Fourier Definition and 1000 Threads

1. Expand the function f(x)=x^3 in a Fourier sine series on the inteval 0≤ x ≤ 1 2. I was thinking of using these equations in an attempt to find the solution f(x)=∑b_{n}sin(nx) and b_n=\frac{2}{∏}∫f(x)sin(nx)dx where n=1,2,...,I am somewhat lost in what to do exactly, could anyone help...

Hi! I am taking a second look on Fourier transforms. While I am specifically asking about the shape of the Fourier transform, I'd appreciate if you guys could also proof-read the question below as well, as I've written down allot of assumptions that I've gained, which might be wrong. OK...

Hi bros, so I feel like I am very close, but cannot find out how to go further. Q.1 Compute the DTFT of the following signals, either directly or using its properties (below a is a fixed constant |a| < 1): for $x_n = a^n \cos(\lambda_0 n)u_n$ where $\lambda_0 \in (0, \pi)$ and $u_n$ is the...

Hi, My aim is to get a series of images in 2D space that run over different timestamps and put them through a 3D Fourier Transform. So my 3D FT has 2 spatial axes and one temporal axis. However I have never done anything like this before, and I have a very basic knowledge of Python. So...

Fourier said that any periodic signal can be represented as sum of harmonics i.e., containing frequencies which are integral multiples of fundamental frequncies. Why did he chose the basis functions i.e., the functions which are added to make the original signal to be sinusoidal? I know...

HI please help me this could someone verify it for me please find attachement clc; clear all; k=0; s=0; N=inf; for i=1:N s=s+(1/(k^2+1)); k=k+1; end syms x n a0=1/pi*int(cosh(x),-pi,pi); an=1/pi*int(cosh(x)*cos(n*x),-pi,pi); bn=1/pi*int(cosh(x)*sin(n*x),-pi,pi); fs=0...

Homework Statement OK, we're given to practice Fourier transforms. We are given f(x) = \int^{+\infty}_{-\infty} g(k) e^{ikx}dk and told to get a Fourier transform of the following, and find g(k): f(x) = e^{-ax^2} and f(x) = e^{-ax^2-bx} Homework Equations The Attempt at a Solution For...

Homework Statement Sketch the waveform and develop its Fourier series. f(\omega t)= \begin{cases} 0 & if & 0 \leq \omega t \leq \frac{π}{2} \\ V*sin(\omega t) & if & \frac{π}{2} \leq \omega t \leq π\\ 0 & if & π \leq \omega t \leq \frac{3π}{2} \\ V*sin(\omega t) & if & \frac{3π}{2} \leq...

Is there a time domain function whose Fourier transform is the Dirac delta with no harmonics? I.e. a single frequency impulse

I have this expression: f(\tau) = 4 \pi \int \omega ^2 P_2[\cos (\omega \tau)] P(\omega) \, \mathrm{d}w \quad [1] where P_2 is a second order Legendre polynomial, and P(\omega) is some distribution function. Now I am told that, given a data set of f(\tau), I can solve for P(\omega) by either...

We know that a function f(x) over an interval [a, b] can be written as an infinite weighted sum over some set of basis functions for that interval, e.g. sines and cosines: f(x) = \alpha_0 + \sum_{k=1}^\infty \alpha_k\cos kx + \beta_k\sin kx. Hence, I could provide you either with the function...

When I sample a certain digital signal with increasing sampling frequency, the fast Fourier transform of the sampled signal becomes finer and finer. (the image follows) Previously I thought higher sampling frequency makes the sampled signal more similar to the original one, so the Fourier...

I'm currently reading Tolstov's "Fourier Series" and in page 58 he talks about a criterion for the convergence of a Fourier series. Tolstov States: " If for every continuous function F(x) on [a,b] and any number ε>0 there exists a linear combination σ_n(x)=γ_0ψ_0+γ_1ψ_1+...+γ_nψ_n for which...

So I've been self-studying from Griffiths Intro to QM to get back in shape for graduate school this fall, and I guess I'd just like some confirmation that I'm on the right track... So while I am sure there are many other applications, the one I am dealing with is eigenfunctions of an operator...

Homework Statement Hello guys, I have to solve one basic problem, but I got the result twice smaller that it should be. So, I am thinking that I must have missed something basic. The problem is f\left(x\right) = 2x-1 for ##0<x<1##. I have to find the Fourier coefficients. I have found A_n...

Hi All, I have a problem I've been thinking about for a while, but I haven't come up with a really satisfactory solution: I want to do a discrete Fourier transform on data that has been sampled at 2 different sampling frequencies. I've attached a picture of what my data will look like...

