Fourier transform Definition and 950 Threads

Magic of Fourier Transform? Hello everyone,i am doing my project in image processing... i have done video sementation using the Fourier transform . I applied 3-D fft on video frames ((gray image(2D)+no of video frames(1D)=3D) and Obtained magnitude and phase spectrum and reconstructed video...

Homework Statement f(p) is the Fourier transform of f(x). Show that the Fourier Transform of eipox f(x) is f(p- p0).Homework Equations I'm using these versions of the Fourier transform: f(x)=1/√(2π)∫eixpf(p)dx f(p)=1/√(2π)∫e-ixpf(x)dx The Attempt at a Solution I have...

2D Fourier Transform on a non-rectangular area Is it possible to perform a Fourier transform on a shape instead of a rectangular region? To be specific I am attempting to make a linear zoom function that doesn't produce any pixelation and that mimics natural blur that occurs with distance...

hey pf! physically, what does a Fourier transform do? physically what comes out if i put velocity in? thanks! josh

calculate the Fourier transform of the function g(x) if g(x) = 0 for x<0 and g(x) = ##e^{-x}## otherwise. putting g(x) into the transform we have: ##\tilde{g}(p) \propto \int_{0}^{inf} e^{-ipx} e^{-x} dx## which we can write: ##\tilde{g}(p) \propto \int_{0}^{inf} e^{-x(ip+1)} dx##...

Hi all, Now naturally after completing a physics degree I am very familiar with the form and function of the Fourier Transform (FT) but never have grasped it quite conceptually. I understand that given a function f(x) I can express every functional value as a linear combination of complex...

Suppose I have a function of the type: h(t) = g(t)f(t) where g(t) is a periodic function. Are there any nice properties relating to the Fourier transform of such a product? Edit: If not then what about if g(t) is taken as the complex exponential?

Which is the difference between the Fourier integral and Fourier transform? Or they are the same thing!? Fourier integral:

Homework Statement Find possible momentum, and their probabilities. Find possible energies, and their probabilities. Homework Equations The Attempt at a Solution First, we need to Fourier transform it into momentum space: \psi_k = \frac{1}{\sqrt{2\pi}} \int \psi_x e^{-ikx} dx =...

Homework Statement Let FT(f) = Fourier transform of f, (f*g)(x) = convolution of f and g. Given FT(f*g) = FT(f)FT(g), the first part of the convolution theorem, show that FT[fg] = [FT(f)*FT(g)]/2pi. Homework Equations Duality: FT2f(x) = (2pi)f(-x) Convolution: (f*g)(x) =...

Homework Statement Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F_13 for i, j = 0,1,2, 3. Compute F(hat) and verify that F(hat)F = I Homework Equations The matrix F(hat) is called the inverse discrete Fourier transform of F. The Attempt at a Solution I found that e = 4...

Homework Statement (i) Verify that 5 is a primitive 4th root of unity in F13. (ii) Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F13 for i, j = 0,1,2, 3. Compute F(hat) and verify that F(hat)F= I. Homework Equations The matrix F(hat) is called the inverse discrete Fourier...

Hey! :o Could you give me a hint how to prove the following property of the Fourier transform, when $F[f(x)]=\widetilde{f}(x)$, where $F[f(x)]$ is the Fourier transform of $f(x)$? $$F[ \widetilde{f}(x) ]= \frac{f(-k)}{2 \pi}$$ We know that: $ \widetilde{f}(k)=\int_{- \infty}^{+ \infty}{...

Homework Statement Use the Fourier transform directly to solve the heat equation with a convection term u_t =ku_{xx} +\mu u_x,\quad −infty<x<\infty,\: u(x,0)=\phi(x), assuming that u is bounded and k > 0. Homework Equations fourier transform inverse Fourier transform convolution thm The...

Homework Statement Take the inverse Fourier Transform of 5[\delta(f+100)+\delta(f-100)]\bigg(\frac{180+j2\pi f*0.0135}{1680+j2\pi f*0.0135}\bigg)Homework Equations g(t)=\int_{-\infty}^{\infty} G(f)e^{j2\pi ft}dt The Attempt at a Solution g(t)=\int_{-\infty}^{\infty}...

Homework Statement Without solving the differential equation, find the differential equation that solves Fourier transformation of given differential equation for ##a>0##. a) ##y^{'}+axy=0## b) For what ##a## is the solution of part a) an eigenfunction of Fourier Transform Homework Equations...

