Fourier transform Definition and 951 Threads

Could you explain a bit about the relationship between locality and uncertainty in Fourier pairs? Many pages talk about uncertainty principle stating that the precision at which we can measure time duration of signal cannot unlimitedly grow without affecting precision on bandwidth. Many other...

Homework Statement [/B] I am trying to match each of the following 28-point discrete-time signals with its DFT: Set #1: Set #2: Homework EquationsThe Attempt at a Solution Set #1 We have already established (here) that: ##Signal 1 \leftrightarrow DFT3## ##Signal 4 \leftrightarrow...

Hi! 1. Homework Statement From the website http://www1.uprh.edu/rbaretti/MomentumspaceIntegration8feb2010.htm we can see the Fourier transform of the ground state hydrogenic wave function : Φ(p) = ∫ ∫ ∫ exp(-i p r) (Z3/π )1/2 exp(-Zr) sin(θ) dθ dφ r² dr (1.1) After intregation...

Homework Statement Match each discrete-time signal with its DFT: Homework EquationsThe Attempt at a Solution I am mainly confused about Signal 7 and Signal 8. Signal 1 is the discrete equivalent to a constant function, therefore its DFT is an impulse (Dirac ##\delta##), so it corresponds...

Hi! I am currently trying to derive the Fourier transform of a 2D HgTe Hamiltonian, with k_x PBC and vanishing boundary conditions in the y direction at 0 and L. Here is the Hamiltonian: H = \sum_{k}\tilde{c_k}^{\dagger}[A\sin{k_x}\sigma_x + A\sin{k_y}\sigma_y + (M-4B+2B[\cos{k_x} +...

Homework Statement $$u_{xx}=u_t+u_x$$ subject to ##u(x,0)=f(x)## and ##u## and ##u_x## tend to 0 as ##x\to\pm\infty##. Homework Equations Fourier Transform The Attempt at a Solution Taking the Fourier transform of the PDE yields $$ (\omega^2-i\omega) F\{u\}=...

Homework Statement Hi guys, I am going to calculate the following integral: $$\int_0^{f_c+f_m} |Y(f)|^2\, df$$ where:$$Y(f)=\frac{\pi}{2} \alpha_m \sum_{l=1}^{L} \sqrt{g_l}\left [ e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0) + e^{-j(\omega \tau_l + \theta_m)} \delta(\omega +...

Apart from the fact that it is, what is the physical significance of the fact that you can get the momentum distribution of a particle by taking the Fourier transform of its position distribution?

Homework Statement F(t) = sqrt(π/2)e-t for t>0 F(t) = sqrt(π/2)et for t<0 In other words the question asks to solve this integral: 1/sqrt(2π) ∫F(t)eitxdt and show that it equals 1/(1+x2) Homework Equations F(t) = sqrt(π/2)e-t for t>0 F(t) = sqrt(π/2)et for t<0 1/sqrt(2π) ∫F(t)eitxdt The Attempt...

Homework Statement I want to find the Fourier series of the sawtooth function in terms of real sine and cosine functions by using the formula: $$f_p (t)=\sum^\infty_{k=-\infty} c_k \exp \left(j2\pi \frac{k}{T}t \right) \tag{1}$$ This gives the Fourier series of a periodic function, with the...

Hello, I am trying to find an expression for the signal-to-noise ratio of an oscillating signal on top some white noise. In particular I would like to know how the SNR scales with the integration time. It is well known that during some integration time ##T##, the SNR increases as ##T^{1/2}##...

Homework Statement Homework Equations I'm not sure. The Attempt at a Solution I started on (i) -- this is where I've gotten so far. I am asked to compute the Fourier transform of a periodic potential, ##V(x)=\beta \cos(\frac{2\pi x}{a})## such that...

Hello All, I would like to convert a partial diff equation in time domain into frequency domain, however there is a term of the form: Re(∇(E1.E2*) exp(j[ω][/0]t)) where E1 and E2 are the magnitudes of the electric field and [ω][/0] is the angular frequency. Can someone please help me to...

I'm studying Quantum Field Theory and the first example being given in the textbook is the massless Klein Gordon field whose equation is just the wave equation \Box \ \phi = 0. The only problem is that I'm not being able to get the same solution as the book. In the book the author states that...

Homework Statement The problem is from an optics text, however I believe the problem to be a mathematical one. I'm trying to take the Fourier transform of P(t) = ε0∫ X(t-τ)E(τ) dτ which should equal P(ω) = ε0X(ω)E(ω) where ε0 is a constant X is the susceptibility E is the...