Homework Statement what type of waveform would this make ? Homework Equations V(t)=2/π(sin(ωt)+1/2sin(2ωt)+1/3sin(3ωt)+1/4sin(4ωt)+...) 5sin(ωt)+5sin(2ωt)+5sin(3ωt)+5sin(4ωt)... The Attempt at a Solution

Hello all, I realize this isn't exactly the correct place to post this, but I can't start a thread in the mathematics learning forum, I'm not sure if I am supposed to be able to or not. I realized that I threw away one of my instructors math books on Fourier Theory I was borrowing over the...

Hi there, I'm having trouble understanding the Fourier transform of a function where the result in the frequency domain has imaginary components. For example, if you take the Fourier transform of Sin[t] , the result is I Sqrt[\[Pi]/2] DiracDelta[-1 + \[Omega]] - I Sqrt[\[Pi]/2]...

Homework Statement Find the FT of the following signal The function is: f(t) = t(\frac{sen(t)}{t\pi})^2 Homework Equations Fourier transform: F(\omega)= \int_{-\infty}^\infty f(t)e^{-jt\omega} My attempt began with this Fourier transform, and that's my goal: F[tf(t)]=...

I was wondering if anyone could help me with this integral. I've heard of contour integration but I'm unsure of how it would be used for this integral.

Homework Statement Is the function even, odd, or neither y(t) = \frac{2At}{w} for 0<t<\frac{w}{2} y(t) = \frac{-2At}{w}+2A for \frac{w}{2}<t<w Homework Equations even function f(-t) = f(t) off function f(-t) = -f(t) The Attempt at a Solution I just don't understand the concept, any help...

1. Hi! I am new at this forum, and english is not my native language, so, I hope I can make myself clear. A teacher send us a list of activities, but he did not give us the theory about it (the theoretical class). So, I have read a few things on the internet and I have solved some exercises. I...

Homework Statement \int_{-\infty}^{\infty} |k|^{2\lambda} e^{ikx} dkHomework Equations The Attempt at a Solution As you can guess, this is the inverse Fourier transform of |k|^{2\lambda}. I've tried splitting it from -infinity to 0 and 0 to infinity. I've tried noting that |k| is even, cos is...

I'm participating in research this summer and it's has to do with the Fourier Series. My professor wanted to give me practice problems before I actually started on the research. He gave me a square wave and I solved that one without many problems, but this triangle wave is another story. I've...

We have a wave ψ(x,z,t). At t = 0 we can assume the wave to have the solution (and shape) ψ = Q*exp[-i(kx)] where k = wavenumber, i = complex number The property for a Fourier transform of a time shift (t-τ) is FT[f(t-τ)] = f(ω)*exp[-i(ωτ)] Now, assume ψ(x,z,t) is shifted in time...

Can anybody help in in finding the Fourier transform of f(x) = xe^-x where -1<x<0 and f(x)= 0 otherwise?

Magic of Fourier Transform? Hello everyone,i am doing my project in image processing... i have done video sementation using the Fourier transform . I applied 3-D fft on video frames ((gray image(2D)+no of video frames(1D)=3D) and Obtained magnitude and phase spectrum and reconstructed video...

Homework Statement f(p) is the Fourier transform of f(x). Show that the Fourier Transform of eipox f(x) is f(p- p0).Homework Equations I'm using these versions of the Fourier transform: f(x)=1/√(2π)∫eixpf(p)dx f(p)=1/√(2π)∫e-ixpf(x)dx The Attempt at a Solution I have...

2D Fourier Transform on a non-rectangular area Is it possible to perform a Fourier transform on a shape instead of a rectangular region? To be specific I am attempting to make a linear zoom function that doesn't produce any pixelation and that mimics natural blur that occurs with distance...

hey pf! physically, what does a Fourier transform do? physically what comes out if i put velocity in? thanks! josh

calculate the Fourier transform of the function g(x) if g(x) = 0 for x<0 and g(x) = ##e^{-x}## otherwise. putting g(x) into the transform we have: ##\tilde{g}(p) \propto \int_{0}^{inf} e^{-ipx} e^{-x} dx## which we can write: ##\tilde{g}(p) \propto \int_{0}^{inf} e^{-x(ip+1)} dx##...

Our Fluid Mechanics professor gave us a challenge: to find the shape of a vessel with a hole at the bottom such that the water level in the vessel will change at a constant rate (i.e. if z is the height of the water in the tank dz/dt=constant). I presented a solution assuming that the vessel...

