Fourier Definition and 1000 Threads

Homework Statement Periodic function P=3 f(t) = 0 if 0<t<1 1 if 1<t<2 0 if 2<t<3 a) Draw the graph of the function in the interval of [-3,6] b) Calculate the Fourier series of f(x) by calculating the coefficient. Homework EquationsThe Attempt at a Solution a) in attached...

Homework Statement Homework Equations The Attempt at a Solution Since P=2L, L=1 ? a_o = 1/2 [ ∫(from -1 to 0) -dx + ∫(from 0 to 1) dx ] = 1/2 [ (0-1) + (1-0) ] = 1/2(0) = 0 a_n = - ∫ (from -1 to 0) cosnπx dx + ∫ (from 0 to 1) cosnπx dx = 0 b_n = - ∫ (from -1 to 0) sinnπx dx...

I'm not sure whether to put this here or in Linear Algebra, if any Mod feels it should go in Linear Algebra I won't mind. I've just been introduced to Fourier Series decompositions in my Linear Algebra text, and I understand all the core concepts so far from the Linear Algebra side of it (a...

The text does it thusly: imgur link: http://i.imgur.com/Xj2z1Cr.jpg But, before I got to here, I attempted it in a different way and want to know if it is still valid. Check that f^{*}f is finite, by checking that it converges. f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x +...

1. Homework Statement Find the Fourier series of ##f(x) = \delta (x) - \delta (x - \frac{1}{2})## , ## - \frac{1}{4} < x < \frac{3}{4}## periodic outside. Homework Equations [/B] ##\int dx \delta (x) f(x) = f(0)## ##\int dx \delta (x - x_0) f(x) = f(x_0)##The Attempt at a Solution...

Fourier transform is defined as $$F(jw)=\int_{-\infty}^{\infty}f(t)e^{-jwt}dt.$$ Inverse Fourier transform is defined as $$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(jw)e^{jwt}dw.$$ Let ##f(t)=e^{-at}h(t),a>0##, where ##h(t)## is heaviside function and ##a## is real constant. Fourier...

Homework Statement Two infinitely grounded metal plates at y=0 and y=a are connected at x=b and x=-b by metal strips maintained at a constant potential V. Find the potential inside the rectangular pipe.Homework Equations Laplaces EquationThe Attempt at a Solution I posted a photo of what I've...

Homework Statement Using the CTFS table of transforms and the CTFS properties, find the CTFS harmonic function of the signal 2*cos(100*pi(t - 0.005)) T = 1/50 Homework Equations To = fundamental period T = mTo cos(2*pi*k/To) ----F.S./mTo---- (1/2)(delta[k-m] + delta[k+m]) The Attempt at...

Homework Statement [/B] I was using the Fourier transform to solve the following IVP: \frac{\partial^2 u}{\partial t \partial x} = \frac{\partial^3u}{\partial x^3} \\ u(x,0)=e^{-|x|} Homework Equations [/B] f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{f}(\omega)e^{i\omega...

1. Find the Fourier series of : $$g(t)=\frac{t+4}{(t^2+8t+25)^2}$$ 2. I have been trying to write the function to match the formula $$\mathcal{F} [\frac{1}{1+t^2}] = \pi e^{-\mid(\omega)\mid}$$ 3. I have simplified the function to $$(t+4)(\frac{1}{9}(\frac{1}{1+\frac{(t+4)^2}{9}})^2)$$...

Mod note: Moved from technical math section, so no template was used. Hey! So the complex Fourier transform of the square wave $$ f(x) = \begin{cases} 2 & x \in [0,2] \\ -1 & x \in [2,3] \\ \end{cases}, \space \space f(x+3) = f(x)$$ is ##C_k = \frac{3j}{2 \pi k}( e^{-j \frac{4 \pi k}{3}}...

What is the relationship between the Fourier transform of a periodic function and the coefficients of its Fourier series? I was thinking Fourier series a special version of Fourier transform, as in it can only be used for periodic function and only produces discrete waves. By this logic, aren't...

I've been trying to answer this question for several days now with no results. Here is the question Imgur: The most awesome images on the Internet Now, I know the answer is -4/npi, but after integrating the function piece-wise (broke it into 3 separate integrals) I got 4sin(npi/2)/npi...

Homework Statement I'm supposed to be using the similarity theorem and the shift theorem to solve: cos(πx) / π(x-.5) has transform e^(-iπs)*Π(s) Homework Equations similarity theorem f(ax) has transform (1/a)F(s/a) shift theorem f(x-a) has transform e^(-i2πas)F(s) The Attempt at a Solution...

Homework Statement I have question on doing the following indefinite integral: $$\int{d^3x(\nabla^2A^{\mu}(x))e^{iq.x}}$$ Homework Equations This is part of derivation for calculating the Rutherford scattering cross section from Quarks and Leptons by Halzen and Martin. This books gives the...

I read about how MRI works briefly, by flipping the water molecules using a magnetic field to the correct state then send the radio wave to these atoms and have it bounces back to be received by receiver coils and apply Fourier Transform to figure out the imaging. My question is, how does...

