Fourier transform Definition and 950 Threads

Multi-Variable / Dimension Fourier Decomposition Say we have f(x, y). We can Fourier decompose it in terms of f1(y, v) and e^{\ x\ v}, f2(x, u) and e^{\ u\ y}, or both variables simultaneously f3(u, v) and e^{\ x\ v\ +\ u\ y}. Similarly for any greater number of variables or dimensions. Now, is...

Hi, I know this topic is more suited for Computing & Technology, but it has even more to do with general questions about Fourier transform capabilities. I have a question about sample restoration in Discrete Fourier Transform. Suppose we have a signal with the stack of frequencies from 1 Hz...

Homework Statement The Attempt at a Solution I don't understand this step. It's got to be some sort of identity that I missed. I also don't understand why the limits of integration change.

Fourier Transform help! (bit urgent) Hi there, I'm having a recurring problem with my Fourier transforms that I have tried really hard to figure out but I feel like I'm missing something important. It keeps popping up in my communications and signal processing papers. I keep getting FTs...

I understand that if you have a system that is linear and time invariant, that you can perform a Fourier transform on it. But that doesn't mean you need to Fourier transform it. Or does it? Is a linear, time invariant system equivalent to or in some way implies a Fourier transform? Or is the...

Homework Statement F{f(t)} is the Fourier transform of f(t) and L{f(t)} is the Laplace transform of f(t) why F{f(t)} = L{f(t)} where s = jw in L{f(t)} The Attempt at a Solution I suppose the definition of F{f(t)} is ∫[f(t)e^-jwt]dt where the lower integral limit is -∞...

Just began a serious study of the Fourier transform with a couple of books. One of them defines the Fourier transform on \mathbb R as \hat f(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-i\xi x}dx. Another defines it as \hat f(\xi) = \int_{-\infty}^\infty f(x)...

I'm trying to evaluate the following intergral using complex function theory: \begin{equation} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{e^{i(ap+aq+b\sqrt{k^2-p^2-q^2})}}{\sqrt{k^2-p^2-q^2}}dpdq \end{equation}I though that it is possible if i can calculate: \begin{equation}...

I guess this is programming and physics all combined into one but hopefully I can get some help anyway. I am doing some signal analysis of real-time streaming sensor data. I would like to do a DFFT on the data in real time as it streams in. So far pretty easy, however, there are a number of...

Hi I am trying to analytically calculate the Fourier transform attached. I am getting really stuck with the integral, can anyone help? I've attached how far I've got with it, any help much appreciated! Kind Regards, Mike

Question: Derive the relationship \int^t_{- \infty} f(\tau) d \tau \Leftrightarrow \frac{F(\omega)}{j \omega} + \pi F(0) \delta (\omega) (where \Leftrightarrow means "Fourier transforms into"). Attempt: I have already proved the relationship \frac{dg(t)}{dt} \Leftrightarrow j \omega G(...

What is the Fourier transform of $f'(ax)$, where a>0 is a constant? Firstly, I reasoned that (lets say $F[f]$ is the Fourier transform of f) $F[f('x)]=\frac{1}{a}F[f](\frac{k}{a})$ by scaling theorem, then using the derivative rule we get $F[f'(ax)]=\frac{ik}{a}F[f(x)](\frac{k}{a})$. But when I...

Ohno Potential is modeled by v(r)=\frac{U}{\alpha ^{2}r^{2}+1}. U and \alpha are constants. I try to Fourier transform it V(q)=\int V(r) e^{iqr\cos \theta}r^{2} \sin \theta d \phi d \theta dr It gives V(q) = 2 \pi U \int \frac {r \sin qr}{\sqrt{\alpha ^{2} r^{2}+1}} dr The...

Homework Statement a) Find the normalization constant N for the Gaussian wave packet \psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}. b) Find the Fourier Transform and verify it is normalized. 2. The attempt at a solution a) I think I've got \psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}} \int...

Hi everyone, I have a question on the discrete Fourier transform. I already know its a change of basis operator on C^N between the usual orthonormal basis and the "Fourier" basis, which are vectors consisting of powers of the N roots of unity. But if i recall correctly from complex...

I'm trying to find \frac{1}{2\pi}\int \limits_{-\infty}^{\infty}e^{-itx}\frac{1}{a^2+x^2}\mathrm{d}x where 'a' is a constant. First I noticed that there is \frac {\partial \arctan x}{\partial x} in this and using a substitute got \int \limits_0^{\pi / 2}\cos( t \tan x )\mathrm{d}x with some...

Homework Statement f(t)=t*e^(-2t^2) Find the Fourier Transform F(w) of f(t). It is given that when f(t)=e^[(-at^2)/2] F(w)=√(2*pi/a)*e^[(-w^2)/2a] Homework Equations The Attempt at a Solution The transform of e^(-2t^2) is easily obtained from the given information, and I got...

Hello Everyone, Actually my question is related to Window Fourier transform (WFT). I have studied that with the help of WFT we can easily determine the phase of the image. Like by multiplying the window to only a specific part of the input and considering the outside part of the window equals...

Homework Statement Just something I am working through and am a bit stuck on. Homework Equations I have taken the Fourier transform of an RC circuit which gives me : Y(ω)=((X(ω))/(1+iωτ)) If i take the voltage across the circuit as white noise then i get: Y(ω)=σ^²/2π/(1+iωτ)) How...

\geqHomework Statement Find the Fourier Transform of y = exp(^{}-at)sin(\omega_{}0t) for t ≥ 0 and = 0 for t < 0 Find the amplitudes C(\omega, S(\omega), and energy spectrum \Phi' for \omega > 0 if the term that peaks at negative frequency can be disregarded for pos frequency...

I haven't had differential equations yet, so I am struggling in your math methods class. I understand what a Fourier Transform is, but I'm having trouble with this particular problem. Homework Statement Here's a screenshot. Better than I can write it. http://i.imgur.com/PQ6tB.png The...

I'm confused about the DFT of the data, fn = cos(3\pin/N) for n=0,1,...,N. It is definitely an even function, and I read that the Fourier coefficients of an even function is real. But when I take the FFT of this in Matlab I get complex numbers, not real numbers. What am I missing? Thanks ...

Hello all, I want to extract the period out of a complex discrete signal. Currently I have the Matlabscript of the attachement. However, the values I get out of this script are not correct. There is some kind of systematic bias in it. I think it has something to do with index *...

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 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 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 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...

$u_t-u_{xx}=0,$ $x\in\mathbb R,$ $t>0$ 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!