...

Homework Statement By applying the Gram–Schmidt procedure to the list of monomials 1, x, x2, ..., show that the first three elements of an orthonormal basis for the space L2 (−∞, ∞) with weight function ##w(x) = \frac{1}{\sqrt{\pi}} e^{-x^2} ## are ##e_0(x)=1## , ##e_1(x)= 2x## ,##e_2(x)=...

Suppose that a parameter y= 123. That parameter is somehow "perturbed" and its instantaneous value is: y(t)= 123 + sin(t - 50°) * 9 + sin(t * 3 + 10°) * 3 + sin(t * 20 + 60°) * 4 Suppose that I don't know the above formula, but I can calculate y(t) for any t. Hence I decide to use the...

Here is the Problem Statement : Find Fourier Transform of the piecewise function Can someone sheds some lights on how to start solving this? Thanks

I think this is probably a very basic question: why does the Fourier transform of a wavefunction describing position probabilities gives us a function describing momentum probabilities ? Is there a fairly simple explanation for this ? What leads us to this relation ?

Hi guys, I have been trying to solve the Helmholtz equation with no luck at all; I'm following the procedure found in "Engineering Optics with MATLAB" by Poon and Kim, it goes something like this: Homework Statement Homework Equations Let's start with Helmholtz eq. for the complex amplitude ##...

Hello everyone. I'm trying to better understand structured illumination microscopy and in the literature, I keep coming across bits of text like this. Source: http://www.optics.rochester.edu/workgroups/fienup/PUBLICATIONS/SAS_JOSAA09_PhShiftEstSupRes.pdf From Fourier analysis, if I take the...

Hi, I have a a Fourier transformed variable \hat{\eta}(k) defined as the following: \hat{\eta}(k)=\frac{e^{-k^{2}}\tanh k}{kU^{2}+(-B+\Omega U+E_{b}|k|-k^{2})\tanh k} The parameters U,B,\Omega,E_{b} have all been defined previously. I have naively tried the following: \eta...

Does anyone know if it is possible to solve an equation of the type u_x=(sin(x))*(u) on a periodic domain using the fft. I have tried methods using convolutions but have had no success thanks in advance

Good afternoon people. Recently I started taking a course at my college about Fourier series but I got extremely confused. Here's what's going on. In school we were asigned to use the symmetry formulas to find the Fourier series of the following: f\left ( t \right )=\begin{cases} 1 & \text{ if...

In an image processing paper, it was explained that a 2D Gabor filter is constructed in the Fourier domain using the following formula: $$ H(u,v)=H_R(u,v) + i \cdot H_I(u,v)$$ where HR(u,v) and HI(u,v) are the real and imaginary components, respectively. It also mentions that the real and...

Homework Statement Here is my question Homework Equations I don't know with what formula does the book find Fourier series? The Attempt at a Solution

Today I had a maths exam with a question which was worded something like: Write ##sin(3x-x_0)## as its Fourier representation. By doing a suitable integral or otherwise, find the possible values of its Fourier coefficients. You may find the following useful: ##sin(\alpha-\beta) =...

Homework Statement Homework EquationsThe Attempt at a Solution I think I'm ok with the first part. I start with: ##\widetilde{f}(p) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \! e^{-ipx}f(x) \, \mathrm{d}x## Then moving on to the transform for ##e^{ip_0 x}f(x)## I get...

Homework Statement Homework EquationsThe Attempt at a Solution First write ##\phi(x,t)## as its transform ##\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \! e^{ipx} \widetilde{\phi}(p,t) \, \mathrm{d}p## which I then plug into the PDE in the question to get...

Homework Statement How are the coefficients of the Fourier series modified for a function with a period 2πT? Homework Equations a0 = 1/π ∫π-π f(x) dx an = 1/π ∫π-π f(x) cos(nx) dx bn = 1/π ∫π-π f(x) sin(nx) dx The Attempt at a Solution I tried letting x= t/T so dx = dt/T and the limits x = ±...

Homework Statement The following function is periodic between -π and π: f(x) = |x| Find the Coefficients of the Fourier series and, by examining the Fourier series at x=π or otherwise, determine: 1 + 1/32 + 1/52 + 1/72 ... = Σ∞j=1 1/(2j - 1)2 Homework Equations f(x) = a0/2 + ∑∞n=1 ancos(nx) +...

Homework Statement Homework EquationsThe Attempt at a Solution So we want sine in terms of the exponentials when we take the Fourier transform F(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}dx where f(x)=\sin(3\pi x/L). Let a=3pi/L. Then \sin(ax)=\frac{e^{iax}-e^{-iax}}{2i}. (Is this correct?) Then we...

Homework Statement So i have a function f(x)=x^2 that is periodic -a<x<a and need to sketch this function from -3a<x<a. I know how to find the Fourier coefficients though. Homework Equations f(x)=x^2 sketch it periodically The Attempt at a Solution I know that a function is only periodic...

Hi everyone, in this days i was seeing a little of Fourier series and transform, and i wondered if it was necessary to better understand before the measure and Lebesgue integral before studying it. Or it's not necessary?

Given the Laplace's equation with several boundary conditions. finally i got the general solution u(x,t). One of the condition is that: u(1,y)=y(1-y) After working on this I finally got: ∑An sin(π n y )sinh (π n) = y(1-y) However, i was asked to find An, by not using Fourier series...

Homework Statement I'm trying to calculate the Fourier Series for a periodic signal defined as: y = x 0<x<2Π y = 0 2Π≤x<3Π Homework Equations Fn = 1/T ∫T f(t)cos(kwοt + θk)[/B] cn/2 + ∑k=1k=∞(cn)cos(kwοt+θk) cn= 2|Fn| θk=∠Fn The Attempt at a Solution I got Cn =...

In lectures, I have learned that F(k)= \int_{-\infty}^{\infty} e^{-ikx}f(x)dx where F(k) is the Fourier transform of f(x) and the inverse Fourier transform is f(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx}f(k)dk . But on the same chapter in the lecture notes, there is an example solving...