3d Fourier transform of function which has only radial dependence ##f(r)##. Many authors in that case define \vec{k} \cdot \vec{r}=|\vec{k}||\vec{r}|\cos\theta where ##\theta## is angle in spherical polar coordinates. So \frac{1}{(2\pi)^3}\int\int_{V}\int e^{-i \vec{k} \cdot...

I'm learning digital signal processing in my engineer class, but I'm more interested in apply these things into Astrophysics, so i know a little bit about for what is useful the Fourier Transform, so i thought why not use this in Analyzing the sun spectra! But what do you think!? Is it useful...

Given the following signal, find the Fourier representation, ##V(jf)= \mathfrak{F}\left \{ v(t) \right \}##: ## v(t)=\left\{\begin{matrix} A, & 0\leqslant t\leqslant \frac{T}{3}\\ 2A, & \frac{T}{3}\leqslant t\leqslant T\\ 0, & Else \end{matrix}\right. ## Then sketch ##V(jf)##. Homework...

Homework Statement Find a function ##u## such that ##\int_{-\infty}^\infty u(x-y)e^{-|y|}dy=e^{-x^4}##. Homework Equations Not really sure how to approach this but here's a few of the formulas I tried to use. Fourier transform of convolution ##\mathscr{F} (f*g)(x) \to \hat f(\xi ) \hat g(\xi...

Consider the following article: https://en.wikipedia.org/wiki/Fourier_series At definition, they say that an = An*sin() and bn = An*cos() So with these notations you can go from a sum having sin and cos to a sum having only sin but with initial phases. Why can I write an = An*sin() and bn =...

In a book the Fourier transform is defined like this. Let g(t) be a nonperiodic deterministic signal... and then the integrals are presented. So, I understand that the signal must be deterministic and not random. But why it has to be nonperiodic (aperiodic). The sin function is periodic and we...

Homework Statement Find the Fourier transform F(w) of the function f(x) = [e-2x (x>0), 0 (x ≤ 0)]. Plot approximate curves using CAS by replacing infinite limit with finite limit. Homework Equations F(w) = 1/√(2π)*∫ f(x)*e-iwxdx, with limits of integration (-∞,∞). The Attempt at a Solution I...

Homework Statement Use the Fourier transform to compute \int_{-\infty}^\infty \frac{(x^2+2)^2}{(x^4+4)^2}dx Homework Equations The Plancherel Theorem ##||f||^2=\frac{1}{2\pi}||\hat f ||^2## for all ##f \in L^2##. We also have a table with the Fourier transform of some function, the ones of...

Homework Statement I've gotten myself mixed up here , appreciate some insights ... Using Fourier Transforms, shows that Greens function satisfying the nonhomogeneous Helmholtz eqtn $$ \left(\nabla ^2 +k_0^2 \right) G(\vec{r_1},\vec{r_2})= -\delta (\vec{r_1} -\vec{r_2}) \:is\...

I've gotten myself mixed up here , appreciate some insights ... Using Fourier Transforms, shows that Greens function satisfying the nonhomogeneous Helmholtz eqtn $ \left(\nabla ^2 +k_0^2 \right) G(\vec{r_1},\vec{r_2})= -\delta (\vec{r_1} -\vec{r_2}) $ is $ G(\vec{r_1},\vec{r_2})=...

Homework Statement The complex amplitudes of a monochromatic wave of wavelength ##\lambda## in the z=0 and z=d planes are f(x,y) and g(x,y), redprctively. Assume ##d=10^4 \lambda##, use harmonic analysis to determine g(x,y) in the following cases: (a) f(x,y)=1 ... (d) ##f(x,y)=cos^2(\pi y / 2...

Considering two functions of ##t##, ##f\left(t\right) = e^{3t}## and ##g\left(t\right) = e^{7t}##, which are to be convolved analytically will result to ##f\left(t\right) \ast g\left(t\right) = \frac{1}{4}\left(e^{7t} - e^{3t}\right)##. According to a Convolution Theorem, the convolution of two...

If I have a wave function given to me in momentum space, bounded by constants, and I have to find the wave function in position space, when taking the Fourier transform, what will be my bounds in position space?

My book says the expansion of $f(x)=x, -\pi \lt x \lt \pi = \sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{n}$, I get double that so please tell me where this is wrong: f(x) is odd, so $a_n=0$ $ b_n=\frac{1}{\pi} \int_{-\pi}^{\pi}x Sin(nx) \,dx = \frac{1}{\pi} [\frac{1}{n^2}Sin(nx) - \frac{x}{n}...

Homework Statement When ##f## and ##g## are ##2\pi##-periodic Riemann integrable functions define their convolution by ##(f*g)(x) = \frac{1}{2\pi} \int_0^{2\pi} f(y)g(x-y)dy## Denoting Fourier coefficients by ##c_n(f)## show that ##c_n(f * g) = c_n(f)c_n(g)##. Homework Equations ##c_n =...

