Transform Definition and 1000 Threads

I found this formula in a paper: \int exp( \frac{x1 + i x2}{ \sqrt 2} \eta^* - \frac{x1 - i x2}{ \sqrt 2} \eta) D(\eta)/ \pi d^2 \eta the author calls it the Fourier transform of D. It is the first time thar i see this formula. How common is this notation? Can we use it without problem?

Homework Statement g(t) = (t+1)us(t) - (t-1)us(t-1) - 2us(t-1) - (t - 2)us(t-2) + (t-3)us)(t - 3) + us(t-3) Homework Equations unit step function us(t-3) is same as u3 (t) Shift in time: L[f(t - T)us(t-T)] = e-TsF(s) us(t) ↔ 1/s t ⇔ 1/s2 The Attempt at a Solution 1/s2 + 1/s - e-s/s2 + e-s/s -...

Homework Statement Diagram for a vehicle suspension is given. Displacement of wheel is given by 'x' and and displacement of body is 'y'. Spring constant, k = (7*10^4) Nm Damping coefficient, c = (3*10^3) N/m/s mass,m = 250kg a) Make a Laplace Transform of system and utilize it to predict 'y'...

I don't know if it is the right section to post in. I have a problem with a "simple" Fourier transform. This is the function to transform: $$f(t)=\frac{\sin\left({2\pi t}\right)}{t}$$. My first idea was to write that as $$\sin\left({2\pi t}\right)\cdot\frac{1}{t}$$ but then my fantasy crashed...

Homework Statement Show that Ccos(wt+phi) = Acos(wt)+Bsin(wt) Homework Equations Trig identity that states cos(wt+phi) = cos(wt)cos(phi)-sin(wt)sin(phi) The Attempt at a Solution Ccos(wt+phi)=(Ccos(phi))cos(wt)+(-Csin(phi))sin(wt) let A = Ccos(phi) Let B = -Csin(phi) Ccos(wt+phi) =...

Show that the 3-D FT of a radially symmetric function may be rewritten as a Fourier sin transform i.e. $ \frac{1}{({2\pi})^{{3}_{2}}} \int_{-\infty}^{\infty}f(r)e^{ik \cdot r} \,d^3x = \frac{1}{k} \sqrt{\frac{2}{\pi}} \int_{-\infty}^{\infty} \left[ rf(r) \right] sin(kr) \,dr $ The example...

Homework Statement Solve the following initial value problem using Laplace transforms: y' + 4y = 3t3 e−4t ; y(0) = 0 . Useful information: Recall that the Laplace transform of y 0 is pY − y(0), where Y is the Laplace Transform of y. The Laplace transform of tk e−at is k!/(p + a)k+1 . Confirm...

Hello, My name is Thibaut. I am looking to improve my code in python in order to have a better look a my Fourier transform. as you can see on the image, we barely see any detail of the peaks on the image. Also it's not centred. the zero order peak in on the corner, not in the centre. Any idea...

Homework Statement Use Laplace transform to solve the following ODE Homework Equations xy'' + y' + 4xy = 0, y(0) = 3, y'(0) = 0 The Attempt at a Solution L(xy'') = -\frac{dL(y'')}{ds} L(4xy) = -\frac{4dL(y)}{ds} L(y'') = s²L(y) - sy(0) - y'(0) = s²L(y) -3s L(y') = sL(y) - sy(0) - y(0) =...

I was reading this paper: http://dinamico2.unibg.it/recami/erasmo%20docs/SomeOld/RevisitingSLTsLNC1982.pdf It is on superluminal Lorentz transformations and is too advanced for me. :confused: But anyway, take a look at equation(s) (11). For the y' and z' transformations, there is an imaginary...

Homework Statement What is the Fourier transform of the function graphed below? According to some textbooks the Fourier transform for this function must be: $$ab \left( \frac{sin(\omega b/2)}{\omega b /2} \right)^2$$ Homework EquationsThe Attempt at a Solution I believe this triangular...

Hello, I am searching for some kind of transform if it is possible, similar to a Fourier transform, but for an arbitrary function. Sort of an inverse convolution but with a kernel that varies in each point. Or, like I say in the title of this topic a sort of continuous equivalent of fitting a...

Find the Fourier Transform of $ e^{-a|t|}Cosbt $ I'd like to simplify this using $Cosbt = Re\left\{e^{ibt}\right\}$ $\therefore \hat{f}(\omega) = Re\left\{ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\left(-a+ib+iw\right)|t|} \,dt \right\} = Re\left\{ \frac{1}{\sqrt{2\pi}}...

Hello, I hope I am posting this in the correct forum topic. It really is more of a "mathy" type of question, but I am posting it here because it deals with radar, and this type of math is used a lot in radar. To the mods, feel free to move it to a more suitable location if desired. I have...

