Fourier Definition and 1000 Threads

I understand that the Uncertainty Principle relates the variances of Fourier conjugates. I am having trouble finding: 1) the mathematical relationship between the expectation values of Fourier conjugates generally; 2) and then specifically for a normalized Gaussian. Any suggestions or insights?

I have come across a paper where it is stated that if the infinity assumption in the FT is removed, the uncertainty doesn't hold. Is this a sensible argument? Thank you.

I am having trouble following a step in a book. So we are given that $$\varphi (x) = \int \frac {d^3k}{(2\pi)^3 2\omega} [a(\textbf{k})e^{ikx} + a^*(\textbf{k})e^{-ikx}] $$ where the k in the measure is the spatial (vector) part of the four-momentum k=(##\omega##,##\textbf{k}##) and the k in the...

Homework Statement By using Fourier transform, I want to calculate power of signal. I confuse that f(x) in attached equation represents voltage or power. Is that possible when f(x) means power to use Fourier transform. Homework Equations The Attempt at a Solution

I've a system of partial diff. eqs. in thermo-elasticity, I can solve it using normal mode analysis method but I need to solve it using laplace or Fourier

Homework Statement Calculate ##F(\frac 1 {1+x^4})##. Homework Equations ##\hat f (ξ) = \int_ℝ \frac 1 {1+x^4} e^{-2\pi i ξ x} dx## and Residue Theorem The Attempt at a Solution I know the function has to be real and even because ##\frac 1 {1+x^4}## is real and even, but I can't work out the...

Hi, I have to show that if ##f \in L^1(ℝ^n)## then: $$ ||\hat f||_{C^0(ℝ^n)} \le ||f||_{L^1(ℝ^n)}$$ Since ##|f(y)e^{-2 \pi i ξ ⋅y}| \le |f(y)|##, using the dominated convergence theorem, it is possible to show that ##\hat f \in C^0(ℝ^n)## but now I don't know how to go on. Thanks is advance.

It is often reported that the Fourier transform of a constant is δ(f) : that δ denotes the dirac delta function. ƒ{c} = δ(f) : c ∈ R & f => Fourier transform however i cannot prove this Here is my attempt:(assume integrals are limits to [-∞,∞]) ƒ{c} = ∫ce-2πftdt = c∫e-2πftdt = c∫ƒ{δ(f)}e-2πftdt...

Fourier Transform problem with f(t)=cos(at) for |t|<1 and same f(t)=0 for |t|>1. I have an answer with me as F(w)=[sin(w-a)/(w-a)]+[sin(w+a)/(w+a)]. But I can't show it.

Homework Statement Solve the following partial differential equation , using Fourier Transform: Given the following: And a initial condition: Homework EquationsThe Attempt at a Solution First , i associate spectral variables to the x and t variables: ## k ## is the spectral variable...

I understand that the solutions to the time-independent Schrodinger equation are complete, so a linear combination of the wavefunctions can describe any function (i.e. ##f(x) = \sum_{n = 1}^{\infty}c_n\psi_n(x) = \sqrt{\frac{2}{a}} \sum_{n=1}^{\infty} c_n\sin\left(\frac{n\pi}{a}x\right)## for...

This is a rather simple question, but am I understanding the following correctly? 1. Homework Statement The Attempt at a Solution This isn't really the problem, but I have a feeling my problem the assignments, is me misunderstanding the function description. I don't see how this 2 pi...

Homework Statement find the Fourier transform of the following function in two ways , once using direct computation , and second using the convolution therom . Homework Equations Acos(w0t)/(d2+t2) The Attempt at a Solution I tried first to solve directly . used Euler's identity and got...

Hi All! I've been looking at this Fourier Transform integral and I've realized that I'm not sure how to integrate the exponential term to infinity. I would expect the result to be infinity but that wouldn't give me a very useful function. So I've taken it to be zero but I have no idea if you can...

Homework Statement The Fourier transfrom of the wave function is given by $$\Phi(p) = \frac{N}{(1+\frac{a_0^2p^2}{\hbar^2})^2}$$ where ##p:=|\vec{p}|## in 3 dimensions. Find N, choosing N to be a positive real number. Homework Equations $$\int d^3\vec{p}|\Phi(p)|^2=1$$ , over all p in the 3...

