Fourier Definition and 1000 Threads

Hi everyone. I ran into a problem while attempting my Fourier Series tutorial. I don't really understand the "L" in the general formula for a Fourier Series (integration form). I shall post my question and doubts as images. Thank you for any assistance rendered. <I am solving Q3 in the image.>

when i use this MATLAB code but with rectangular pulse shape instead of chirped pulse i didn't get the predicted output , so is there a limitation with this method in case of rectangular pulse or the MATLAB code is wrong...

Homework Statement Homework EquationsThe Attempt at a Solution I'd like to see if I have the right line of thinking in my solutions: a. The sampling frequency should be such that no aliasing or folding occurs, so it should be twice the frequency of the original signal. $$x(t) = -17...

Hi everyone, do you know how to calculate the Fourier transform for the infinitely deep circular well (confined system)? The radial wave function is given by R=N_m J_m (k r). k=\alpha_{mn}/R. R is the radius of the circular well. R(k R)=0. Thanks. Another question is that The k in J_{m}(k r)...

Does anyone know how to calculate the error between a function and its Fourier series representation as a function of the partial sums of the series? So far I haven't been able to find anything in the literature that talks about this. I'm also interested in looking at how well a Fourier series...

Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ $$a_0=\frac{2}{b-a}\int_{a}^{b}f(t)dt,$$ $$a_n=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi t}{b-a}dt,$$...

Hi, I have a simple harmonic oscillation problem whose Green function is given by $$\Bigl[\frac {d^2}{dt^2}+ \omega_{0}^{2}\Bigl] G(t, t') = \delta(t-t')$$ Now I found out the Fourier transform of $G(t, t')$ to be $$G(\omega)= \frac{1}{2\pi} \frac{1}{\omega_{0}^{2}-\omega^2}$$ which has poles...

Homework Statement An #a*b*c box is given in x,y,z (so it's length #a along the x axis, etc.). Every face is kept at #V=0 except for the face at #x=a , which is kept at #V(a,y,z)=V_o*sin(pi*y/b)*sin(pi*z/c). We are to, "solve for all possible configurations of the box's potential" Homework...

I' m trying to solve something as apparently simple like this cos ax/sin pi*x which appears solved in https://archive.org/details/TheoryOfTheFunctionsOfAComplexVariable in the page 157, exercise 9. second part. I'm trying by Fourier series, but by the moment I can't achieve it. Thanks.

Homework Statement The major problem I am facing while solving for Fourier series is about the limits to be taken while integrating..! In the general equation of Fourier series the upper & lower limits are t1 & t1+T respectively..while solving for even functions we take t1 =-T/2..! Why is it...

Hello, Let's suppose we are given a function f:\mathbb{R}\rightarrow \mathbb{R}, and we assume its Fourier transform F=\mathcal{F}(f) exists and has compact support. What sufficient condition could we impose on f, in order to be sure that F is also bounded?

Let's say I have Fourier series of some function, f(t), f(t)=\frac{a0}{2}+\sum_{n=1}^{\infty}(an\cos{\frac{2n\pi t}{b-a}}+bn\sin{\frac{2n\pi t}{b-a}}), where a and b are lower and upper boundary of function, a0=\frac{2}{b-a}\int_{a}^{b}f(t)dt, an=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi...

Whais is the effect of a multiplication by (-1)^n in the DTFT ?? In other words, what's is this transform : x(n)* (-1)^n ??

Hi, I am totally a non-math guy. I had to attend a training (on automobile noise signals) that had a session discussed about Fourier Transform (FT). Let me pl. write down what I understand: "The noise signal observed at any point in the transmission line can be formed using a sum of many sine...

Homework Statement Silly question, but I can't seem to figure out why, in e.g. Peskin and Schroeder or Ryder's QFT, the Fourier transform of the (quantized) real scalar field \phi(x) is written as \phi (x) = \int \frac{d^3k}{(2\pi)^3 2k_0} \left( a(k)e^{-ik \cdot x} + a^{\dagger}(k)e^{ik...