Hello everybody! I'm sorry if it's not the right section to post in. I'm trying to solve this exercise: $$\frac{1}{2i\pi}*\int_{8-i\infty}^{8+i\infty}\frac{e^{s(t-5)}}{(s+4)^2}ds$$ The request is to find the result in function of $$t$$ I know i must use the Riemann inversion formula, and so the...

Alonso has in fact raised a relevant issue. Given any clock synchronization in the embankment frame, there is a unique pair of points at the embankment with coordinates x=a and x=b, and a unique time t=T, such that at t=T in the embankment frame, these points are next to the back end and the...

Homework Statement Using Parseval's theorem, $$\int^\infty_{-\infty} h(\tau) r(\tau) d\tau = \int^\infty_{-\infty} H(s)R(-s) ds$$ and the properties of the Fourier transform, show that the Fourier transform of ##f(t)g(t)## is $$\int^\infty_{-\infty} F(s)G(\nu-s)ds$$ Homework Equations...

In the picture taken from my book, in the bottom red box, it states that the equivalent resistance seen between terminals 1 and 2 is R1 + R3, implying R1 and R3 are in series. But clearly, there is a third resistor R3 at the same node where R1 and R2 meet. Then that means R1 and R3 cannot be in...

If I cut my image into several portions and use the Fast Fourier Transform on each portioned image, will I achieve the same result as if I used Fast Fourier Transform on the whole image? I have this concern because I need to process a large image using the Fast Fourier Transform, the problem is...

Hello everyone, So, i have a big test tomorrow and my professor said i should study the DC level in Fourier transform , in the frequency domain. So, i did a little research and found out that the dc level is the percentage of the time a signal is active, and that's all. Can't see how that's...

Did I do this right?View image: 20151201 152557

Did I do this right? View image: 20151201 143310 1

I'm working on a few problems to find Laplace transforms and I got stuck on this one. ${U}_{3}(t){(t-3)}^{5/2}$ It looks different from the other I've been doing so I don't really know how to get started

Homework Statement xy''+y'+xy=0, y'(0)=0, y(0)=0 Using the method of Laplace transforms, show that the solution is the Bessel function of order zero. Homework Equations -(d/ds)L{f(x)} The Attempt at a Solution The only thing I got out of this when trying to solve it was y=0. Obviously not...

So I have to solve an initial value problem involving the Laplace Transformation method. I have all the terms in Y(t) besides one term, I cannot figure how to change it from frequency domain back into time domain.Not sure how to type in Latex, so i uploaded a picture, using the whiteboard...

Show that the eigenvalues of any matrix are unaltered by a similarity transform - the book says this follows from the invariance of the secular equation under a similarity transform - which is news to me. The secular eqtn is found by $$Det(A-\lambda I)=0$$ and is a poly in $$\lambda $$, so I...

Homework Statement L-1{(2s2+3)/(s2+3s-4)2} The Attempt at a Solution I factored the denominator f(t)=(2s2+3)/((s-1)(s+4))2 now I've tried partial fractions to get (2s2+3)/((s-1)(s+4))2 = A/(s-1)2 + B(s+4)2 (2s2+3)=A(s+4)2 + B(s-1)2 by substitution, s=1 and s=-4 5=A(25) A=1/5 35=B(25)...

Homework Statement L-1{3/s1/2} Homework EquationsThe Attempt at a Solution 3L-1{1/s1/2} 3L-1{(1/sqrt(π))(sqrt(π)/(sqrt(s))} 3/(sqrt(π))L-1{(sqrt(π))/(sqrt(s))} 3/(sqrt(π))(1/(sqrt(t)) This is what I got from the solution for this problem. What tipped them off to multiply by sqrt(π)? And...

Homework Statement The differential equation given: y''-y'-2y=4t2 Homework EquationsThe Attempt at a Solution I used the laplace transform table to construct this equation,and then I did partial fraction for finding the inverse laplace transform.But I'm now stuck at finding the inverse laplace...

Fourier transform is defined as $$F(jw)=\int_{-\infty}^{\infty}f(t)e^{-jwt}dt.$$ Inverse Fourier transform is defined as $$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(jw)e^{jwt}dw.$$ Let ##f(t)=e^{-at}h(t),a>0##, where ##h(t)## is heaviside function and ##a## is real constant. Fourier...

How would I take the laplace transform of f(t)= te^tsin^2(t)?

Homework Statement [/B] I was using the Fourier transform to solve the following IVP: \frac{\partial^2 u}{\partial t \partial x} = \frac{\partial^3u}{\partial x^3} \\ u(x,0)=e^{-|x|} Homework Equations [/B] f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{f}(\omega)e^{i\omega...