I am trying to recover a laser beam profile using numerical differentiation of the data obtained from a "knife-edge scan". I am trying to select between two different methods to smooth out the numerical noise. Here is my raw data and the derivative: Here, I arbitrarily chose the 13 points...

Please, can anyone explain how formula (5) is obtained in J.J. Barton article ''Approximate translation of screened spherical waves" . Phys.Rew. A ,Vol.32,N2, 1985. ? https://doi.org/10.1103/PhysRevA.32.1019 The same formula are given in the book Pendry J.B. "Low enrgy electron diffraction. The...

When you do a discrete Fourier transform (DFT) of a one-dimensional signal, I understand that the second half of the result is the complex conjugate of the first half. If you threw out the second half of the result, you're not actually losing any data and you would be able to recreate the entire...

Above is a question I need some help with. I can get to a certain point, but I can't get beyond. Can somebody with knowledge about this help?

Hi, outside the mathematical proof that shows that sines of different frequency are orthogonal... is there geometric interpretation/picture of this phenomena?

Homework Statement Q1. a) In relation to Fourier analysis state the meaning and significance of 4 i) odd and even functions ii) half-wave symmetry {i.e. f(t+π)= −f(t)}. Illustrate each answer with a suitable waveform sketch. b) State by inspection (i.e. without performing any formal analysis)...

Homework Statement Given a continuous non-periodic function, its Fourier transform is defined as: $$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(x) e^{-ikx} dx$$ The problem is proving this is true by evaluating the...

A common use of the Fourier transform in physics is to transform between momentum-space and position-space. But what physical variables am I allowed to transform between? For instance can I use the Fourier transform to go from momentum space to frequency space or whatever?

Homework Statement Find the Fourier spectrum ##C_k## of the following function and draw it's graph: Homework Equations 3. The Attempt at a Solution [/B] I know that the complex Fourier coefficient of a rectangular impulse ##U## on an interval ##[-\frac{\tau}{2}, \frac{\tau}{2}]## is ##C_k =...

Homework Statement Find the admissible current density Jadm for a wire that has no insulation and also for a wire that has two layers of insulation and compare it to Jadm for the case when the wire has only one layer of insulation.2. The attempt at a solution and equations In the image I've...

The orientation of frequency components in the 2-D Fourier spectrum of an image reflect the orientation of the features they represent in the original image. In techniques such as nonlinear microscopy, they use this idea to determine the preferred (i.e. average) orientation of certain features...

Working on some microwave stuff, read about this but can't understand the explanations online.

$\displaystyle \begin{align*} F \left( \omega \right) &= \mathcal{F} \left\{ f \left( t \right) \right\} \\ &= \int_{-\infty}^{\infty}{ f\left( t \right) \mathrm{e}^{-\mathrm{j}\,\omega \, t}\,\mathrm{d}t } \\ &= \int_{-\infty}^{-2}{ 0\,\mathrm{d}t } + \int_{-2}^0{ \left( 1 + \frac{t}{2}...

Homework Statement Consider the action of ##T_2## acting on ##M_k(\Gamma_{0}(N)) ##, and show that ##\theta^4(n)+16F ## and ##F(t)## are both eigenfunctions. Functions are given by: Homework Equations For the Hecke Operators ##T_p## acting on ##M_k(\Gamma_{0}(N)) ##, the Hecke conguence...

Homework Statement I am supposed to compute the Fourier transform of f(x) = integral (e-a|x|) Homework Equations Fourier transformation: F(p) = 1/(2π) n/2 integral(f(x) e-ipx dx) from -infinity to +infinity The Attempt at a Solution My problem is, that I do not know how to handle that there...

Homework Statement Given the Fourier transformation pair ##f(t) \implies F(jw)## where ##f(t) = e^{-|t|}## and ##F(jw)=\frac{2}{w^2+1}## find and make a graph of the Fourier transform of the following functions: a) ##g(t)=\frac{2}{t^2+1}## b) ##h(t) = \frac{2}{t^2+1}\cos (w_ot)## Homework...

Homework Statement Homework Equations The Attempt at a Solution I am stuck trying to figure out why there are three different alphas and why in the equation we are supposed to use has a theta and what that means. If I can set up the Fourier series I can properley I know how to solve it for...

https://en.wikipedia.org/wiki/Discrete_Fourier_transform Why is the signal obtained from a DFT periodic? The time signal x[n] is finite and the number of sinusoids being correlated with it is finite, yet its said the frequency spectrum obtained after the DFT is periodic. I've also read the...