Homework Statement Determine the Fourier-transfroms of the functions \begin{equation*} a) f : f(t) = H(t+3) - H(t-3) \text{ and } g : g(t) = \cos(5t) f(t) \end{equation*} and \begin{equation*} b) f : f(t) = e^{-2|t|} \text{ and } g : g(t) = \cos(3t) f(t) \end{equation*}Homework Equations The...

Homework Statement This is a combination of two questions, one being the continuation of the other 3) Calculate the DFT of the sequence of measurements \begin{equation*} \{ g \}_{k=0}^{5} = \{ 1,0,4,-1,0,0 \} \end{equation*} 4a) Draw the DFT calculated in question 3 on the complex plane. 4b)...

Hello, for a function f∈L2(ℝ), are there known necessary and sufficient conditions for its Fourier transform to be zero only on a set of Lebesgue measure zero?

[##f^*## represents complex conjugate of ##f##. ] [##\widetilde{f}(k)## represents Fourier transform of the function ##f(x)##.] $$\begin{align} \int_{-\infty}^{\infty}f^*(x)e^{ikx}\,dx&=\int_{-\infty}^{\infty}f^*(x)\left(e^{-ikx}\right)^*\,dx\\...

We know, $$\delta(x) = \begin{cases} \infty & \text{if } x = 0 \\ 0 & \text{if } x \neq 0 \end{cases}$$ And, also, $$\int_{-\infty}^{\infty}\delta(x)\,dx=1$$ Using Fourier Transformation, it can be shown that, $$\delta(x)=\lim_{\Omega \rightarrow \infty}\frac{\sin{(\Omega x)}}{\pi x}$$ Let's...

Given the Fourier conjugates ##\vec{r}## and ##\vec{k}## where ##\vec{r} = [r_1,r_2,r_3]## and ##\vec{k} = [k_1,k_2,k_3]## , are ##r_1## /##k_1##, ##r_2##/##k_2##, ##r_3##/##k_3## also Fourier conjugates, such that: ##\begin{equation} \begin{split} f(\vec{r})&=[f_1(r_1),f_2(r_2),f_3(r_3)] \\...

Dear "General Math" Community, my goal is to calculate the following integral $$\mathcal{I} = \int_{-\infty }^{+\infty }\frac{f\left ( \mathbf{\vec{x}} \right )}{\left | \mathbf{\vec{c}}- \mathbf{\vec{x}} \right |}d^{3}x $$ in the particular case in which f\left ( \mathbf{\vec{x}} \right...

Homework Statement Solve ut+3ux=0, where -infinity < x < infinity, t>0, and u(x,0)=f(x).Homework Equations Fourier Transform where (U=fourier transform of u) Convolution Theorem The Attempt at a Solution I've used Fourier transform to get that Ut-3iwU=0 and that U=F(w)e3iwt. However, I'm...

Homework Statement Link: http://i.imgur.com/JSm3Tqt.png Homework Equations ##\omega=2\pi t## Fourier: ## Y(f)=\int ^{\infty}_{-\infty}y(t)\mathrm{exp}(-j\omega t)dt## Linearity Property: ##ay_1(t)+by_2(t)=aY_1(f)+bY_2(f)##, where a and b are constants Scaling Property...

We have a waveform that is composed of several waves, maybe something like this: If we Fourier transform the graph we get something like this: My question is, does the value of the largest column represent the peak to peak voltage of the waveform pictured above?

Homework Statement Assume a wave packet is has contributions from various frequencies, give by g(ω)=C for |ω|<ω0, and g(ω) =0 for elsewhere. a)What is the signal strength as a function of time, i.e., V(t)=? b) Sketch g(ω) and V(t); You can use fooplots.com, for example, or python. c)...

Hi Guys, I'm having trouble with the following: A finite-time signal is the result of a filter G(t) applied to a signal. The filter is simply “on” (1) for t ∈ [0,T] and off (“0”) otherwise. If x(t) is the signal, and x(ω),its Fourier transform, compute the Fourier transform of the filtered signal...

Can anyone explain what does the author mean by the statement below? page 27 of this paperI don't understand the relation between the Fourier transform and translational invariance. Thanks

Problem F denotes a forward Fourier transform, the variables I'm transforming between are x and k - See attachment Relevant equations So first of all I note I am given a result for a forward Fourier transform and need to use it for the inverse one. The result I am given to use, written out...

In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...

