Fourier Definition and 1000 Threads

Hello! (Wave) The following problem shall show the way with which the Fourier series can be used for the solution of initial value problems.Find the solution of the initial value problem $$y''+ \omega^2 y=\sin{nt}, y(0)=0, y'(0)=0$$ where $n$ is a natural number and $\omega^2 \neq n^2$. What...

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 +...

Homework Statement WE have a thermally insulated metallic bar (from enviroment/surroundings) . It has a temperature of 0 ºC. At t=0 two thermal sources are applied at either end, the first being -10 ºC and the second being 10 ºC. Find the equation for the temperature along the bar T(x,t), in...

Homework Statement I need to expand this piecewise function f(x) = h for a<x<L and f(x) = 0 for 0<x<a. I am told that this is a square wave so ao and an in the expansion are 0 (odd function). Therefore I only need to worry about bn. The limits on the integral are from a to L, but what about the...

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?

Hello All, Briefly on the exposition; I'm an undergraduate assistant to a professor. We contribute to the Muon g-2 experiment in Fermilab, designing and optimizing the magnetic-measurement equipment. As you might imagine I utilize the Fourier Transform often to analyze data. The data I'm...

Homework Statement [/B] I am looking for help with part (d) of this question 2. Homework Equations The Attempt at a Solution I have attempted going through the integral taking L = 4 and t0 = -2. I was able to solve for a0 but I keep having the integrate by parts on this one. I've tried it...

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, can we make a Fourier series expansion of a (increasing or decreasing) step function ? like the one that I attached here. I just want to know the idea of that if it is possible.

Homework Statement Let ## f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx) ## What can be said about the coefficients ##a_n## and ##b_n## in the following cases? a) f(x) = f(-x) b) f(x) = - f(-x) c) f(x) = f(π/2+x) d) f(x) = f(π/2-x) e) f(x) = f(2x) f) f(x) = f(-x) =...

Homework Statement f(x) = -1, -π ≤ x ≤ 0 2, 0 ≤ x ≤ π Given this find the Fourier series using both $$a) \sum_{n=-∞}^\infty a_n e^{inx}$$ $$b) \sum_{n=0}^\infty [A_n cos(nx) + B_n sin(nx)]$$ Homework Equations $$a_o = \frac {1} {2L} \int_{-L}^L f(t) \, dt $$ $$a_n = \frac {1} {L}...

Homework Statement A rectangular box measuring a x b x c has all its walls at temperature T1 except for the one at z=c which is held at temperature T2. When the box comes to equilibrium, the temperature function T(x,y,z) satisfies ∂T/∂t =D∇2T with the time derivative on the left equal to zero...

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 Derive the expression for coefficients of Fourier series in exponential form for the sequence of rectangular pulses (with amplitude A, period T and duration θ) shown in this image: Derive the expression for signal power depending on the coefficients of Fourier series...

So a little bit of background: I work in an undergraduate lab at UMass Amherst and am currently building/optimizing a faraday magnetometer for use in the Muon g-2 experiment at Fermilab. The magnetometer works as follows. A laser is shone through a crystal with a particular Verdet Constant at...

Homework Statement i have this function \begin{equation} f(t) = e^t \end{equation} Homework Equations [/B] the Fourier seria have the form \begin{equation} f(t) = \sum C_{n} e^{int} \end{equation}The Attempt at a Solution } [/B] so i need to find the coeficients $c_{n}$ given by...

Where , rho 1 and rho 2 are two dimensional position vectors and K is a two dimensional vector in the Fourier domain. I encountered the above Eq. (27) in an article and the author claimed that after integration the right hand side gives the following result: I tried to solve this integral but...

Homework Statement A violin string is plucked to the shape of a triangle with initial displacement: y(x,0) = { 0.04x if 0 < x < L/4 (0.04/3)(L-x) if L/4 < x < L Find the displacement of the string at later times. Plot your result up to the n = 10...

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 Homework Equations All Fourier series trigonometric equations. I think we are required to use sigma function, integrals, etc.[/B]The Attempt at a Solution We are currently working through our Fourier series revision studying integrals of periodic functions within K.A...

Q: Suppose ##u(x,t)## satisfies the heat equation for ##0<x<a## with the usual initial condition ##u(x,0)=f(x)##, and the temperature given to be a non-zero constant C on the surfaces ##x=0## and ##x=a##. We have BCs ##u(0,t) = u(a,t) = C.## Our standard method for finding u doesn't work here...

