Fourier Definition and 1000 Threads

Homework Statement Given ∑^{∞}_{n=1} n An sin(\frac{n\pi x}{L}) = \frac{λL}{\pi c} σ(x-\frac{L}{2}) + A sin(\frac{\pi x}{2}), where L, λ, c, σ and A are known constants, find An. Homework Equations Fourier half-range sine expansion. The Attempt at a Solution I understand I...

I have done a test on a 4 piston test engine which is expected to exhibit torsional resonance at 800RPM and a vertical translational resonance at 1200RPM. The data we gathered from the test bed machine was as follows: Theta | Signal 0 | -5 60 | -1 120 | 7 180...

Hey! :o Could you give me a hint how to prove the following property of the Fourier transform, when $F[f(x)]=\widetilde{f}(x)$, where $F[f(x)]$ is the Fourier transform of $f(x)$? $$F[ \widetilde{f}(x) ]= \frac{f(-k)}{2 \pi}$$ We know that: $ \widetilde{f}(k)=\int_{- \infty}^{+ \infty}{...

Hello, Homework Statement Develop in Fourier series 1/cos(z) and cotan(z) for Im(z)>0 Homework Equations The Attempt at a Solution I really don't know how to do this, i was looking at my notes and we just saw Fourier transform and there is no example for complex functions. I...

Homework Statement Use the Fourier transform directly to solve the heat equation with a convection term u_t =ku_{xx} +\mu u_x,\quad −infty<x<\infty,\: u(x,0)=\phi(x), assuming that u is bounded and k > 0. Homework Equations fourier transform inverse Fourier transform convolution thm The...

I have been trying to follow how the complex Fourier coefficients are obtained; the reference I am using is at www.thefouriertransform.com. However I am unable to follow the author's working exactly and wondered if anyone could help me see where I am going wrong. First, I understand that the...

Homework Statement Take the inverse Fourier Transform of 5[\delta(f+100)+\delta(f-100)]\bigg(\frac{180+j2\pi f*0.0135}{1680+j2\pi f*0.0135}\bigg)Homework Equations g(t)=\int_{-\infty}^{\infty} G(f)e^{j2\pi ft}dt The Attempt at a Solution g(t)=\int_{-\infty}^{\infty}...

May i know to obtain Fourier series representation for trigonometric and complex form base on magnitude spectrum and phase spectrum?? what i found is that to get trigonometric form is from phase spectrum, but i don't know how.. can anyone help

Homework Statement Without solving the differential equation, find the differential equation that solves Fourier transformation of given differential equation for ##a>0##. a) ##y^{'}+axy=0## b) For what ##a## is the solution of part a) an eigenfunction of Fourier Transform Homework Equations...

Hello everyone . I need to ask if I want to get the right units for each parameter to be able to get the right results Can anyone define them properly? Like the gamma the initial power of pump or signal dispersion and non linear factors Thank you

Homework Statement Calculate (from the definition, no tables allowed) the Fourier Transform of e^{-a*|t|}, where a > 0. Homework Equations Fourier Transform: G(f) = \int_{-\infty}^{\infty} g(t)e^{-j\omega t} dt The Attempt at a Solution I thought I'd break up the problem into the two cases...

I'm having some problem in determining the phase of an exponential Fourier series. I know how to determine the coefficient which in turn gives me the series after I multiply by e^-(jωt) I can determine the amplitude by dividing the coefficient by 2 |Dn| = Cn/2 Now my question is how to...

hey pf! if we have a piecewise-smooth function ##f(x)## and we create a Fourier series ##f_n(x)## for it, will our Fourier series always have the 9% overshoot (gibbs phenomenon), and thus ##\lim_{n \rightarrow \infty} f_n(x) \neq f(x)##? thanks!

Homework Statement Suppose f(x), -\infty<x<\infty, is continuous and piecewise smooth on every finite interval, and both \int_{-\infty}^\infty |f(x)|dx and \int_{-\infty}^\infty |f'(x)|dx are absolutely convergent. Show the Fourier transform of f'(x) is i\mu F(\mu).Homework Equations...

A necessary condition that a function f(x) can be Fourier transformed is that f(x) is absolutely integrable. However, some function, such as |t|, still can be Fourier transformed and the result is 1/w^2, apart from some coefficients. This can be worked out, as we can add a exponential...

Homework Statement "Suppose x[n] is a DT (discrete time) periodic signal with fundamental period N. Let us define x_{n}[n] to be x[n] for n ε {0, 1,2, ... , N-1} and zero elsewhere. Denote the Fourier transform of x_{n}[n] with X_{n}[e^jω]. How can one find the Fourier Series coefficients...

1. Given: The DTFT over the interval |ω|≤\pi, X\left ( e^{jω}\right )= cos\left ( \frac{ω}{2}\right ) Find: x(n) 2. Necessary Equations: IDTFT synthesis equation: x(n)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}X\left ( e^{jω} \right ) e^{j\omega n}d\omega Euler's Identity...

Homework Statement Determine the Fourier Transform of the function shown. Plot the result using excel, MathCad, or Matlab. See attachment for figure of triangle above x-axis from -X0/2 tp X0/2 with a max height of 1 at x=0. Homework Equations The answer is F(k) = X0/2 [sin(kX0/4) /...

Homework Statement Using the formal limit definition of the derivative, derive expressions for the Fourier Transforms with respect to x of the partial derivatives \frac{\partial u}{\partial t} and \frac {\partial u}{\partial x} . Homework Equations The Fourier Transform of a function...

Hello, I have a problem synthesising the complex Fourier series using Matlab. The time domain periodic function is: -1, -1.0 ≤ t < -0.5 1 , -0.5≤ t <0.5 -1, 0.5 ≤ t < 1 The single non zero coefficient is: Cn = \frac{2}{\pi n}, Co is 0 (average is 0). f(t)= \sum Cn...

Homework Statement (a) Find the Fourier transform f(ω) of: f(x) = cos(x) between -pi/2 and pi/2 (b) Find the Fourier transform g(ω) of: g(x) = sin(x) between = -pi/2 and pi/2 (c) Without doing any integration, determine f(ω)/g(ω) and explain why it is so Homework Equations f(ω) =...

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...

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...