Fourier transform Definition and 1000 Threads

How you get Fourier transform from Fourier series? Do Fourier series becomes Fourier transform as L --> infinity? http://mathworld.wolfram.com/FourierTransform.html I don't understand where discrete A sub n becomes continuous F(k)dk ( where F(k) is exactly like A sub n in Fourier series)...

Hi Do Fourier Transforms give us actual amplitude/phase of the particular frequency (ejωt) just like Fourier series? Thanks Salil

I'm taking the Fourier transform of a signal. This integral has bounds from -∞ to ∞, but since the signal is 0 for negative t, the bounds become 0 to ∞ doing the integration, the antiderivative I get is et*(-3-jω+2j) where j is sqrt(-1) Now I have to evaluate this at t=infinity (since it is a...

I need to use the Fourier transform to solve the wave equation: $\begin{aligned} & {{u}_{tt}}={{c}^{2}}{{u}_{xx}},\text{ }x\in \mathbb{R},\text{ }t>0, \\ & u(x,0)=f(x), \\ & {{u}_{t}}(x,0)=g(x). \end{aligned} $ So I have $\dfrac{{{\partial }^{2}}F(u)}{\partial...

Homework Statement So this is a physics problem, but this question doesn't really have to do with the "physics" part of it as much as simply calculating the Fourier transform. (This is a second year physics course and our prof is trying to briefly teach us math tools like this in learning...

Hello all, first time here and I have really silly problem... I am working on something in MATLAB, in which I have to make discrete Fourier transform of gaussian distributed variable. i.e. array of numbers which are taken from f(x)~exp(-x^2). I know that when you Fourier transform it with...

It is fairly easy to demonstrate that the Dirac delta function is the Fourier transform of the plane wave function, and hence that: \delta(x)=∫_{-∞}^{∞}e^{ikx}dk (eg Tannoudji et al 'Quantum Physics' Vol 1 p101 A-39) Hence it should be the case that ∫_{-∞}^{∞}e^{ik}dk = \delta(1) = 0...

I have a question, specifically to physics people, on their definition of the Fourier Transform (and its inverse by proxy). I'm an EE and math person, so I've done a lot of analysis of (real/complex) and work with (signal processing) the transform. In a physics class I'm taking, the professor...

Homework Statement Calculate the Fourier transform of f(x) = (1 + |x|2)-1, x\inℝ3 The attempt at a solution As far as I can tell, the integral we are supposed to set up is: Mod note: Fixed your equation. You don't want to mix equation-writing methods. Just stick to LaTeX. $$\int...

Find the Fourier transform of the unit rectangular distribution f(t) = 1 for ltl<1 else 0 Since e-iωt is zero except for t in ]-1;1[ it must be an integral over this interval. But should I take the boundaries as -1 and 1? Because they are not included in the interval where e-iωt is not zero but...

Homework Statement Find the Fourier transform of f(t)=exp(-ltl)Homework Equations The expression for the Fourier transform. The Attempt at a Solution Applying the Fourier transform I get an expression, where I have to take the limit of t->-∞ of exp(-i\omegat) - how do I do that?

In my book the dirac delta is described by the equation on the attached picture. This realtion is derived from the Fourier transform, but I'm not sure that I understand what it says. If u=t it is clear that one gets f(u) in the Fourier inversion theorem. But why wouldn't u=t? In the derivation...

Homework Statement Hi, the question is from a piece of coursework and before hand we were asked to find the Fourier transform G(K) of the function g(x)= e^(-∏(x^2)) (where g(x)= ∫ G(K)e^2∏ikx dx (integral from -∞ to ∞)). We were told to find G(K) by forming a differential equation in H(K)...

Homework Statement Computer the Fourier transform of U(t), where U(t) = 1 for |t| < 1, and U(t) = 0 for |t| > 1. Homework Equations Fourier Transform: F(w) = ∫U(t)e-iwtdt (bounds: ∞, -∞) The Attempt at a Solution If |t| < 1, obviously F(w) = 0. If |t| > 1, F(w) = (-1/wt)*[cos(-wt) + i...

Just curious, how does one switch from the frequency version of a Fourier transform to the ω version. I know that ω = 2∏*f but looking at the variations of the table it seems like there is more than just this difference

I have a set of N data points defined over a periodic interval, 0\le x \le 1. I made a discrete fast Fourier transform of these data points and I get a discretized function in the Fuorier space. My question is what are the coordinate of these data points in the Fourier space? I mean, in the real...

Homework Statement Show that |\tilde{I}(\vec{k})|^2\leq CP^2 Where C denotes a constant, Using this inequality: \int f(\vec{r})^*g(\vec{r})\,\text{d}\vec{r}\leq \int f^*f\text{d}\vec{r}\,\int gg^*\text{d}\vec{r} Where k denotes the Fourier transform from r->k(in 3d) R is assumed positive...

Homework Statement if y(ω) = F{x(t)}, what is F{y(t)} (F is the Fourier transform operation)Homework Equations non The Attempt at a Solution I tried finding F^-1{y(ω)}, which is equal too x(t), but I could not go on with finding F{y(t)}

Hi! I have a function for which I need to calculate or at least approximate the 2D Fourier transform, that is, the Fourier transform applied twice on the function but on different variables. The function is tanh(w)/w, where w is the absolute value of the vector (wx, wy). So the function can be...