I was given a function that is periodic about 2π and I need to plot it. I was wondering if there is a way to input a value and have mathematica generate a new graph with the number of iterations. The function is: $$\sum_{n=1}^{N}\frac{sin(nx)}{n}$$ where n is an odd integer. I guess a better...

Homework Statement Fourier series expansion of a signal f(t) is given as f(t) = summation (n = -inf to n = +inf) [3/(4+(3n pi)2) ) * e j pi n t A term in expansion is A0cos(6 pi ) find the value of A0 Repeat above question for A0 sin (6 pi t) Homework Equations Fourier expansion is summation...

Homework Statement b) state by inspection (i.e. without performing any formal analysis) all you can about each of the periodic waveforms shown in FIGURE 1 in terms of their Fourier series when analysed about t = 0 Homework Equations 3. The Attempt at a Solution Hi could someone please be...

Homework Statement Find the Fourier series of the function f(x) =√(x2) -pi/2<x<pi/2 , with period pi Homework EquationsThe Attempt at a Solution I have tried attempting the question, but couldn't get the answer. uploaded my...

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?

Self Study 1. Homework Statement Consider a periodic function f (x), with periodicity 2π, Homework Equations ##A_{0} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)dx## ##A_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)cos\frac{2\pi rx}{L}dx## ##B_{n} =...

[##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...

Homework Statement Find the values of the constant a for which the problem y''(t)+ay(t)=y(t+π), t∈ℝ, has a solution with period 2π which is not identically zero. Also determine all such solutions Homework Equations With help of Fourier series I know that : Cn(y''(t))= -n2*Cn(y(t)) Cn(y(t+π)) =...

Homework Statement I think I am being stupid, I am trying to show that ## \int^{T}_{0} e^i\frac{2\pi(n-m)t}{T} dt = 0 ## [1] if ## n \neq m## ## = T ## if ##n=m##, ##T## the period. Homework Equations [/B] I am using the following ##cos## and ##sin## orthogonal...

Hello everybody. I am currently comparing fourier's transformation of one physical phenomena and a two models which seek to emulate it. One of the models nails the frecuency and the other one even though it's displaced to higher frequencies the power (defined as 2* absolute value of fourier's...

Homework Statement I have ## f(t) = \sum\limits^{\infty}_0 a_{n} e^{2 \pi i n t} ## [1] and ## g(t) = \sum\limits^{\infty}_0 b_{n} e^{ \pi i n t} ## [2] I want to show that ##b_n = a _{2n} ## Homework Equations see above. The Attempt at a Solution [/B] So obviously you want to use the...

Hello everybody. I am triying to calculate a band-pass filter using the Fourier transform. I have a vector with 660 compomponents; one for each month. I am looking for a phenomenon which has a periodicity between 3 and 7 years (it's el niño, on the souhtern pacific ocean). I want to make zero...

Svein submitted a new PF Insights post Using the Fourier Series To Find Some Interesting Sums Continue reading the Original PF Insights Post.

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)] \\...

< Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown >[/color] Hi all. I'm completely new to these forums so sorry if I'm doing anything wrong. Anyway, I have this question... Find the Fourier series for the periodic function f(x) = x^2 (-pi < x < pi)...

Hi, I'm starting to studying Fourier series and I have troubles with one exercises of complex Fourier series with f(t) = t: $$t=\sum_{n=-\infty }^{\infty } \frac{e^{itn}}{2\pi }\int_{-\pi}^{\pi}t\: e^{-itn} dt$$ $$t=\sum_{n=-\infty }^{\infty } \frac{cos(tn)+i\, sin(tn)}{2\pi...

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?

Could someone explain the intuition behind the variables of the FT and DTFT? Do I understand it correctly ? For FT being X(f), I understand that f is a possible argument the frequency, as in number of cycles per second. FT can be alternatively parameterized by \omega = 2 \pi f which...

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}...

has Fourier used sin(x) and cos(x) in his series because "there must be such interval [a,b] where integral of "some function"*sin(x) on that interval will be zero?" so based on that he concluded that any function can be represented by infinite sum of sin(x) and cos(x) cause they are "orthogonal"...