Can someone tell me if the continuous Fourier transform of a continuous (and vanishing fast enough ) function is also a continuous function?

Hi all, Now naturally after completing a physics degree I am very familiar with the form and function of the Fourier Transform (FT) but never have grasped it quite conceptually. I understand that given a function f(x) I can express every functional value as a linear combination of complex...

Hello! My problem consists of : there is a representation of an uneven surface in terms of Fourier series with random coefficients: The random coefficients are under several conditions: W - function is undefined. Maybe you've confronted with such kind of expressions. The...

Can anybody helps in suggesting books on Fourier transforms and applications. I have seen many applications of Fourier transforms. But, I'm not able to visualize what's going on. Fourier transformations are there in Quantum mechanics also. It will be helpful in learning quantum mechanics. Thanks

Hey! :o I have to solve the following initial and boundary value problem: $$u_t=u_{xx}, 0<x<L, t>0 (1)$$ $$u(0,t)=u_x(L,t)=0, t>0$$ $$u(x,0)=x, 0<x<L$$ I did the following: Using the method separation of variables, the solution is of the form: $u(x,t)=X(x)T(t)$ Replacing this at $(1)$, we...

Hi, The following contains two questions that I encountered in the books of Claude Cohen-Tannoudji, "Atom-Photon Interactions" and "Atoms and Photons: Introduction to Quantum Electrodynamics". The first one is about how to calculate two Fourier transforms, and the second one is a example of...

Suppose I have a function of the type: h(t) = g(t)f(t) where g(t) is a periodic function. Are there any nice properties relating to the Fourier transform of such a product? Edit: If not then what about if g(t) is taken as the complex exponential?

Hi all, I have a problem in evaluating Fourier coefficient. I have an array of numbers (which is a magnetic field) taken on a circle of radius 8mm and I want to know the harmonics of this field. I have written a code in matlab, I have used some in built function in MATLAB and mathcad, but...

Homework Statement utt=a2uxx Initial conditions: 1)When t=0,u=H,1<x<2 and u=0,x\notin(1<x<2) 2)When t=0,ut=H,3<x<3 and u=0,x\notin(3<x<4) The Attempt at a Solution So I transformed the first initial condition \hat{u}=1/\sqrt{2*\pi} \int Exp[-i*\lambda*x)*H dx=...

Hi I wanted to check how to do a Fourier synthesis to recreate a signal from a frequency spectrum. So I basically have the frequency spectrum so I have the power of the fundamental frequency and the harmonics. Is there a way I can do a synthesis to create a time signal?

The Fourier integrals and series can be written of 3 forms (possibly of 4): the "real cartesian": a(ω)cos(ωt) + b(ω)sin(ωt) the "real polar": A(ω)cos(ωt - φ(ω)) where: A² = a² + b² sin(φ) = b/A cos(φ) = a/A tan(φ) = b/a the "complex polar" A(ω)exp(iφ(ω))exp(iωt) And my...

Which is the difference between the Fourier integral and Fourier transform? Or they are the same thing!? Fourier integral:

Homework Statement Find possible momentum, and their probabilities. Find possible energies, and their probabilities. Homework Equations The Attempt at a Solution First, we need to Fourier transform it into momentum space: \psi_k = \frac{1}{\sqrt{2\pi}} \int \psi_x e^{-ikx} dx =...

Homework Statement Let FT(f) = Fourier transform of f, (f*g)(x) = convolution of f and g. Given FT(f*g) = FT(f)FT(g), the first part of the convolution theorem, show that FT[fg] = [FT(f)*FT(g)]/2pi. Homework Equations Duality: FT2f(x) = (2pi)f(-x) Convolution: (f*g)(x) =...

Homework Statement Find the value of An and given that f(x) = 1 for 0 < x < L/2, find the sum of the infinite series. Homework Equations The Attempt at a Solution The basis is chosen to be ##c_n = \sqrt{\frac{2}{L}}cos (\frac{n\pi }{L}x)## for cosine, and ##s_n = \sqrt{\frac{2}{L}}sin...

Homework Statement Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F_13 for i, j = 0,1,2, 3. Compute F(hat) and verify that F(hat)F = I Homework Equations The matrix F(hat) is called the inverse discrete Fourier transform of F. The Attempt at a Solution I found that e = 4...

Homework Statement (i) Verify that 5 is a primitive 4th root of unity in F13. (ii) Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F13 for i, j = 0,1,2, 3. Compute F(hat) and verify that F(hat)F= I. Homework Equations The matrix F(hat) is called the inverse discrete Fourier...