Fourier transform Definition and 1000 Threads

So its been awhile since I've taken PDE, and forgot a lot about Fourier transforms. Anyways I'm trying to understand what the inverse of the Fourier transform actually represents. I understand perfectly how the infinite sum of periodic functions can be used to create any periodic function when...

Hello, I'm dealing with the proof of the theorem below: x(t) \in L^1(\mathbb{R}), tx(t) \in L^1(\mathbb{R}) \Rightarrow \mathfrak{F}[x(t)] \in C^1(\mathbb{R})and\frac{\mathrm{d} }{\mathrm{d} \omega}\mathfrak{F}[x(t)](\omega) = \mathfrak{F}[-jtx(t)]I'm going to write down an interesting proof...

The result is well known, but i need more details about the integral below \int \mathrm{d}^2x \frac{1}{|\mathbf{x}|} e^{- \mathrm{i} \mathbf{q} \cdot \mathbf{x}} = \frac{2 \pi}{q} I've done the Fourier transform of the Coulomb potential in 3D. But failed to get the right answer in 2D...

I understand Fourier Series fairly well and how to use them to approximate functions (I even wrote a C program to do it) but Transforms are really confusing. If I was to take the value of each Fourier coefficient and plot it on the y-axis against the angular frequency on x-axis (obv. there...

Hi i am stuck with a program in MATLAB to find the time domain impulse response of a system from its continuous transfer function in the frequency domain. Here is the program- delt=1.5625e-9; %definition of delta t(sampling time).To be taken sufficiently small depending upon the time...

Hi all, I have discrete data of a signal but I do not know the periods of the signal. The signal is like a "beat" I guess, but not really sure. I plan to use fft in MATLAB to get it's frequency spectrum and get the 0Hz value as the average of the signal. Is this a bad idea? Any other ways...

Hello, I read somewhere that if a function f decays rapidly (e.g. \lim_{x \to \infty}f(x)=0 ), then its Fourier transform F is smooth. How can I prove this? (Reference to some sources are welcome too). Thanks.

Homework Statement In lab, I obtained a single slit diffraction pattern and recorded an image of it. The slit width is known to be 0.000134 m. We are supposed to compare our experimentally-obtained diffraction pattern to the result of taking a Discrete Fourier Transform of the aperture in...

From my undergraduate textbook: Circuits, Signals, and Systems by Siebert, p 453 ==================================================== Consider the two principal waveform representations schemes ... x(t) = \int x(\tau)\delta(t - \tau)d\tau x(t) = \int X(f)e^{j2\pi f t}df...

Hi all I have a function F, which depends on a discrete variable x, and I need to Fourier Transform it. I have put all the values of F in a table. Then I have used the command "Fourier" on the table, which - according to http://reference.wolfram.com/mathematica/ref/Fourier.html - results...

Given; \frac{d^{2}u}{dx^{2}} = \frac{1}{c^{2}} \frac{d^{2}u}{dt^{2}} and; u(x,0) = \phi (x) \frac{d^{2}u(x,0)}{dt^{2}} = \theta(x) Show that the Fourier Transform of the u(x,t) w.r.t. to x is; \tilde{u}(k,t) = \tilde{\phi} (k) cos(ckt) +...

Homework Statement The problem is to obtain the inverse Fourier transform of the following 2D functions F(\mathbf{k})=\frac{k_{x}k_{y}}{k^{2}} Homework Equations The relevant equations are the 2d Fourier transform formulas described...

Homework Statement Hi, I need to find the Fourier Transform of: g(t)=\frac{1}{x}e^{\frac{-\pi t^2}{x^2}} Homework Equations G(f)=\int_{-\infty}^{\infty }g(t)e^{-j2\pi ft}dt \therefore G(f)=\int_{-\infty}^{\infty }\frac{1}{x}e^{\frac{-\pi t^2}{x^2}-j2\pi ft}dt The Attempt at a Solution...

Why are they useful, what do they denote (physically or otherwise)...

how could i calculate the Fourier transform \int_{-\infty}^{\infty}dx \frac{e^{iux}}{(a^{2}+x^{2})^{s}} if i try contour integral i find 2 poles at x=a and x=_a but of order 's' which can not be an integer, is there another definition or faster way to calculate the Fourier transform of...

