Fourier transform Definition and 1000 Threads

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 ?

I was reading up on (discrete) Fourier transform when my mind started to think of an what-if scenario: Assumed I'm sampling a signal of the form a1*sin(b1+c1) + a2*sin(b2+c2) + a3*sin(b3+c3) + ... + aN*sin(bN+cN) + some noise where the a's represents magnitudes, b's represents frequencies and...

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...

Hello I'm doing some problems in QM scattering regarding the Green's function. Homework Statement Determine the differential equation of G(\vec{r},\vec{r}',\omega) Homework Equations I've been given the Fourier transform for the case where the Hamiltonian is time independent...

I have a complex signal eg: cos(wt + phase1) + i*cos(wt + phase2) the frequency of both the waves is same. When i have a look at the phase spectrum of the above signal, i am not able to interpret the phase values. They are making no sense. I tried to determine phase shift for real signals and...

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 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 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 Show that the Hilbert transform of ##\frac{\sin(at)}{at}## is given by $$\frac{\sin^2(at/2)}{at/2}.$$ Homework Equations The analytic signal of a function is given by ##f_a(t) = 2 \int^\infty_0 F(\nu) \exp(j2 \pi \nu t) \ d\nu,## where ##F(\nu)## is the Fourier transform...

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...

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...

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

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 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...

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...

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 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 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 Consider a LTI system defined by the following difference equation: ##\mathrm{y}[n]=-2x[n]+4x[n-1]-2x[n-2]## a) Determine the impulse response of the system b) Determine the frequency response of the system Homework Equations DTFT...

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?

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...

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...

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...

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...

Homework Statement Using Parseval's theorem, $$\int^\infty_{-\infty} h(\tau) r(\tau) d\tau = \int^\infty_{-\infty} H(s)R(-s) ds$$ and the properties of the Fourier transform, show that the Fourier transform of ##f(t)g(t)## is $$\int^\infty_{-\infty} F(s)G(\nu-s)ds$$ Homework Equations...

If I cut my image into several portions and use the Fast Fourier Transform on each portioned image, will I achieve the same result as if I used Fast Fourier Transform on the whole image? I have this concern because I need to process a large image using the Fast Fourier Transform, the problem is...

Hello everyone, So, i have a big test tomorrow and my professor said i should study the DC level in Fourier transform , in the frequency domain. So, i did a little research and found out that the dc level is the percentage of the time a signal is active, and that's all. Can't see how that's...

Fourier transform is defined as $$F(jw)=\int_{-\infty}^{\infty}f(t)e^{-jwt}dt.$$ Inverse Fourier transform is defined as $$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(jw)e^{jwt}dw.$$ Let ##f(t)=e^{-at}h(t),a>0##, where ##h(t)## is heaviside function and ##a## is real constant. Fourier...

Homework Statement [/B] I was using the Fourier transform to solve the following IVP: \frac{\partial^2 u}{\partial t \partial x} = \frac{\partial^3u}{\partial x^3} \\ u(x,0)=e^{-|x|} Homework Equations [/B] f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{f}(\omega)e^{i\omega...

1. Find the Fourier series of : $$g(t)=\frac{t+4}{(t^2+8t+25)^2}$$ 2. I have been trying to write the function to match the formula $$\mathcal{F} [\frac{1}{1+t^2}] = \pi e^{-\mid(\omega)\mid}$$ 3. I have simplified the function to $$(t+4)(\frac{1}{9}(\frac{1}{1+\frac{(t+4)^2}{9}})^2)$$...

What is the relationship between the Fourier transform of a periodic function and the coefficients of its Fourier series? I was thinking Fourier series a special version of Fourier transform, as in it can only be used for periodic function and only produces discrete waves. By this logic, aren't...

Homework Statement Evaluate the Following integrals 1. http://www4b.wolframalpha.com/Calculate/MSP/MSP10141fif9b428c5bab0b00005dc489hi851d28h7?MSPStoreType=image/gif&s=37&w=164.&h=35. Homework Equations...

Homework Statement I'm supposed to be using the similarity theorem and the shift theorem to solve: cos(πx) / π(x-.5) has transform e^(-iπs)*Π(s) Homework Equations similarity theorem f(ax) has transform (1/a)F(s/a) shift theorem f(x-a) has transform e^(-i2πas)F(s) The Attempt at a Solution...

Homework Statement I have question on doing the following indefinite integral: $$\int{d^3x(\nabla^2A^{\mu}(x))e^{iq.x}}$$ Homework Equations This is part of derivation for calculating the Rutherford scattering cross section from Quarks and Leptons by Halzen and Martin. This books gives the...

I read about how MRI works briefly, by flipping the water molecules using a magnetic field to the correct state then send the radio wave to these atoms and have it bounces back to be received by receiver coils and apply Fourier Transform to figure out the imaging. My question is, how does...

3d Fourier transform of function which has only radial dependence ##f(r)##. Many authors in that case define \vec{k} \cdot \vec{r}=|\vec{k}||\vec{r}|\cos\theta where ##\theta## is angle in spherical polar coordinates. So \frac{1}{(2\pi)^3}\int\int_{V}\int e^{-i \vec{k} \cdot...