Fourier transform Definition and 950 Threads

Homework Statement Using properties of the Fourier transform, calculate the Fourier transform of: sgn(x)*e^(-a*abs(x-2)) Homework Equations FT(f(x))= integral from -∞ to +∞ of f(x)*e^(-iwx) dx The Attempt at a Solution I've realized that with the signum function, the boundaries...

Homework Statement I need to find the Fourier transform of v(t)=A*e(-t) such that t≥0. Homework Equations ∫v(t)*e(-j*2*∏*f*t)dt,t,0,∞) The Attempt at a Solution ∫v(t)*e(-j*2*∏*f*t)dt,t,0,∞)=A/(4*f2*∏2+1)-i*(2*A*f*∏)/(4*f2*∏2+1) the answer should be A/(1+i*2*∏*f). It seems...

Homework Statement http://dl.dropbox.com/u/11341635/IntegrationProblem.jpg Homework Equations http://dl.dropbox.com/u/11341635/Fourier%20Transform%20Equations.jpg The Attempt at a Solution http://dl.dropbox.com/u/11341635/1st%20part%20of%20attempt.png...

Hello, this time it's hard to tell whether this is the right forum to post this thread. Suppose I have a continuous function f:\mathbb{R}\rightarrow [0,100), whose Fourier transform exists and is known. Note that the codomain of the function is composed by all the real numbers between 0 and...

Homework Statement Question: Find the Fourier series for f(x) = x(2π-x) 0<x<2π f(x) = f(x+2π)hope the pi is clear as π The Attempt at a Solution this is in the attachments

Consider the Fourier transform of a complex function f(t): f(t)=\int_{-\infty}^\infty F(\omega)e^{-i\omega t} Here t and \omega are on real axis. Let's suppose f(t) is square integrable. Here are my questions: 1) Since f(t) is square integrable, so we have...

I would be grateful if someone could help me out with the problem that I have attached. I believe I have successfully answered part (a) of the question but am completely unsure of how to approach part (b). I realize it must have to do with specific properties of the delta function but I am lost...

Homework Statement For α > 0, determine u(x) by the inverse Fourier transform u(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\ \frac{e^{ikx}}{ik+\alpha}\ dk Homework Equations The Attempt at a Solution This seemed like a relatively simple residue problem. You just note that...

Homework Statement (part of a problem) Find the inverse Fourier of F(w) = (3jw+9)/((jw)^2+6jw+8) where w is the angular frequency, w=2pi * f = 2*pi/T Homework Equations The fourier transfrom and its properties i guess. Also the exponential FT common pair exp(-at)u(t) <-> 1/(jw+a) where...

I want to do a discrete Fourier transform of the solution I have found using NDSolve, however, because the NDSolve creates Interpolating functions rather than numbers I can't do this. Any help is appreciated. I've attatched the file I'm working with. Catrin

Homework Statement calculate the Fourier Transform of the following function: Homework Equations x(t) = e-|t| cos(2t) The Attempt at a Solution ∫0-∞ et ((e2jt + e-2jt) / 2) e-jωt + ∫∞0 e-t ((e2jt + e-2jt) / 2) e-jωt The first integral is easy to calculate and equals: (1/2) *...

Fourier transform help urgent? dear friends, sorry to bug you all with things that are lengthy and rather tedious. please help me solve these questions if possible. i have tried them but i just can't do the integrals. please help FIND THE FOURIER TRANSFORMS OF THE FOLLOWING. f(t) =...

Both Fourier transform and z transform can convert discrete time domain to frequency spectrum domain. Then why do we use Fourier transform rather than z transform? What is the reason behind it? Both give us the frequency spectrum we want.

So this is a very simple question that I am having some trouble figuring out: Let s(t) be a finite energy signal with Fourier Transform S(w). Show that \lim_{w \to \infty } S(w) = 0 We know by defintion that the FT of this signal is \ints(t)e^{-jwt}dt and also that ∫|s(t)|2dt < ∞. I'm a...

Homework Statement Solve the above F.T. Homework Equations http://en.wikipedia.org/wiki/Euler%27s_formula http://en.wikipedia.org/wiki/Fourier_transform The Attempt at a Solution I use euler's formula and apply the definition of the F.S. and i get to zero, not surprisingly, as the sine is...

Hi, I have a question about the Fourier transform of \frac{1}{|\mathbf{r_1} - \mathbf{r_2}|} over a finite cube of unit volume. Where |\mathbf{r_1} - \mathbf{r_2}| is \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2} I know it looks like \sum_\mathbf{k} f_k e^{-i\mathbf{k}\cdot...

Hi, I'm pretty much an amateur in quantum mechanics. If anyone could clarify the following, that would be greatly appreciated! When you write a wave-function (phi or "amplitude" for example) in terms of basis states (either position or momentum), does it undergo a Fourier decomposition? If...