Homework Statement Homework Equations All I know is the a's have something to do with the integrals. The Attempt at a Solution I used FFT analysis in Matlab but I do not know what I am looking for. How do the a0s relate to the f(t) in the question and how would I even do run that equation in...

I would like to express that when I am viewing the repetitive Fourier transform on Internet I encounter that for instance twice Fourier transform may lead the same value at the end of first Fourier transform. When does repetitive( twice or third... consecutively)fourier transform be same with...

Homework Statement I am reading the book of Gerry and Knight "Introductory Quantum Optics" (2004). In page 60, Chapter 3.7, there is two equation referring Fourier Transformation in the complex plane as follows: $$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$...

Hi i’m having problems with the following equations: X(w)=2/(-1+iw)(-2+iw)(-3+iw) This then becomes the following equation according the the tutorial, although there is no explanation as to how: X(w)=1/-1+iw, -2/-2+iw, +1/-3+iw The commas indicated the end of each fraction to make it easier...

Hi, I'm struggling with a conceptual problem involving the Fourier transform of distributions. This could possibly have gone in Physics but I suspect what I'm not understanding is mathematical. The inverse Fourier transform of a Cauchy distribution, or Lorentian function, is an exponentially...

Hi everybody. There has been a thread about this on physics forums, where the Fourier transform X(w) of x(t) volts (with time units in seconds) could be considered as volt second, or volt per Hz. So when we see tables of Fourier transform pairs, we might see Fourier transform plots associated...

Joseph Fourier turned 250 this week: https://www.nature.com/articles/d41586-018-03389-w Joyeux anniversaire, Mr Fourier !

Homework Statement I am to solve an ODE using the Fourier Transform, however I am quite inexperienced in using this method so I'd like some advice: Homework Equations a) The Fourier Transform b) The Inverse Fourier Transform The Attempt at a Solution I started by applying the Fourier...

Hello, I tried to compute the Fourier series coefficients for the Dirac comb function. I did it using both the "complex" formula and the "real" formula for the Fourier series, and I got : - complex formula : Cn = 1/T - real formula : a0 = 1/T, an = 2/T, bn = 0 This seems to be valid since it...

1. The problem statement, all variables, and given/known data Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) =...

Homework Statement I've written a program that calculates the discrete Fourier transform of a set of data in FORTRAN 90. To test it, I need to "generate a perfect sine wave of given period, calculate the DFT and write both data and DFT out to file. Plot the result- does it look like what you...

Homework Statement Find the Fourier series of the function ##f## given by ##f(x) = 1##, ##|x| \geq \frac{\pi}{2}## and ##f(x) = 0##, ##|x| \leq \frac{\pi}{2}## over the interval ##[-\pi, \pi]##. Homework Equations From my lecture notes, the Fourier series is ##f(t) = \frac{a_0}{2}*1 +...

Homework Statement I am only interested in 9 (a) Determine the Fourier Cosine series of the function g(x) = x(L-x) for 0 < x < L Homework Equations The Answer for 9 a. g(x) = (L^2)/6 - ∑(L^2/(nπ)^2)cos(2nπx/L) This is the relevant equation given where ω=π/L f(t) = a0+∑ancos(nωt) a0=1/L...

Homework Statement Homework Equations Scaling property and property of dual. I got the answer. The Attempt at a Solution I got the answer using scaling property and using property of dual. x1(t)---> X2(W)----(another Fourier transform)--->2(3.14) x1(-w) But I think the final answer should be...

Does the complex form of Fourier series assume even or odd half range expansion?

Hi, let's take this ode: y''(t) = f(t),y(0)=0, y'(0)=0. using the FT it becomes: -w^2 Y(w) = F(w) Y(w)=( -1/w^2 )F(w) so i can say that -1/w^2 is the Fourier transorm of the green's function(let's call it G(w)). then y(t) = g(t) * f(t) where g(t) = F^-1 (G(w)) (inverse Fourier transorm) how can...

I'm doing a research project over the summer, and need some help understanding how to construct an inverse Fourier transform (I have v. little prior experience with them). 1. Homework Statement I know the explicit form of ##q(x)##, where $$ q(x) = \frac{M}{2 \pi} \int _{- \infty}^{\infty} dz...