Homework Statement A the density of a gas \rho obeys the modified diffusion equation \frac{\partial \rho(x,t)}{\partial t}-D\frac{\partial^2 \rho(x,t)}{\partial x^2}=K\delta(x)\delta(t) A) Express \rho in terms of its 2D Fourier transform \widetilde{\rho}(p,\omega) and express the right...

i have found the general solution which is, u(x)= (C1 + C2x)e^ax + (1/2a)[tex]\int f(x-y) e^\left|y\right| dy is this correct?? now, i just want your help to guide me for justifying f(x)=x^5... is that wrong if i solve the integration and just substitute the integral which is the...

I have run the following command: c = wavread('sample.wav'); amplitude = log (abs(fft(c))); and obtained the following plot: http://img179.imageshack.us/img179/8733/withoutplusone.jpg however, i was told to use this instead: amplitude = log (1+abs(fft(c))); and obtained the...

ok well I'm pretty much home and dry in this problem the aim of this problem is to get the general solution for the ode below.. 2u'' - xu' + u = 0 = g(x) i started to solve it by rearranging the equation.. 2u'' + u = xu' apply Fourier transform.. 2F(u'') + u^ = g^ (-2k^2)u^ + u^...

Here's a question solved: rect (t/2) = 2 sinc (w) T=3: f(t+3)+f(t-3) <=> 4 sinc (w) cos 3w I don't understand how "4 sinc (w) cos 3w" comes in the final answer. Is there any step missing? Thanks.

Homework Statement This is an example provided by my lecturer in his notes. He puts practically zero working in. When i work the problem through i do not get the same answer as he does. In this section i have copied the exact text from the problem: Find the Fourier transform of...

hey there.. i really don't know how to start answering this question.. can someone please guide me to solve this question..

Hi everybody, I'm in the process of writing a discrete Fourier transform program using the algorithm on the DFT wikipedia page. When I throw in functions that I know the frequency domain signal of it gives the predicted shape but I have absolutely know idea how to generate a frequency axis...

Short Time Fourier Transform -- invertible? On Wikipedia, http://en.wikipedia.org/wiki/Short-time_Fourier_transform" However, it's also said That is to say, Gabor is invertible, it's able to obtain the original signal, but modulated. original signal is obviously x(t), w(t-τ)...

I have read that the time dependent wavefunction is related to the Fourier transform of the wavefunction for the angular wavenumber like so \bar{\psi}(k,t) = \frac{1}{\sqrt{2\pi}}\int \psi(x,t)e^{-ikx}dx Can anyone explain why it is relevant to take the Fourier transform of the...

I've been asked to write a function (.m file) in Matlab to calculate the discrete Fourier transform coefficient for an arbitrary function x. So far this is what I've done: function a = mydft(x,N) %MYDFT Calculates the discrete Fourier transform %usage: %[a]=mydft(x) %x=[ x[0] x[1] ... x[N-1] ]...

Homework Statement Determine the Fourier transform on the tempered distribution: \langle f, \varphi \rangle Where f can be given by they taylor series representation: f = i\sum_{n=0}^{\infty} \frac {x^{3n+2}}{(2n)!} The Attempt at a Solution Fourier transform on tempered distribution...

Homework Statement Hi, I'm taking a Chemistry NMR & MRI class and were going over the part where the time domain graph is converted to the frequency domain graph by way of the Fourier Transform. I took Applied Mathematics last semester and we spent a week on the Fourier Transform but no time...

Homework Statement The solution of Schrodinger’s equation for a free particle can be written in the form: \psi(x,t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi(k)e^{i(kx-wt)}dk [Q1]: Explain why the function \phi(k) is given by: \phi(k) =...

Homework Statement A differential equation is given by: \frac{\partial^{3}u}{\partial x^{3}} + 2 \left( \frac{\partial u}{\partial x} \right) = \frac{\partial u}{\partial t} By first Fourier transforming the equation (*) with respect to x, show by substitution that: u(k,t) =...

Homework Statement The Fourier transform of a function f(x) is given by the product of the Fourier transforms of cos(\alpha x) and e^{-|x|} ; f^{~} = F^{~}\left[cos[\alpha x]\right]F^{~}\left[e^{-|x|}\right] Find f(x) and show that it can be written as a real function. Note: Do...

Homework Statement One has a function g(x) that has a periodic nature, but the period is unknown (the term 'period' is used a bit loosely). To be specific, the g(x) (i.e. a signal) appears to oscillate, but the displacement between each oscillation is unknown (nothing is known about whether...

My math professor doesn't include the correcting (normalizing factor) neither in the Fourier transform nor in the inverse Fourier! He says that it's optional. It is weird because I have seen it used in all the textbooks! Is it a big deal? He defines the Fourier transform as:-...