1. Find the Fourier series of : $$g(t)=\frac{t+4}{(t^2+8t+25)^2}$$ 2. I have been trying to write the function to match the formula $$\mathcal{F} [\frac{1}{1+t^2}] = \pi e^{-\mid(\omega)\mid}$$ 3. I have simplified the function to $$(t+4)(\frac{1}{9}(\frac{1}{1+\frac{(t+4)^2}{9}})^2)$$...

What is the relationship between the Fourier transform of a periodic function and the coefficients of its Fourier series? I was thinking Fourier series a special version of Fourier transform, as in it can only be used for periodic function and only produces discrete waves. By this logic, aren't...

Homework Statement Evaluate the Following integrals 1. http://www4b.wolframalpha.com/Calculate/MSP/MSP10141fif9b428c5bab0b00005dc489hi851d28h7?MSPStoreType=image/gif&s=37&w=164.&h=35. Homework Equations...

The ordinary differential equation, with initial values,shall be solved using Laplace transform. The ODE looks like this \begin{equation} y''(t')+2y''(t)-2y(t)=0 \end{equation} And the initial conditions are \begin{equation} y(0)=y'(0)=0, y''(0)=0 \end{equation} The problem is with the first...

Homework Statement I'm supposed to be using the similarity theorem and the shift theorem to solve: cos(πx) / π(x-.5) has transform e^(-iπs)*Π(s) Homework Equations similarity theorem f(ax) has transform (1/a)F(s/a) shift theorem f(x-a) has transform e^(-i2πas)F(s) The Attempt at a Solution...

Kindly help me find the Inverse laplace transform of the attached.

Homework Statement I have question on doing the following indefinite integral: $$\int{d^3x(\nabla^2A^{\mu}(x))e^{iq.x}}$$ Homework Equations This is part of derivation for calculating the Rutherford scattering cross section from Quarks and Leptons by Halzen and Martin. This books gives the...

Homework Statement Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teacher is against using it..) y'' + 2y' + 2y = 2 ; y(0)= y'(0) = 0 Homework Equations Lf'' = ((s^2)*F) - s*f(0) - f'(0) Lf' = sF - f(0) Lf = F(s) The...

Homework Statement Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teach is against using it..) y'' - 4y' + 3y = 0 ; y(0)=2 y'(0) = 8 Homework Equations Lf'' = ((s^2)*F) - s*f(0) - f'(0) Lf' = sF -...

I read about how MRI works briefly, by flipping the water molecules using a magnetic field to the correct state then send the radio wave to these atoms and have it bounces back to be received by receiver coils and apply Fourier Transform to figure out the imaging. My question is, how does...

3d Fourier transform of function which has only radial dependence ##f(r)##. Many authors in that case define \vec{k} \cdot \vec{r}=|\vec{k}||\vec{r}|\cos\theta where ##\theta## is angle in spherical polar coordinates. So \frac{1}{(2\pi)^3}\int\int_{V}\int e^{-i \vec{k} \cdot...

I'm learning digital signal processing in my engineer class, but I'm more interested in apply these things into Astrophysics, so i know a little bit about for what is useful the Fourier Transform, so i thought why not use this in Analyzing the sun spectra! But what do you think!? Is it useful...

Homework Statement So I know 1/(s-a)=e^(a1), but why is say, 2/((s+4)^2) equal to 2xe^-4x? Do I just simply add an X if the numeration is a constant other than 1?

Homework Statement Find a function ##u## such that ##\int_{-\infty}^\infty u(x-y)e^{-|y|}dy=e^{-x^4}##. Homework Equations Not really sure how to approach this but here's a few of the formulas I tried to use. Fourier transform of convolution ##\mathscr{F} (f*g)(x) \to \hat f(\xi ) \hat g(\xi...

In a book the Fourier transform is defined like this. Let g(t) be a nonperiodic deterministic signal... and then the integrals are presented. So, I understand that the signal must be deterministic and not random. But why it has to be nonperiodic (aperiodic). The sin function is periodic and we...

Homework Statement Let ##f(t)= \begin{cases} \sin t , \; \; 0 \le t < \pi \\ 0 , \; \; \; \; \; \text{else.} \end{cases}## Use Laplace transform to solve the initial value problem ##x'(t)+x(t)=f(t), \; \; \; x(0)=0.## Homework Equations Some useful Laplace transforms...

Homework Statement Find the Fourier transform F(w) of the function f(x) = [e-2x (x>0), 0 (x ≤ 0)]. Plot approximate curves using CAS by replacing infinite limit with finite limit. Homework Equations F(w) = 1/√(2π)*∫ f(x)*e-iwxdx, with limits of integration (-∞,∞). The Attempt at a Solution I...