Fourier transform Definition and 1000 Threads

Homework Statement F(w) is the Fourier transform of f(t). Write down the equation for F(w) in terms of f(t). A rectangular pulse has height H and total length t0 in time. Show that as a function of w, the amplitude density is propertional to sinc(wt0/2). Homework Equations F(w) =...

Homework Statement Hey guys. I need to find the Fourier transform of sin, is this right? http://img156.imageshack.us/img156/5531/scan0004r.jpg I searched the internet but all I could find is the answer with the dirac delta and I don't need that. Thanks. Homework Equations...

Homework Statement The Attempt at a Solution I'm really at a loss on this question, which is why i have achieved so little on it so far. I think i more or less understand what a Fourier transform does (transpose amplitude vs time to amplitude vs frequency, ie the Fourier transform...

I am new to wavelets. I was reading about wavelets and Fourier transforms. So the main disadvantage of Fourier Transform is that you cannot use it on a non-uniform signal. Even though you use it you have to use a window and select your region of interest. If the window is small enough you can...

Homework Statement Using Matlab, I need to record vowels (a, e, o, i, etc) of my voice and apply a digital Fourier transform (DFT) to the recorded sound signal. The resulting signal should be a periodic train of 'spikes', with modulated amplitude. The modulation of the amplitude is what forms...

To find the frequency, Why do you need to consider the signal over long period of time? For example - if you look at a sine wave from 0-360 with two cycles, isn't it enough to get the frequency? I get the second part - you need a short time window to see sudden changes in frequency.

Homework Statement Find the discrete Fourier transform X[k] = DFTn {x[n]} of the following periodic sequences x[n] = x[n - N] with period N: (a) For n = 0 . . .N - 1 we have x[n] =\delta[n]. (b) For n = 0 . . .N - 1 we have x[n] = \mu[n] -\mu[n - K] with K < N. (c) x[n] = cos( (2*pi*M*n)/N...

Hi all, How is the Fourier transform applied to non-periodic functions, such as the Rect function? Any help would be greatly appreciated, Cheers, Jamie :)

Taking a Fourier-transform of a real signal, gives me a spectrum that has symmetry. If I take the FFT of a real signal, then throw away half of the spectrum, and then do an inverse transform I get a complex-signal. I go from r(t) to rc(t) where rc(t) is a complex-signal. Now this...

Homework Statement an exponentially decaying sinusoid is defined as f (t) = a exp (-t/towel) exp (i2(pie)vt) ; t greater than or equal to 0 0 ; t less than zero Homework Equations i have to...

Confusion with Fourier Transform and Step Function...clarification needed please :) I am required to find the Fourier Transform of (without integration): s(t) = 1 for 0 < t < 4; -t/2 for 4 < t < 6. I understand that for: s(t) = t for 0 < t < 1; 1 for t > 1 that this is the same as...

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) =...

Homework Statement Which of the following things are true given the sound data that is represented in the two graphs above? Select ALL that are true. (One graph of Sound Pressure v Time with what looks like a repeating pattern & one graph of Amplitude v Frequency with 2 large spikes at 512Hz...

Homework Statement I'm trying to do the Fourier Transform of "x*f(x)". According to some web sites (http://everything2.com/index.pl?node=fourier%20transform), it is simply a "property of integration", but I've having some issues. Homework Equations I'm using the form: 1/2pi int ( x...

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...

Determine the Fourier transform of the following signals: x(t) = \frac{1}{(1+t^2)} I start off by doing X(f) = \int{x(t)*e^{-j2\pi{}ft} dt} So i plug in x(t) into that equation, but I'm lost as to how to integrate it. Am i going in the right direction? Also: x(t)=abs(t) for...

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...

I'm reading the text Understanding Digital Signal Processing Second Edition and in the text they give the IDFT without any proof and so I tried to do a quick proof, but I have not been able do it here is my attempted steps: X(m)=\sum^{N-1}_{n=0}x(n)e^{-i2\pi mn/N} \\ Considering x(1)...

Homework Statement \begin{subequations} \begin{eqnarray} \dot{\hat{{\cal E}}}(t) &=& -\kappa \hat{{\cal E}}(t) + i g\int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta)\, \Bigg\{ e^{-(\gamma + i\Delta)(t-t_{0})}\hat{\sigma}_{ge}(t_{0},\Delta)+ ig\int_{t_{0}}^{t} d t' \hat{{\cal...

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 Let a function f on (-\infty, \infty) be defined as f(x) = cos x, if |x|<1; f(x) = 0, otherwise Find the Fourier transform of f and then evaluate the integral \int ^{\infty}_{\infty} \frac{sin 2w}{w} cosw dw 2. The attempt at a solution I calculate the Fourier...

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...

Homework Statement Find the Fourier transforms of the following distributions: log|x|, d/dx log|x|. The Attempt at a Solution I'm starting with the second distribution: \langle F(pf(\frac{1}{x})),\phi \rangle = \langle pf(\frac{1}{x}),F(\phi) \rangle where F() is the Fourier transform...

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...

Hi, Did anyone know how to do the Fourier transform of the hyperbolic secant? I know the answer; it's given in the text (I'm reading Ablowitz, Fokas, Complex Variables), it's another hyperbolic secant, but I want to know how to do it. My dilemma is: a) what contour to use? I'm having...

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...