Hi guys, I'm having some issues understanding something about the Fourier transform. In my first signals and systems class we used the angular frequency omega. Doing it like that you end up with a weighing factor or 1/(2pi) when you take the transform. Now in the dsp class I am taking now we...

Homework Statement I'm reviewing my Fourier transforms (useful in quantum mechanics, in this case 1-dimensional representation), and I'm having a heck of time *explicitly* solving the Fourier transform of \psi(x) = sinc(x) \phi(p) = F \left\{ \psi(x) \right\} = \frac{1}{\sqrt{2 \pi \hbar}}...

Hi everyone, I uploaded a solution about Fourier transform. At the solution of this problem, it states that make convolution. But i tried to do convolution but my result is not same with this result. When you do the convolution with 2.10 and 2.11, is the result 2.13 correct ? How is it done ...

Homework Statement f(x) = {-1, -pi<x<0 ; 1, 0<x<pi ; 0, |x|>pi} Find the exponential Fourier transform of the given f(x) and write f(x) as a Fourier integral. Homework Equations The Attempt at a Solution I have the equations for the Fourier transforms and I know how to find...

Guys, how do u get the Fourier transform of a product of Greens Functions?I have to get Fourier transform of: G_{el}(k+q,\tau-\tau1)*G_{el}(k,\tau1) where \tau and \tau1 are two different times (\tau>\tau1) and q is phonon momentum and k is electron momentum... Thanks

Homework Statement Determine the Fourier transform of the following: u(t) - u(t-T) where u(t) is a unit step function. Homework Equations The Attempt at a Solution I know that the Fourier transform of u(t) is pi*delta(w) + 1/jw but when u(t-T) comes into the picture, the...

Homework Statement Find the Fourier transform of f(t) = e-at2, with a > 0 using the residue theorem. The Attempt at a Solution The problem I have is the function g(t) = f(t).e-iwt, which is holomorph in all C. Is there another way to do it without residues?

fourier transform simple question?? pls help can anyone help me in finding the Fourier transform of this signal?? i need quick help

Homework Statement Find the Fourier transform of f(x)=\frac{1}{(x^2+a^2)^2},\ a>0, and show by direct calculation that with inverse Fourier transform you'll get the original function f(x)! Homework Equations Fourier transform and it's inverse...

Homework Statement Im having a real problem trying to imagine what happens inside the machine can anyone help? I thought that at the interferometer a beam is split into two and then recombined with a path difference to introduce interference. Because the IR source is broadband would all of the...

Homework Statement The sufficient condition of the existence of the Fourier transform of a function is that the function is absolutely integrable. I have identified a function that is absolutely integrable, but not square-integrable f(t) = \frac{1}{\pi}\frac{1}{1+t^2}|t|^{\frac{-1}{2}}...

Consider the following H1 NMR signal from the human brain obtained at 1.5 T S(t)= S(0) { 3 exp (-iw_NAA*t)* exp (-t/T2_NAA*) + exp(-iw_Cr*t)*exp(-t/T2_Cr*) + exp(-iw_Cho*t)*exp(-t/T2_Cho*)} where the 3 terms shown rep. the contributions from the major peaks of total...

The "Free Induction Decay signal" (FID) is a particular type of NMR signal observed in both MRI and MRS. An idealized representation of the signal Sf(t) is given by Sf(t)= Sf(0) exp (-i2pi(f_0)(t))*exp(-t/T2*) t>=0 Sf(t)= 0 it was proven that Gf(f) corresponding to this signal is given...

\frac{u^3}{u^4+4} what is the transform of this function?

cant understand this transformation i know that each derivative pops iw and \hat{y}' ->-ixy(x) \hat{y}'(\omega) ->-ixy(x) x is a signs of derivative but i don't know how its been done in here -ixy'(x)=(i\omega \hat{y}(w))' how they decided that is the derivative of this whole...

Fourier Transform question... please help.. work shown! The "Free Induction Decay signal" (FID) is a particular type of NMR signal observed in both MRI and MRS. An idealized representation of the signal S(t) is given by: S(t)= S(0)exp (i*w_0*t)exp(-t/T2*), t>=0 S(t)=0 , t<0 You showed...

The "Free Induction Decay signal" (FID) is a particular type of NMR signal observed in both MRI and MRS. An idealized representation of the signal S(t) is given by: S(t)= S(0)exp (i*w_0*t)exp(-t/T2*), t>=0 S(t)=0 , t<0 You showed the spectrum G(w) corresponding signal is given by...