Hi! Here is my m-file for Fourier series plot: clear clc syms n a0=input('Enter coefficient a0: '); an=input('Enter coefficient an: '); bn=input('Enter coefficient bn: '); a=input('Enter lower boundary: '); b=input('Enter upper boundary: '); t=linspace(a,b,10000); sum=0; for n=1:10 %%n could...

E.g., if I have a time independent wavefunction \psi(x) with Fourier transform \tilde{\psi}(k), in computing the expectation of momentum are we calculating the principal value \lim_{R \to \infty} \int_{-R}^{R} dk\,\lvert \tilde{\psi}(k)\lvert^2\, \hbar k instead of the improper integral...

Hi all, I have a somewhat qualitative understanding of image Fourier transforms and what they represent which for the most part is sufficient for me. However i am interested to know how when i use an image analysis program to produce the Fourier transform of a real image, what is actually...

Homework Statement Problem: a) Find the Fourier transform of the Dirac delta function: δ(x) b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves). c) test by integration, that the delta function represented by a Fourier integral integrates...

Is it possible to represent some signal in terms of Fourier series in Multisim? For example, Fourier series of sawtooth voltage with period T=2pi is $$\sum_{n=1}^{\infty }\frac{2}{n}(-1)^{n+1}sin{(nt)}=2sin{(t)}-sin{(2t)}+\frac{2}{3}sin{(3t)}-\frac{1}{2}sin{(4t)}+...$$. These terms on right side...

I have been given this y(t)=\frac{sin(200πt)}{πt} All I want is to find, is how the rectangular pulse will look like if I take the transformation of the above. That "200" kind of confusing me, because it isn't a simple sinc(t)=\frac{sin(πt)}{πt} I need somehow to find the height of the...

In Kittel's solid state text, problem 2.3, he says that the volume of the Brillouin zone is the same as a primitive parallelepiped in Fourier space. Somehow I can't see why this is true. Can someone help me see why this is true? Also, is the same relationship true between Wigner-Seitz cells and...

Hi! I have a question for you. At the end of the post there's a link. There's the homework which I have to do for an exam. I have to study the Fourier Integral Operator that there is at the begin of the paper. I did almost all the homework but I can't do a couple of things. First: at the point...

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...

Here we will use the following transforms: $\displaystyle \begin{align*} \mathcal{F}^{-1} \left\{ \frac{n!}{ \left( a + \mathrm{i}\,\omega \right) ^{n+1} } \right\} = t^n\,\mathrm{e}^{-a\,t}\,\mathrm{H}(t) \end{align*}$ and $\displaystyle \begin{align*} \mathcal{F}^{-1} \left\{...

a) Start by writing $\displaystyle \begin{align*} \frac{ 1}{ \omega ^2 - 7\mathrm{i}\,\omega - 10 } = \frac{1}{ \omega ^2 - 7\mathrm{i} \,\omega + 10\mathrm{i}^2 } = \frac{1}{ \left( \omega - 5\mathrm{i} \right) \left( \omega - 2\mathrm{i} \right) } \end{align*}$ Now applying partial fractions...

Completing the square gives $\displaystyle \begin{align*} \frac{2\mathrm{i}\,\omega}{\omega ^2 + 10\omega + 29} &= \frac{2\mathrm{i}\,\omega}{ \omega ^2 + 10\omega + 5^2 - 5^2 + 29} \\ &= \frac{2\mathrm{i}\,\omega}{ \left( \omega + 5 \right) ^2 + 4 } \\ &= \frac{2\mathrm{i}\,\omega}{ \left(...

The first thing you need to do is take the Fourier Transform of both sides of the equation. $\displaystyle \begin{align*} \mathcal{F} \left\{ f(t) + 3\int_{-\infty}^{\infty}{f(t-u)\,\mathrm{e}^{-2u} \, \mathrm{H}(u)\,\mathrm{d}u} \right\} &= \mathcal{F} \left\{ 15\mathrm{e}^{-2t^2} -...