Fourier Definition and 1000 Threads

We have: Period T = 4, so fundamental frequency w0 = pi/2. This question seems sooo easy. But when I use the integral: x(t) = Σa[k] * exp(i*k*pi/2*t). I get 1 + sum(cos(k*pi/2*t)), which does not converge. Where did I went wrong ? Thanks a lot for your help.

Homework Statement I know that we can write ## \int_{-\infinity}^{\infinity}{e^{ikx}dx}= 2\pi \delta (k) ## But is there an equivalent if the interval which we are considering is finite? i.e. is there any meaning in ##\int_{-0}^{-L}{e^{i(k-a)x}dx} ## is a lies within 0 and L? Homework...

Homework Statement Homework Equations if x(t) --> X(W) then x(-t) --> X(-W) and x(t+a)-->ejwX(W) The Attempt at a Solution I'm getting right answer for 1st part. For second part book says right answer is C. Where am I wrong?[/B]

Homework Statement There is a sawtooth function with u(t)=t-π. Find the Fourier Series expansion in the form of a0 + ∑αkcos(kt) + βksin(kt) Homework Equations a0 = ... αk = ... βk = ... The Attempt at a Solution After solving for a0, ak, and bk, I found that a0=0, ak=0, and bk=-2/k...

Homework Statement Show, by completing the square in the exponent, that the Fourier transform of a Gaussian wavepacket ##a(t)## of width ##\tau## and centre (angular) frequency ##\omega_0##: ##a(t)=a_0e^{-i\omega_0t}e^{-(t/\tau)^2}## is a Gaussian of width ##2/\tau##, centred on ##\omega_0##...

I'm kinda just hoping someone can look over my work and tell me if I'm solving the problem correctly. Since my final answer is very messy, I don't trust it. 1. Homework Statement We're asked to find the Fourier series for the following function: $$ f(\theta)=e^{−\alpha \lvert \theta \rvert}}...

Homework Statement In the following problem I am trying to extend the function $$f(x) = x $$ defined on the interval $$(0,\pi)$$ into the interval $$(-\pi,0)$$ as a even function. Then I need to find the Fourier series of this function.Homework EquationsThe Attempt at a Solution So I believe I...

Homework Statement Considering the function $$f(x) = e^{-x}, x>0$$ and $$f(-x) = f(x)$$. I am trying to find the Fourier integral representation of f(x). Homework Equations $$f(x) = \int_0^\infty \left( A(\alpha)\cos\alpha x +B(\alpha) \sin\alpha x\right) d\alpha$$ $$A(\alpha) =...

In the following question I need to find the Fourier cosine series of the triangular wave formed by extending the function f(x) as a periodic function of period 2 $$f(x) = \begin{cases} 1+x,& -1\leq x \leq 0\\ 1-x, & 0\leq x \leq 1\\\end{cases}$$ I just have a few questions then I will be able...

I am studying online course notes from University of Waterloo on 'Analytical mathematics in geology' in which the author describes a 'modified Fourier transform' which can be used to incorporate 3rd kind of boundary conditions. The formula is ## \Gamma \small[ f(x) \small] = \bar{f}(a) =...

Homework Statement The question below is asking how long it would take for the cooler side of the handle to heat up till its unbearably hot. I'm having a bit of trouble trying to understand the solution and would like some guidance. I can't seem to get how the ##\Delta T ## that represents...

Homework Statement Find trigonometric Fourier series for ##f(x)=|x|##, ##x∈[−\pi, \pi]##, state points where ##F(x)## fail to converge to ##f(x)##. Homework Equations ##F(x) = \frac{a_0}{2}+\sum\limits_{n=1}^\infty a_ncos(\frac{n\pi x}{L})+b_nsin(\frac{n\pi x}{L})##...

Homework Statement Consider the Fourier series of a signal given by $$x(t)=\sum_{k=-\infty}^{\infty} a_ke^{jk\omega_0t}$$ Let's consider an approaches to this series given by the truncated series. $$x_N(t)=\sum_{k=-N}^{N} a_ke^{jk\omega_0t}$$ a- Show that if $x(t)$ is real then the series...