Homework Statement Calculate (from the definition, no tables allowed) the Fourier Transform of e^{-a*|t|}, where a > 0. Homework Equations Fourier Transform: G(f) = \int_{-\infty}^{\infty} g(t)e^{-j\omega t} dt The Attempt at a Solution I thought I'd break up the problem into the two cases...

Homework Statement Suppose f(x), -\infty<x<\infty, is continuous and piecewise smooth on every finite interval, and both \int_{-\infty}^\infty |f(x)|dx and \int_{-\infty}^\infty |f'(x)|dx are absolutely convergent. Show the Fourier transform of f'(x) is i\mu F(\mu).Homework Equations...

A necessary condition that a function f(x) can be Fourier transformed is that f(x) is absolutely integrable. However, some function, such as |t|, still can be Fourier transformed and the result is 1/w^2, apart from some coefficients. This can be worked out, as we can add a exponential...

1. Given: The DTFT over the interval |ω|≤\pi, X\left ( e^{jω}\right )= cos\left ( \frac{ω}{2}\right ) Find: x(n) 2. Necessary Equations: IDTFT synthesis equation: x(n)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}X\left ( e^{jω} \right ) e^{j\omega n}d\omega Euler's Identity...

Homework Statement Determine the Fourier Transform of the function shown. Plot the result using excel, MathCad, or Matlab. See attachment for figure of triangle above x-axis from -X0/2 tp X0/2 with a max height of 1 at x=0. Homework Equations The answer is F(k) = X0/2 [sin(kX0/4) /...

Hi, I have problems finding out the Fourier transform of the following function, 1/\sqrt{q^2 + m^2}, where m\neq 0 denotes a parameter. It seems easy, but I don't know how. Could anybody show me how to do it ? Thanks in advance. hiyok

Homework Statement Find the inverse Fourier transform of f(w)=1 Hint: Denote by f(x) the inverse Fourier transform of 1 and consider convolution of f with an arbitrary function. Homework Equations From my textbook the inverse Fourier transform of f(w)=\int F(w)e^-iwt dw The...

Dear Physics Buddies, How are well all, okay I hope. I was wondering if I might browse all your infinite intellects and ask you a very simple question. I am working with some medical images in MATLAB and my collaborators would like to know the orientation of the fibres that it contains...

After searching on the web and reading a bit, I found that lenses can perform Fourier transform. All you need to do is put a transparant object in front of it, like a transparant sheet with black stripes on it and a screen behind the lens(so basically a 4f setup). The lens will then perform a...

So the complex exponential Fourier series form an orthonormal basis for the space of functions. A periodic function can be represented with countably many elements from the basis, and an aperiodic function requires uncountably many elements. Given a signal, we can find the coefficients of the...

Using geometric evaluation of the magnitude of the Fourier transform from the corresponding pole-zero plot, determine, for each of the following Laplace transforms, whether the magnitude of the corresponding Fourier transform is approximately lowpass, highpass, or bandpass. \[ H_1(s) =...

Homework Statement the function is Exp[-w^2]/w^2, how to solve the Fourier transform analytically with Residue theorem? It is better if there is more general results. Mathematica can solve it analytically, but I need a human-soluable way. Homework Equations The Attempt at a...

##\varphi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int^{\infty}_{-\infty}dx\psi(x)e^{-\frac{ipx}{\hbar}}##. This ##\hbar## looks strange here for me. Does it holds identity ##\int^{\infty}_{-\infty}|\varphi(p)|^2dp=\int^{\infty}_{-\infty}|\psi(x)|^2dx=1##? I'm don't think so because this ##\hbar##. So...

Hello, I was wondering if one can give meaning to the Fourier transform of the linear function: \int_{-\infty}^{+\infty} x e^{ikx}\, dx I found that it is \frac{\delta(k)}{ik} , does this make sense?

Okay, I am trying to determine the Fourier transform of cos (2\pix)=f(x) Where F(k)=^{\infty}_{\infty}\intf(x)exp^{-ikx} dx, So I use eulers relation to express the exponential term in terms of cos and sin, and then I want to use sin/cos multiplication integral results, such as...

Homework Statement Homework Equations sinc(x) = \frac{sin(x)}{x} The Attempt at a Solution bit unsure how to get started?? i know transform of rectangular pulse pτ(t)=τ*sinc(τω/2∏) also that sin(ωt)= ejωt-e-jωt / (2) I could also probably sketch sinc(t/2∏), if that helps.

Dear all, I have recently come across the following Fourier transform (FT): I=\int_{-\infty}^{\infty} dx \, e^{-\imath x t} \frac{(1-x^2)}{(1+x^2)^{3/2} (a^2+x^2)}. The integrand contains two branch points on the imaginary axis, plus two poles also residing on the imaginary...

