Fourier Definition and 1000 Threads

Hi, I have problems finding out the Fourier transform of the following function, 1/\sqrt{q^2 + m^2}, where m\neq 0 denotes a parameter. It seems easy, but I don't know how. Could anybody show me how to do it ? Thanks in advance. hiyok

Homework Statement Find the inverse Fourier transform of f(w)=1 Hint: Denote by f(x) the inverse Fourier transform of 1 and consider convolution of f with an arbitrary function. Homework Equations From my textbook the inverse Fourier transform of f(w)=\int F(w)e^-iwt dw The...

Dear Physics Buddies, How are well all, okay I hope. I was wondering if I might browse all your infinite intellects and ask you a very simple question. I am working with some medical images in MATLAB and my collaborators would like to know the orientation of the fibres that it contains...

After searching on the web and reading a bit, I found that lenses can perform Fourier transform. All you need to do is put a transparant object in front of it, like a transparant sheet with black stripes on it and a screen behind the lens(so basically a 4f setup). The lens will then perform a...

Homework Statement Homework Equations here is list of Fourier transforms: http://uspas.fnal.gov/materials/11ODU/FourierTransformPairs.pdfThe Attempt at a Solution so I know the solution but I don't know how to get it. Here is what I think to do: the ramp function r(t) and the rect pτ(t). I...

Homework Statement Hey, the question i have been given reads: By a simple change of variables, show that if g(x) is a periodic real valued function with period L it can be represented as g(x)~ ∑∞n=-∞ An exp(-2\piinx/L) where the complex constants An are given by LAm =[L/2,-L/2]...

So the complex exponential Fourier series form an orthonormal basis for the space of functions. A periodic function can be represented with countably many elements from the basis, and an aperiodic function requires uncountably many elements. Given a signal, we can find the coefficients of the...

So I'm currently busy studying a Digital Micromirror Device which is used for top-hat beam generation. Programming the input pattern and error diffusion needed for optimal top-hat generation is heavily based on Fourier Optics. The problem however is: I don't know Fourier optics. I know this...

Using geometric evaluation of the magnitude of the Fourier transform from the corresponding pole-zero plot, determine, for each of the following Laplace transforms, whether the magnitude of the corresponding Fourier transform is approximately lowpass, highpass, or bandpass. \[ H_1(s) =...

Homework Statement the function is Exp[-w^2]/w^2, how to solve the Fourier transform analytically with Residue theorem? It is better if there is more general results. Mathematica can solve it analytically, but I need a human-soluable way. Homework Equations The Attempt at a...

##\varphi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int^{\infty}_{-\infty}dx\psi(x)e^{-\frac{ipx}{\hbar}}##. This ##\hbar## looks strange here for me. Does it holds identity ##\int^{\infty}_{-\infty}|\varphi(p)|^2dp=\int^{\infty}_{-\infty}|\psi(x)|^2dx=1##? I'm don't think so because this ##\hbar##. So...

Hi. I have an electric field E(r) which can be equivalently characterized by its Fourier spectrum \tilde{E}(k) through E(r)\propto\int\tilde{E}(k)exp[ik\cdotr]dk The Maxwell equation states that in a homogeneous and isotropic medium ∇\cdotE=0 So, applying this equation to my Fourier...

Homework Statement (f*g)(x) = integral from -pi to pi of (f(y)g(x-y))dy f(x) = ∑cneinx g(x) = ∑dneinx en is defined as the Fourier Coefficients for (f*g) {the convolution} an is denoted by: en = 1/(2pi) integral from -pi to pi of (f*g)e-inx dx Evaluate en in terms of cn and dn...

Any boolean function on n variables can be thought of as a function f : \mathbb{Z}_2^n \rightarrow \mathbb{Z}_2 which can be written as f(x) = \sum_{s \in \mathbb{Z}_2^n} \hat{f}(s) \prod_{i : x_i = 1} (-1)^{x_i} where \hat{f}(s) = \mathbb{E}_t \left[ f(t) \prod_{i : s_i = 1}...

Homework Statement I am going over a practice exam, and I need to find the FSS of f(x)=x(\pi^2-x^2) Homework Equations f(x) \sim \sum^\infty_{n=1}a_n sin\left(\frac{n \pi x}{L}\right) a_n=\frac{2}{L}\int^L_0 f(x)sin\left(\frac{n\pi x}{L}\right)dx The Attempt at a Solution I think I...

Homework Statement a) Show that the Fourier Cosine Series of f(x)=x,\quad 0\leq x<L is x ~ \frac{L}{2}-\frac{4 L}{\pi ^2}\left[\left(\frac{\pi x}{L}\right)+ \frac{\cos\left(\frac{3\pi x}{L}\right)}{3^2}+\frac{\cos\left(\frac{5 \pi x}{L}\right)}{5^2}+\dots\right] b) use the above series to...