Suppose all Dirichlet conditions are met and we have a function that has jump discontinuities. Dirichlet's theorem says that the series converges to the midpoint of the values at the jump discontinuity. What bothers me then is: Dirichlet's theorem is basically telling us the series isn't the...

Hello, I think that I have done this correctly, but this is the first problem I have done on my own and would appreciate confirmation. 1. Homework Statement Find the Fourier series corresponding to the following functions that are periodic over the interval (−π, π) with: (a) f(x) = 1 for...

I have been very briefly introduced to Fourier transformations but the topic was not explained especially well (or I just didn't understand it!) We were shown the graphs with equations below and then their Fourier transformation (RHS). I understand the one for cos(2pist) but NOT the sin(2pist)...

Homework Statement Compute ##\int_0^\infty \frac{\sin x}{x}dx## using that ##\frac{\sin x}{x} = \frac{b_0}{2} +\sum_1^\infty b_n \cos nx \; \; , \; \; 0 < x < \pi## with ##b_n = \frac{1}{\pi} \int_{(n-1)\pi}^{(n+1)\pi} \frac{\sin x}{x}dx##. Homework Equations Perhaps the following...

Hello. I'm studying quantization of electromagnetic field (to see photon!) and on the way to reach harmonic oscillator Hamiltonian as a final stage, sudden transition that the Fourier components of vector potential A become quantum operators is observed. (See...

Hi I am trying to program excel to take the DFT of a signal, then bring it back to the time domain after a low pass filter. I have a code that can handle simple data for example t = [ 0, 1, 2, 3] y = [2, 3, -1, 4] So I think everything is great and so I plug in my real signal and things go off...

I am learning Fourier series and have come across the sine, cosine, and imaginary exponential expressions. To my knowledge, these individual terms form a basis since they are all orthogonal to each other. I am just wondering: can a Fourier sine series be used to model a purely even function...

I have been studying Fourier transforms lately. Specifically, I have been studying the form of the formula that uses the square root of 2π in the definition. Now here is the problem: In some sources, I see the forward and inverse transforms defined as such: F(k) = [1/(√2π)] ∫∞-∞ f(x)eikx dx...

Hello everyone, The question that I have may not be fully relevant to the title, but I thought that could be the best point to start the main question! I'm working on 2-D data which are images. For some reason, I have converted my data to a 1-D vector, and then transformed them to the...

Homework Statement Derive ##\sum_{n=1}^\infty \frac{1}{n^2+b^2} = \frac{\pi}{2b}\coth b\pi - \frac{1}{2b^2}## from either ##e^{b\theta} = \frac{e^{2\pi b}-1}{2\pi} \sum_{-\infty}^\infty \frac{e^{in\theta}}{b-in}## for ##0 < \theta < 2\pi##. or ##e^{b\theta} = \frac{\sinh...

There are many waves and oscillations books out there that also include Fourier analysis but very few give the subject a thorough treatment, they just pass it in a few pages. If anybody has any sources(particularly books) that have Fourier analysis and particularly Fourier Transforms, I would...

With Dirac Comb is defined as follow: $$III(t)=\sum_{n=-\infty}^\infty\delta(t-nT)$$ Fourier Transform from t domain to frequency domain can be obtained by: $$F(f)=\int_{-\infty}^{\infty}f(t)\cdot e^{-i2\pi ft}dt$$ I wonder why directly apply the above equation does not work for the Dirac Comb...

I have the following problem:

i have read many of the answers and explanations about the similarities and differences between laplace and Fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of laplace. Some signals have Fourier but laplace is not defined , for instance cosine or sine...

My professor in Classical Electrodynamics is great and all, but sometimes he has trouble understanding what it is that I don't understand. So here I am. Let's say we have the some sort of (monochromatic) radiating system generating a electric field with Fourier amplitude Eω(x) and want to...

Is learning Fourier analysis useful for a high school student? If so, which book should I refer for learning the basics of Fourier analysis? This topic is not in my syllabus. But will it be useful for solving problems? (even if its not, it seems interesting to me). I have learned single variable...

Hi, I have a general function u(x,y,z,t). Then, (1) what would be the space-time Fourier transform of G⊗(∂nu/∂tn) and (2) would the relation G⊗(∂nu/∂tn) = ∂n(G⊗u)/∂tn hold true? Here, note that the symbol ⊗ represents convolution and G is a function of (x,y,z) only (i.e. it does not depend on...

Homework Statement Suppose a horizontally stretched string is heavy enough for the effects of gravity to be significant, so that the wave equation must be replaced by ##u_{tt} = c^2u_{xx} - g## where ##g## is the acceleration due to gravity. The boundary conditions are ##u(0,t) = u(l,t) = 0##...

< Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown >[/color] hi I've got a problem that I've partially worked but don't understand the next part/have made a mistake? f(x)=0 for -π<x<0 and f(x)=x for 0≤x≤π i got a0=π/4 and an=0 and bn=0 if n is even and...