I understand that the Fourier transform maps one function onto another. So it is a mapping from one function space onto another. My question is, why is it often referred to as a mapping from time domain to the frequency domain? I don't understand why the image of the Fourier transform...

Hi guys, I've been using this site for a while now, but this is going to be my first post. I want to pick your brains to get some insight on this problem I'm tackling. I'm trying to take a Fourier Transform of a function. My function is a function of (r,phi) and it is a piecewise function...

Hello, I was wondering if such a thing even exists, so here it goes... Let's say I have a function x(s) (it is real, smooth, differentiable, etc.) defined on (0,1). In addition, dx/ds = 0 on the boundary (s=0 and s=1). I can compute its Fourier transform (?) as a_p = \int_0^1 x(s)...

Hi, I hope somebody can help me with this one. Homework Statement Compute the Fourier Transform of the distribution x-a Homework Equations The Fourier Transform of a distribution is just the distribution evaluated with the Fourier Transform of a test function.The Attempt at a Solution See...

I have an exercise with a function of the form: h(t) = f(t)g(t) and f(t) and g(t) both have discrete Fourier series, which implies that h does too. I want to find the Fourier series of h, so my teacher said I should apply the convolution theorem which would turn the product above into a...

Homework Statement Part (a): State inverse Fourier transform. Show Fourier transform is: Part (b): Show Fourier transform is: Part (c): By transforming LHS and RHS, show the solution is: Part(d): Using inverse Fourier transform, find an expression for T(x,t) Homework Equations The Attempt...

http://calclab.math.tamu.edu/~fulling/m412/f07/airywkb.pdf Can someone walk me through this derivation of the Airy integral by Fourier transform? I have tried it but failed

I'm trying to prove that the discrete form of the Fourier transform is a unitary transformation So I used the equation for the discrete Fourier transform: ##y_k=\frac{1}{\sqrt{N}}\sum^{N-1}_{j=0}{x_je^{i2\pi\frac{jk}{N}}}## and I put the Fourier transform into a N-1 by N-1 matrix form...

From -infinity to infinity at the extreme ends do Fourier transforms always converge to 0? I know in the case of signals, you can never have an infinite signal so it does go to 0, but speaking in general if you are taking the Fourier transform of f(x) If you do integration by parts, you get a...

Homework Statement What is the Fourier transform of a single short pulse and of a sequence of pulses? The Attempt at a Solution In class we haven't dealt with the mathematics of a Fourier transform, however my professor has simple stated that a Fourier transform is simply a equation...

I have a tutorial question for maths involving the heat equation and Fourier transform. {\frac{∂u}{∂t}} = {\frac{∂^2u}{∂x^2}} you are given the initial condition: u(x,0) = 70e^{-{\frac{1}{2}}{x^2}} the answer is: u(x,t) = {\frac{70}{\sqrt{1+2t}}}{e^{-{\frac{x^2}{2+4t}}}} In this course...

I learned how to integrate it using the complex plane and semi circle contours but I was wondering if there is a way using Fourier transforms. I know that the Fourier transform of the rectangle wave form is the sinc function so I was thinking maybe i could do an inverse Fourier on sinc x and get...

Hello, Recently I've learned about Fourier Transform, and the uncertainty principle that is arose from it. According to Fourier Transform, if there is only one pulse in a signal, then it is composed from a lot more frequencies, compared to the number of frequencies that are building a...

Fourier transform of RF signal with a "prism"? We can use a prism to decompose visible light into components of different frequencies. This is a Fourier transform by nature. For an ideal prism, the energy is conserved in the process. How about RF signals? There is no fundamental difference...

Hi guys, I'm not sure how they got from first step to the second. Did they use integration by parts? I tried but I didn't arrive at the same result..

\mathcal{F}\{f(r)\}=\int e^{i\vec{k}\cdot \vec{r}}f(r)d\vec{r} in spherical polar coordinates \mathcal{F}\{f(r)\}=\int^{\infty}_0r^2dr\int^{\pi}_0\sin\theta d\theta\int^{\pi}_0d\varphi e^{ikr\cos \theta}f(r) Why could I take ##e^{ikr\cos \theta}## and to take that ##\theta## is angle which goes...

The windowed Fourier transform on R Definition-Proposition-Theorems (Plancherel formula-Parseval formula-inversion formula-Calderon's formula) http://www.4shared.com/office/b2Ho5n7H/The_windowed_Fourier_transform.html