Hello everyone I am doing my own split step Fourier method (SSFM)code on Matlab to solve the Nonlinear schrodinger equation in nonlinear fiber optics My problem is that in the Nonlinear operator we just multiply it with the initial pulse during SSFM without doing any Fourier transform not...

Homework Statement Doing some exam revision and one of the questions from an old exam has me stuck at the last step, simply need to inverse the following F( \omega ) = \frac{e^{i \omega}}{1+\omega ^2} We're allowed to use a table on the exams but I cannot find anything quite...

Hello, I was wondering if one can give meaning to the Fourier transform of the linear function: \int_{-\infty}^{+\infty} x e^{ikx}\, dx I found that it is \frac{\delta(k)}{ik} , does this make sense?

Ok so I've seen the convolution theorem written as: F(h(x)\otimesg(x))=H(k)G(k) (And this is how it appears when I have a quick google). My book then does a problem in which is uses: F(h(x)g(x))=H(k)\otimesG(k) Where H(k)=F(h(x)) and similarly G(k)=F(g(x)), and F represents a Fourier...

Find the Fourier cosine series of cos(x) from x=0 ~to ~\pi Here the Fourier series is given by f(x)=\frac{1}{2}a_0+\sum_{n=1}^{\inf}a_n cos nx dx where a_n=\frac{2}{\pi}\int_0^\pi f(x)cos nx dx I am facing problem to solve it. I am getting a_0=0 and a_n=0 so the Fourier series becomes...

I believe this is an error minimization problem so I am trying to solve the following equation Min((∑ ( (S(t) - A cos(b t + C)))^2 ) Where S(t) is the input signal, t is time and I will sum over t, A is the amplitude, b is radians per second (frequency), and C is the phase angle. I...

Homework Statement Homework Equations sinc(x) = \frac{sin(x)}{x} The Attempt at a Solution bit unsure how to get started?? i know transform of rectangular pulse pτ(t)=τ*sinc(τω/2∏) also that sin(ωt)= ejωt-e-jωt / (2) I could also probably sketch sinc(t/2∏), if that helps.

Okay the question is to find the Fourier transform of: rect(\frac{x}{5})\otimes(\delta(x+3)-\delta(x-3)) =F^{\infty}_{\infty} \intrect(\frac{x'}{5})(\delta(x+3-x')-\delta(x-3-x')) dx' [1] - where F represents a Fourier transform. My Issue Okay I am fine doing this using the convolution...

Dear all, I have recently come across the following Fourier transform (FT): I=\int_{-\infty}^{\infty} dx \, e^{-\imath x t} \frac{(1-x^2)}{(1+x^2)^{3/2} (a^2+x^2)}. The integrand contains two branch points on the imaginary axis, plus two poles also residing on the imaginary...

Homework Statement For he following Fourier series, which of the answers correctly describes the following function y(t) = 2 - \stackrel{1}{π}∑1inf1/nsin(n*πt/2) a) odd function, period = 2 s b) Even function, period = 2s c) Odd function, period = 4s d) Even functio, period = 4s...

Hi. I want to get a quick overview of the theory of Hilbert spaces in order to understand Fourier series and transforms at a higher level. I have a couple of questions I hoped someone could help me with. - First of all: Can anyone recommend any literature, notes etc.. which go through the...

I understand that the Fourier transform maps one function onto another. So it is a mapping from one function space onto another. My question is, why is it often referred to as a mapping from time domain to the frequency domain? I don't understand why the image of the Fourier transform...

Hi guys, I've been using this site for a while now, but this is going to be my first post. I want to pick your brains to get some insight on this problem I'm tackling. I'm trying to take a Fourier Transform of a function. My function is a function of (r,phi) and it is a piecewise function...

Homework Statement x(t) = 5cos(2*pi*1000*t) and g(t) = ∑ from n=-infinity to infinity delta(t-n/10000) find Fourier transform of x(t) and g(t) and the product of the two The Attempt at a Solution x(w) = 5*sqrt(pi/2) [delta(w-2000pi)+delta(w+2000pi)] g(w) = 1 so would the...

Homework Statement V(t) = 4 for 0<t< 1 and 0 for 1<t<3 and repeats itself for all t (negative and positive) Find the first 5 harmonics of the Fourier series in cosine form and find the power if this is the voltage over 100 ohm resistor The Attempt at a Solutionpower = d_dc ^2 / R + .5sum...

Is correct to define Fourier series like: f(t)=\sum_{k=0}^{\infty}a_k \cos \left (\frac{2 \pi k t}{T} \right ) + b_k \sin \left (\frac{2 \pi k t}{T} \right ) Where ak and bk: a_k=\frac{1}{T} \int_{-T}^{+T} f(t) \cos \left (\frac{2 \pi k t}{T} \right ) dt b_k=\frac{1}{T}...

hey pf! okay, so if you've studied PDEs you know the value of a Fourier series, and the difficulty of determining a Fourier coefficient. my question relates to finding this coefficient. briefly, i'll define a Fourier series as f(x)=\sum_{n=0}^{\infty} A_n\cos\frac{n\pi x}{L}+B_n\sin\frac{n\pi...