I'm confused as to what to expect when I take, for example, the Fourier transform of a sequence of 16 pulses of varying duty cycles, repeating. That is, after the 16th pulse, the entire sequence repeats. My confusion is in the interaction of the frequency components of each pulse within the...

g is continuous function, g:[-\pi,\pi]\to\mathbb{R} Prove that the Fourier Transform is entire, G(z)=\int_{-\pi}^{\pi}e^{zt}g(t)dt So, G'(z) = \int_{-\pi}^{\pi}te^{zt}g(t)dt=H(z). Then I need to show that G(z) differentiable for each z_0\in\mathbb{C}. I need to show...

Let $g:[-\pi,\pi]\to\mathbb{R}$ be a continuous function. Define the Fourier transform of $g$ as $$ G(z)=\int_{-\pi}^{\pi}e^{zt}g(t)dt, \quad \text{for all} \ z\in\mathbb{C}. $$ Prove that $G(z)$ is an entire function. That means $G$ has to have no singularities, but other than that I am lost...

Homework Statement From Discrete-Time Signal and Systems 3rd edition. Q2.4 Consider the linear constant-coefficient difference equation Homework Equations y[n] -3/4y[n-1] +1/8y[n-2] = 2x[n-1] Determine y[n] for n >= 0 when x[n] = δ[n] and y[n]=0, n<0. The Attempt at a...

Homework Statement I'm trying to relate phase shift and time shift Fourier Transformers Homework Equations x(t-t_0) → e^(jwt0)X(jw) The Attempt at a Solution I've attached a picture of my work. I'm a bit confused as to how I would be able to make that simplification towards the end...

Homework Statement If F(p) and G(p) are the Fourier transforms of f(x) and g(x) respectively, show that ∫f(x)g*(x)dx = ∫ F(p)G*(p)dp where * indicates a complex conjugate. (The integrals are from -∞ to ∞) Homework Equations F(p) = ∫f(x)exp[2∏ipx]dx G(p) = ∫g(x)exp[2∏ipx]dx G*(p) =...

Homework Statement Use Fourier transforms to calculate the motion of an infinitly large stretched string with initial conditions u(x,0)=f(x) and null initial velocity. The displacements satisfy the homogeneous wave equation. Homework Equations \frac{\partial ^2 u }{\partial t^2...

Homework Statement Fourier Transform f(t)=H(t).cos(ω0t) ,using the transform of H(t) H(t)=Heaviside function (also known as signal function if I ain't wrong) Homework Equations (1) FT[f(t)] = ∫ f(t).e^-(iωt) dt (2) FT[H(t)] = pi.δ(ω) + 1/iω (3) δ(ω) = Delta Dirac Function (4)...

Homework Statement Let f be a suitably regular function on ℝ. (whatever that means). What function do we obtain when we take the Fourier transform of the Fourier transform of f?Homework Equations F(s) = \int_{x=-\infty}^{\infty}f(x)e^{-2\pi isx}dx The Attempt at a Solution...

Homework Statement Using Fourier sine transform with respect to x, show that the solution to \frac{\partial u }{\partial t}=\frac{\partial ^2 u }{\partial x^2} with x and t >0 subject to the conditions u(0,t)=0 and u(x,0)=1 for 0<x<1, u(x,0)=0 otherwise, with u(x,t) bounded gives the solution u...

$u_t-u_{xx}=0,$ $x\in\mathbb R,$ $t>0$ and $u(x,0)=e^{-x^2}.$ By applying Fourier transform on $t$ I have $\dfrac{\partial }{\partial t}F(u)+{{\omega }^{2}}F(u)=0,$ the solution of the latter equation is $F(u)(\omega,t)=ce^{-\omega^2t},$ now by applying the initial condition I have...

I need to apply Fourier transform to solve the following: $t^2u_t-u_x=g(x),$ $x\in\mathbb R,$ $t>0$ and $u(x,1)=0,$ $x\in\mathbb R.$ How do I apply the Fourier transform for $t^2u_t$ ? Thanks!

Homework Statement Find the Fourier transform of A∏((t-T)/(2*T)). Homework Equations V(F)=∫v(t)*e-j*2*pi*f*t The Attempt at a Solution ∫(A*e-j*2*pi*f*t,t,0,2T) = A*sin(4*f*pi*T)/(2*f*pi*T)+j(A*cos(4*pi*f*T)-A)/(2*pi*f) =2*A*T*sinc(4*f*T)+j*(A*cos(4*pi*f*T)-A)/(2*pi*f) The...

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...

I have some questions regarding shifting the phase of a Fourier transformed spectrum : I have a spectrum with flux on the Y-axis and wavelength on the X-axis. What I want to do is take the Fourier transform of this spectrum. Then add a random phase between 0 and 2pi to the phase only. Then...

Homework Statement An atom raised at t=0 to an excited state with energy [itex] E_0= \hbar \omega_0 [itex] has the time dependence [itex] T(t)=\frac{1}{\sqrt \tau}e^{-t/ 2 \tau} [itex] for t>0 and T(t)=0 for t<0. Thus the probability of being in an excited state decays exponentially with...

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

Hi, i do not understand why i can find the FT of sin in mathematica using the built in function but not by integrating, even thoiugh they should be the same: Integrate[Exp[i(ω-ω0) t,{t, -∞, ∞}, Assumptions ->ω0 el Reals && ω el Reals] but the statement FourierTransform[Sin[ω0 x]...

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...