Transform Definition and 1000 Threads

Dear all, In my quantum mechanics book it is stated that the Fourier transform of the Coulomb potential $$\frac{e^2}{4\pi\epsilon_0 r}$$ results in $$\frac{e^2}{\epsilon_0 q^2}$$ Where ##r## is the distance between the electrons and ##q## is the difference in wave vectors. What confuses me...

Hello. I am reviewing the use of the Laplace Transform to do circuit analysis and I am slightly confused about the transform of a constant voltage source. For example, let's say we have a constant voltage source V1(t) applied to a circuit for a long time - let's say it reaches steady state. We...

I know the result: \widehat{\mathscr{H}(f)}(k)=-i\sgn (k)\hat{f}(k) I want to use this to compute the Hilbert transform. I have written code for Fourier transform,inverse Fourier transform and that the Hilbert transform. My code is the following: function y=ft(x,f,k) n=length(k); %See now long...

Suppose that a parameter y= 123. That parameter is somehow "perturbed" and its instantaneous value is: y(t)= 123 + sin(t - 50°) * 9 + sin(t * 3 + 10°) * 3 + sin(t * 20 + 60°) * 4 Suppose that I don't know the above formula, but I can calculate y(t) for any t. Hence I decide to use the...

Homework Statement Homework Equations The equation describing the balance will be f(n+1)=f(n)+R/12*Dm-Cf with f(n)=initial deposit R=Annual Rate Dm=Each mouth Deposit 150 Cf= each month fee The Attempt at a Solution Can someone shed some lights on it? Thanks[/B]

Here is the Problem Statement : Find Fourier Transform of the piecewise function Can someone sheds some lights on how to start solving this? Thanks

Homework Statement FIGURE 4(a) represents a system to measure acceleration (i.e. an accelerometer). It shows a piezoelectric crystal that is connected to an amplifier and display via a length of coaxial cable2.A piezoelectric current is produced when the crystal is distorted by an applied...

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 ?

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...

Hi members, Laplace transform using differential equations.(see attached PDF file) My question d/ds(s^2 y- s Y(0)-Y'(0).)... Y(t)=sin(sqrt(t)) Y(o)=0 Now Y'= cos(sqrt(t)/2sqrt(t) Y'(0)=infinity d/ds (Y'(0)=?? can it be treated as a constant or can we change limit and differentiation??I...

Homework Statement find the laplace transform of (e^-s) / [ (s)(s-3) ] since there's (e^-s) which can be found in L { f(t-a) H(t-a) } = (e^-(as)) F(s) , so , i found a = 1 , then i found F(s) = 1/ [ (s)(s-3) ] , formula : i have attached the working below , is it correct ? btw , the...

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...

Mod note: Moved from a Homework section can i use the Laplace transform to solve a nonhomogeneous equation if i have these Initial condition s(x) and s(-x)

I'm trying to find the distribution of a random variable ##T## supported on ##[t_1, t_2]## subject to ## \mathbb{E}[V(t', T)] = K, \forall t' \in [t_1, t_2]##. In integral form, this is : $$ \int_{t_1}^{t_2} V(t', t).f(t) \, dt = K,\forall t' \in [t_1, t_2], $$ which is just an exotic integral...

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 Using: \mathcal{L}\big\{t^n\big\}=\frac{n!}{s^{n+1}}\text{for all s>0} Give a formula for the Laplace transform of an arbitrary nth degree polynomial p(t)=a_0+a_1t^1+a_2t^2+...+a_nt^n Homework Equations \mathcal{L}=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt The...

How to extract l and L from the following system of equations:

Homework Statement Given that r(t) = L^-1 (Inverse laplace) *H(S) and by making the link between the time-domain and frequency-domain responses of a network, explain in detail why the ideal “brick-wall” lowpass filter is not realisable in practice. [/B]Homework EquationsThe Attempt at...

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...

Hi there, I've recently been doing some studying into time-frequency analysis. I've covered some of the basic materials regarding the Short-Time Fourier Transform (STFT) along with the concepts of temporal and frequency resolution (along with the uncertainty principle of course). I've now...

Homework Statement Given the Laplace transform $$F_L(s) = \frac{1}{(s+2)(s^2+4)},$$ by using the complex inversion formula compute the inverse Laplace transform, ##f(t),## for the following regions of convergence: (i) ##Re(s)<-2;## (ii) ##-2<Re(s)<0;## (iii) ##Re(s)>0.## Homework Equations...

Homework Statement Im trying to understand the Legendre transform from Lagrange to Hamiltonian but I don't get it. This pdf was good but when compared to wolfram alphas example they're slightly different even when accounting for variables. I think one of them is wrong. I trust wolfram over the...

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...

hi, sorry for the bad english. exist a Laplace Transform of tanh(x)? i know math of high school, so sorry if it is a question a little silly thanks

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...

Evening All I have had a go at a laplace transform and got stuck. $$\frac{d^2v}{dt^2}+\frac R L \d v t+\frac 1{LC}v=\frac 1{LC}V_0$$ $$R=12 \Omega, L=0.16H, C=10^{-4}F, V_0=6V, v(0)=0, v'(0)=0$$ so subbing these in i get $$\mathscr L \left[ \frac {d^2v}{dt^2}+75\d v t+62500 v...

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...

Hi, I have a FORTRAN code with an array called Chi that I want to run an inverse FT on. I have defined two spaces X and K which each consist of 3 vectors running across my physical verse and inverse space. My code (If it works??) is extremely slow and inefficient (see below). What is the best...

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...

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 A process can be represented by the first order equation (4δy(t)/δt) + y(t) = 3u(t) Assume the initial state is steady (y = 0 at t = –0). (a) Determine the transfer function of this process in the s domain. (b) If the input is a ramp change in u(t) = 4t, determine the...

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...

The closest Inverse Laplace Transform from my table is $\displaystyle \begin{align*} \mathcal{L}^{-1}\,\left\{ \frac{2\,a\,s\,\omega}{\left( s^2 + \omega ^2 - a^2 \right) ^2 + 4\,a^2\,\omega ^2 } \right\} = \sin{ \left( \omega \, t \right) } \sinh{ \left( a \, t \right) } \end{align*}$ so we...

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 (2s^2) +10s / (s^2 -2s +5 )(s+1) , I have checked the partial fraction , it's correct , but according to the ans it's (e^t)[(3cos2t + 2.5sin2t)] - (e^-t), but my ans is (e^t)[(3cos2t + 4sin2t)] - (e^-t)Homework EquationsThe Attempt at a Solution

Hello everyone, I was studing Heaviside's operators for solving ODE, which I strongly recommend to have a look because it helps a lot when the differential equations have "exotic" inhomogeneous terms, but it is a method that works and you do not know exactly why. Some biographies tell that...

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...

x(t) and y(t) are related by y(t)=1/(x(t) -k), how should I derive Y(s)/X(s)?

for the right part of the notes, why the integral of (e^-su)f(u) from 0 to T will become integral of (e^-st)f(t) from 0 to T suddenly ? why not integral of (e^-s (t-nT) )f(t-nT) from 0 to T ? as we can see, u = t +nTd given/known dataHomework EquationsThe Attempt at a Solution

Hi forum, where can I find useful references on the above topic. I'd appreciate a little help. Thank you.

In derivation of the box-muller transform, the joint distribution p(x,y) = e^(-r^2/2)/(2*pi) is interpreted as the product of a uniform distribution 1/(2*pi) and an exponential distribution e^(-x/2), but isn't an exponential distribution defined as k*e^(-k*x)? What happened to the coefficient?