Homework Statement Let the DTFT (Discrete time Fourier transform) of a signal beY(f)= {1 0≤lfl< \frac{fs}{8} {0 OtherwiseCalc y(k) Homework Equations y(k)=\frac{1}{f_{s}}\int Y(f) e^{jk2\pi fT}df lkl≥0 The Attempt at a Solution So what I understand from this is that my Y(f) is basically 1...

For the Bartlett window below: w(t)=1-|t|/u for -u<t<u w(t)=0 otherwise the books say that the Fourier transform of it is W(f)=1/u*(sin(∏*f*u)/(pi*f)) I use symbolic toolbox of MATLAB and can find the transform of a rectangular window. But I couldn't find it in case of...

Homework Statement Compute the integral \int_{-\infty}^{\infty}\frac{\sin(a\xi)}{(1+i\xi)^3\xi}d \xi by using Parseval's formula for Fourier transform <\overbrace{f}^{\wedge},\overbrace{g}^{\wedge}>=2 \pi<f,g> where \wedge means the Fourier transform of a function The...

Clarification: I have seen in quantum mechanics many examples of wavefunctions and their Fourier transforms. I understand that a square pulse has a Fourier transform which is nonzero on an infinite interval. I am curious to know whether there exists any function which is nonzero on only a...

I would like to know how one finds the Fourier transforms of t, \frac{1}{t} and {t}^{n} with the definition of the Fourier transform as \mathscr{F}\{f(t)\}=\mathcal{F}\{f(t)\}=\frac{1}{ \sqrt{2\pi} }\int\limits_{-\infty}^{\infty}{e}^{-i\omega t}f(t)\mbox{d}t I have tried the definition of...

How does one find the Fourier Transform of 1? \mathscr{F}\{1\}=\mathcal{F}\{1\}=\int\limits_{-\infty}^{\infty}{e}^{-i \omega t} \mbox{d}t=? I tried to solve it and came up with \sqrt{\frac{2}{\pi}}\frac{1}{\omega}\lim_{t \rightarrow \infty}\sin\left(\omega t\right) but that is indeterminate...

Fourier transform of "noise" Hello, when we want to get the magnitude of the Fourier frequency spectrum of a function f we typically calculate F(\omega)=\int_{\mathbb{R}}f(x)e^{-i\omega x}dx and then consider |F(\omega)|. We can do this as long the signal (=function) is deterministic, that...

How do find the inverse Fourier Transform for the following using the transform pairs and properties? X(jw) = 1 / (2 - w^2 + j3w) Thanks!

Homework Statement find the Fourier transform, using the definition of the Fourier transform \widehat{f}(\nu)=∫^{∞}_{-∞}f(t)e^{-2 \pi i \nu t}dt, of the function f(t)=2 \pit^{2}e^{- \pi t^{2}}Homework Equations I have the answer: (1-2{\pi \nu^{2}})e^{- \pi \nu^{2}} The Attempt at a Solution...

Homework Statement Determine whether the assertions are true or false, explain. (a) If (f * g)(t) = f(t), then g(t) must be an impulse, d(t). (b) If the convolution of two functions f1(t) and f2(t) is identically zero, (f1 * f2)(t) = 0 then either f1(t) or f2(t) is identically zero...

Hi I'm trying to understand what we mean when we say that the Fourier transform is used to transform a signal from the time domain to the frequency domain and what we actually have in the frequency domain. In Fourier series we are actually using a different representation of the signal in terms...

Hi, In Fourier analysis, we can decompose a function into sine waves with different wavenumbers that travel at different speeds (i.e., for a given wavenumber k they can have different frequencies ω and therefore different speeds v = ω/k). There is no upper bound on the speed of propagation v...

Homework Statement [PLAIN]http://img600.imageshack.us/img600/161/parcq.png Homework Equations Parseval's Theorem using FT's for this is ∫^{\infty}_{-\infty} |f(t)|^{2}dx = ∫^{\infty}_{-\infty} |\tilde{f}(w)|^{2}dw The Attempt at a Solution From what I know, the Fourier transform...

Homework Statement Hi! I tried to get the inverse Fourier transform of the function: X(j\omega)=1/(jw+a) for a>0, using the integral: x(t)=(1/2\pi)\int_{-\infty}^{+\infty} X(j\omega)e^{j\omega t}d\omega I know that the inverse Fourier transform of X(j\omega) is...

Homework Statement Using that the Fourier transform of e^{|x|} is \frac{2}{\xi^2+1}. Compute the Fourier transform of \frac{x}{(x^2+1)^2}Homework Equations The Attempt at a Solution My first thought was to try and rewrite the problem in a form I recognized, tried a couple of things but what I...

Hi, This is probably trivial, but I don't see it and would therefore appreciate receving inputs. Suppose we define a state |\phi_{lr}\rangle = \sum_{n=0}^{N/r - 1}\sqrt{\frac{r}{N}}|l + n r\rangle How is the quantum Fourier transform of this state equal to...