Homework Statement x(t) = 4 for 0 <x<1 and 0 otherwise and this process repeats for all values including negative. find X_0 and X_n and find the first 6th harmonics of the Fourier series in cosine form Homework Equations The Attempt at a Solution x_0 = 4/3 x_n =...

Hello, I was wondering if such a thing even exists, so here it goes... Let's say I have a function x(s) (it is real, smooth, differentiable, etc.) defined on (0,1). In addition, dx/ds = 0 on the boundary (s=0 and s=1). I can compute its Fourier transform (?) as a_p = \int_0^1 x(s)...

Hi, I hope somebody can help me with this one. Homework Statement Compute the Fourier Transform of the distribution x-a Homework Equations The Fourier Transform of a distribution is just the distribution evaluated with the Fourier Transform of a test function.The Attempt at a Solution See...

I have an exercise with a function of the form: h(t) = f(t)g(t) and f(t) and g(t) both have discrete Fourier series, which implies that h does too. I want to find the Fourier series of h, so my teacher said I should apply the convolution theorem which would turn the product above into a...

Homework Statement Part (a): State inverse Fourier transform. Show Fourier transform is: Part (b): Show Fourier transform is: Part (c): By transforming LHS and RHS, show the solution is: Part(d): Using inverse Fourier transform, find an expression for T(x,t) Homework Equations The Attempt...

Homework Statement The exercise is a) in the attached trial. I have attached my attempt at a solution, but there are some issues. First of all: Isn't the example result wrong? As I demonstrate you get a delta function which yields the sum I have written (as far as I can see), not the sum...

Homework Statement This is a general question, no real problem statement and is connected to solving Fourier series. You know that to solve it, you need to find a_{n}, a_{0} and b_{n}. Homework Equations When solving the above mentioned ''coefficients'' you can get a solution with sin or...

Homework Statement Which of the signals is not the result of Fourier series expansion? options : (a) 2cos(t) + 3 cos(3t) (b) 2cos(\pit) + 7cos(t) (c) cos(t) + 0.5 Homework Equations Dirichlet conditionsThe Attempt at a Solution From observation, I thought all are periodic and so must be...

Hello, I am trying to numerically evaluate a convolution integral of two functions (f*g) using Fourier transform (FT) i.e using FT(f*g) = FT(f) multiplied by FT(g) (1) I am testing for a known case first. I have taken the gaussian functions (eq. 5, 6 and 7) as given in...

Homework Statement Let ##f## be a ##2\pi## periodic function. Let ##\hat{f}(n)## be the Fourier coefficient of ##f## defined by $$ \hat{f}(n)=\frac{1}{2\pi}\int_{a}^{b}f(x)e^{-inx}dx. $$ for ##n\in\mathbb{N}##. If ##\overline{\hat{f}(n)}=\hat{f}(-n)## show that ##f## is real valued. The...

http://calclab.math.tamu.edu/~fulling/m412/f07/airywkb.pdf Can someone walk me through this derivation of the Airy integral by Fourier transform? I have tried it but failed

I'm trying to prove that the discrete form of the Fourier transform is a unitary transformation So I used the equation for the discrete Fourier transform: ##y_k=\frac{1}{\sqrt{N}}\sum^{N-1}_{j=0}{x_je^{i2\pi\frac{jk}{N}}}## and I put the Fourier transform into a N-1 by N-1 matrix form...

Homework Statement Consider the square wave function defined by y(t) = h (constant) when 0 ≤ (t + nT) ≤1, y(t) = 0 elsewhere, where T = 2 is the period of the function. Determine the Fourier series expansion for y(t). Homework Equations Fourier Analysis Coefficients The Attempt...

Simple question; Why isn't it \sum am (from m=1 to infinity) Thanks in advance.

1. We consider the on shell wave packet: \varphi(t,x)=\int\frac{dk}{2\pi}exp(-\frac{(k-k_{0})^{2}}{\Delta k^{2}}+ik(t-x))dk I need to show it is proportional to: exp(ik_{0}(t-x)-\frac{\triangle k^{2}}{4}(t-x)^{2})dk through a Fourier transform of the gaussian 3. I used a Fourier...

From -infinity to infinity at the extreme ends do Fourier transforms always converge to 0? I know in the case of signals, you can never have an infinite signal so it does go to 0, but speaking in general if you are taking the Fourier transform of f(x) If you do integration by parts, you get a...

Find the Fourier series solution to the differential equation x"+x=t It's given that x(0)=x(1)=0 So, I'm trying to find a Fourier serie to x(t) and f(t)=t, and I'm know it must a serie of sin... So here's my question...the limits of integration to the Bn, how do I define them? Will...