Is there any way to convert a continuous, aperiodic spectrum, to a discrete spectrum, in a signal? If so, would part of he energy of this signal be lost, I am this process of conversion, or would it be " distributed" amomg the various frequencies?

I've tried to read the reason for using Fourier transform in wave packets, I don't understand why. Please help me with this.

Hello, PF! I am currently learning Fourier series (and then we'll move on to the Fourier transform) in one of my courses, and I'm having a hard time finding motivation for its uses. Or, in other words, I can't seem to find its usefulness yet. I know one of its uses is to solve the heat...

Hi! I'd like to smear an audio recording, where the frequency content audibly changes, into an audio recording where it does not. Here's a recording of a sampled piano playing a melody, which will serve as an example: https://dl.dropboxusercontent.com/u/9355745/oldmcdonald.wav The frequency...

Hello I am trying to determine the Fourier transform of the hyperbolic tangent function. I don't have a lot of experience with Fourier transforms and after searching for a bit I've come up empty handed on this specific issue. So what I want to calculate is: ##\int\limits_{-\infty}^\infty...

Homework Statement The (computing) task at hand is to take a function f(x) defined at 2N discrete points, and use the Discrete Fourier Transform (DFT) to produce F(u), a plot of the amplitudes of the frequencies required to produce f(x). I have an array for each function holding the value of...

Homework Statement [/B] This is a computing coursework problem. (There is a reasonably long theory preamble). Create a single slit centred on the origin (the centre of your array) width 10 and height 1. The array containing the imaginary parts will be zero and the array containing the real...

Homework Statement Given x(t)=8cos(70\pi t)+4sin(132\pi t)+8cos(24\pi t), find the Fourier transform X(f) in the form of \delta function. Homework Equations X(f)=\int ^{\infty}_{-\infty}x(t)e^{-j\omega _0t}dt cos(\omega t)=\frac{e^{j\omega t}+e^{-j\omega t}}{2} sin(\omega t)=\frac{e^{j\omega...

Homework Statement Evaluate the Fourier Transform of the damped sinusoidal wave g(t)=e^{-t}sin(2\pi f_ct)u(t) where u(t) is the unit step function. Homework Equations \omega =2\pi f G(f)=\int ^{\infty}_{-\infty} g(t)e^{-j2\pi ft}dt sin(\omega _ct)=\frac{e^{j\omega _ct}-e^{-j\omega _ct}}{2j}...

I am using ROOT to calculate the Fourier transform of a digital signal. I can extract the individual parts of the transform, the magnitude and phase in the form of a 1D histogram. I am attempting to reconstruct the transforms from the phase and magnitude but cannot seem to figure it out. Any...

My first thought was simply that the Fourier transform of a sum of Gaussians functions that are displaced from the origin by different amounts would just be another sum of Gaussians: F{G1(x) + G2(x)} = F{G1(x)} + F{G1(x)} where a generalized shifted Gaussian is: G(x) = G0exp[-(x - x0)2 / 2σ2]...

Hello everyone, I was trying to develop a sort of generalized version of the Fourier Transform. My question in particular is: Given a function f(x,u), is there a function g(x,u) with \int_{-\infty}^\infty f(x,u)g(x,u')\mathrm{d}x=\delta(u-u') For f(x,u)=e^{2\pi ixu} the solution would be...

A tad embarrassed to ask, but I've been going in circles for a while! Maybe i'll rubber duck myself out of it. If $$f(t) = f(t+T)$$ then we can find the Fourier transform of $$f(t)$$ through a sequence of delta functions located at the harmonics of the fundamental frequency modulated by the...

I am using a Tascam recorder to record an environmental nuisance noise that is occurring in my home. I then use Virtins Multi Instrument Software, which includes an oscilloscope, band pass filter, and a spectrum analyser. Noise source is probably machinery at a legal marijuana grow op. That...

I am a little familiar with Fourier Analysis, but I don't know where to get tools to get the answer to this question: Consider a discrete signal A[0..N-1], consisting of N samples. Suppose we Fourier transform it and get a series of harmonics. Now, consider the discrete signal A[1..N], that is...

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...

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...

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

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 ?

I was reading up on (discrete) Fourier transform when my mind started to think of an what-if scenario: Assumed I'm sampling a signal of the form a1*sin(b1+c1) + a2*sin(b2+c2) + a3*sin(b3+c3) + ... + aN*sin(bN+cN) + some noise where the a's represents magnitudes, b's represents frequencies and...