Homework Statement If y=(1,0,0,0) and F4*c=y, find c. Homework Equations c=F4-1*y The Attempt at a Solution I'm stuck. I don't know how to get F4-1. F4-1 = (1/N) * [1, 1, 1, 1; 1 -i (-i)^2 (-i)^3; 1 (-i)^2 (-i)^4 (-i)^6; 1 (-i)^3 (-i)^6 (-i)^9] (this...

I want to input the following function so I can find the Fourier Transform of it: tri(\frac{t}{2\pi })Cos(2\pi (\frac{5}{\pi })t) I couldn't find a simple way of doing a tri function so this is what I inputted in matlab: a(t_{1}) = (\frac{t_{1}}{\pi }+1)Cos(2\pi (\frac{5}{\pi })t_{1})...

How can you read the phase spectra from a Fourier Transform? if g(t) = Sin(2\pi f_{c}t) then for the single sided spectrum, you have one frequency component at f=f_{c} with a height of \frac{1}{j} which from looking at the complex plane, corresponds to a phase of \frac{\pi }{2} (ie. g(t) =...

http://en.wikipedia.org/wiki/Square-integrable_function According to the tutorial: it says g*(x) is the complex conjugate of g but I can't get the idea from where this g(x) function comes, than why is it the complex conjugate? And it seems i can't visualize the inner product space...

well, i have to prove that the inv. Fourier transform of a gaussian (e^(-(k^2/2)) is a gaussian, i know some elementary complex analysis(never actually taken a class in it), not well enough, it seems, to find the solution to this. I tried to integrate over a circular contour, and let the radius...

Homework Statement Hi y'all, ran into some trouble with a Fourier transform Im supposed to find the Fourier transform of f(x)=\frac{1}{x^{2}+6x+13} Homework Equations Not that I know The Attempt at a Solution I tried integrating this with no luck. All help is as usual...

Homework Statement Hi I wish to Fourier transform the following expression P(t) = \int\limits_{ - \infty }^\infty {dt_1 dt_2 \chi (t - t_1 ,t - t_2 )E(t_1 )E(t_2 )} What I do is the following \int\limits_{ - \infty }^\infty {P(\omega )e^{ - i\omega t} } = \int\limits_{ -...

Homework Statement Prove; \int_{-\infty}^{\infty} \frac{sin(\gamma)}{cosh(\lambda)-cos(\gamma)} e^{i \omega \lambda}d \lambda= 2 \pi \frac{sinh(\omega(\pi-\gamma))}{sinh(\pi \omega)} Homework Equations Contour Integration/Residue Theorem? The Attempt at a Solution I have messed...

1. Calculate the finite Fourier transform of order m of the following sequences: a) uk = 1, 0\leqk\leqN-1 b) uk = (-1)k, 0\leqk\leqN-1 N even c) uk = k, 0\leqk\leqN-1 2. Homework Equations Uk = (1/N)\sumuke-2pi*i*k*j/N from j=0 to N-1 ; 0<=k<=N-1 Attempt: a) First thing that I tried is that...

Hi.. What is the Fourier transform of 1/r? Is it proportional to 1/(k^2) ? How do you prove this?

Homework Statement I'm trying to get started on a project for a course, which is about Fourier transforms. So I'm trying to find the Fourier transform of sin(2\pif0t) in order to figure something out. http://mathworld.wolfram.com/FourierTransformSine.html I don't really understand the delta...

Hi, I have some silly doubts and i read some articles about FFT but could not able to conclude my self. What are the difference between 1) FFT power sprectrum and Power sprectrum density 2) FFT Phase and magnitude 3) FFT Real and imaginary Can some make it clear to me.

Hi all, I'm trying to compute the Fourier transform of a slightly odd function, a pair of monomials in k cobbled together with heaviside theta functions: f(k)=\theta(1-k) k^{n-2}+\theta(k-1) k^{-2} where n is some integer >2. A complicating factor is that k is really the modulus of a vector...

The form of the Fourier transform I love the most (because it is very symmetric) is: f(x) = \int_{-\infty}^\infty g(\xi)e^{2\pi i x \xi}\,d\xi g(\xi) = \int_{-\infty}^\infty f(x)e^{-2\pi i x \xi}\,dx If we take \xi = p then we get: f(x) = \int_{-\infty}^\infty g(p)e^{2\pi i x p}dp =...

Homework Statement I tried to work out the FT of a sin function with a time delay using first mathematical manipulation, and then using the time shifting property. However I get two very similar, but for some reason not identical answers. Homework Equations Please open the .jpg to...

Homework Statement Hi, I tried to work out the FT of cos(theta), and sin(theta + pi/2) which should both give the exact same FT since they are the same function. However I get two different results as shown in the .jpg. Homework Equations I used the 'time shifting' property to get that...

Homework Statement Compute the Fourier transform of \phi(t)=(e^(-at))H(t) where H(t) is the Heaviside step function Homework Equations The Attempt at a Solution I am stuck in an attempt at the solution, I am confused at how the heaviside step function factors in and think...