I'm trying to use Maxima to examine the error in a Fourier series as the number of terms increases. I've figured out how to produce a Fourier series and plot partial sums, but this has me stumped. If anyone experienced with the Maxima CAS has some insight into this, I would greatly appreciate...

Homework Statement Find Fourier coefficients of the periodic function whose template is x(t) where the Fourier Transform of x(t) is X(f) = (1-f^2)^2 where \left|f\right|<1 and period T_0= 4. Homework Equations FC=\hat x_T(k,T_0)=\sum_{k=-\infty}^\infty\frac{1}{T_0}X\left(k/T_0\right) The...

Homework Statement I have this function ##f(\theta)=cos(n \ sin(\frac{\theta}{2})\pi)## and I need to take the discrete Fourier transform (DFT) numerically. I did so and I attached the result for ##\theta \in [0,2\pi)## and n =2,4,8,16,32, together with the function for a given n. I need to...

Hi, I would like to know how one can simultaneously measure the thickness and refractive index of a sample using Fourier Domain OCT. I have a glue layer on a glass microscope slide and I've calculated the thickness of the glue layer, but I'm unsure of how to the refractive index of the glue. Any...

Homework Statement I need to calculate the derivative of a function using discrete Fourier transform (DFT). Below is a simplified version of my code (just for sin function) in python Homework Equations from __future__ import division import numpy as np from pylab import * pi = np.pi def...

Let's consider a signal which is continuous in both time and amplitude. Now we consider the amplitude of this signal at specific time instants only. This is my understanding of sampling a signal in time domain. When performing a Fourier transform on a time discrete signal, we have to apply the...

Homework Statement Boundary conditions are i) V=0 when y=0 ii) V=0 when y=a iii) V=V0(y) when x=0 iv) V=0 when x app infinity. I understand and follow this problem (separating vars and eliminated constants) until the potential is found to be V(x,y) = Ce^(-kx)*sin(ky) Condition ii...

Homework Statement An LTI system has an impulse response h(t) = e-|t| and input of x(t) = ejΩt Homework Equations Find y(t) the system output using convolution Find the dominant frequency and maximum value of y(t) Ω = 2rad/s The Attempt at a Solution I have tried using the Fourier transform...

Given that position and momentum are Fourier conjugates, what is the derivation for the equation ##\hbar \vec{k} = m \vec{v}##, where momentum-space momentum is defined as ##\hbar \vec{k}## and position-space momentum is defined as ##m \vec{v}##?

Svein submitted a new PF Insights post Further Sums Found Through Fourier Series Continue reading the Original PF Insights Post.

Homework Statement Find the magnitude and phase of the Fourier transform of h(t)=t over the interval 0,1 Homework Equations H(s) = \int^{1}_{0} h(t) e^{-i 2 \pi f t} dt The Attempt at a Solution I found this thread...

Hi, First of all, I want to say that I know how can define and calculate Fourier coefficients but I have some question about the final presentation of Fourier and half-period or unknown period functions. 1)In this function how can we define T? 2)for above diagram, in a book, they define f(t)...

Homework Statement f(x)=x on [0,2) Homework Equations Fourier Series is given as: f(x)=a0/2 + n=1∞∑(an*cos(nπx/L) + bn*sin(nπx/L) a0=1/L*-LL∫f(x)dxThe Attempt at a Solution Basically what I am being taught is that we take the Period, T, to be equal to 2L so, T=2L In this case T=2 and L=1. My...

Hi there, I am also familiar with Hilbert spaces and Functional Analysis and I find your question very interesting. I agree that the Fourier transform is a powerful tool for analyzing LTI systems and diagonalizing the convolution operator. As for your question about whether the same can be...

When you do a Fourier transform of spacetime.. what do you get? (or how does spacetime look in frequency domain? And what applications do this and what results are they looking or solving for?

Homework Statement In Complex Fourier series, how to determine the function is odd or even or neither, as in the given equation $$ I(t)= \pi + \sum_{n=-\infty}^\infty \frac j n e^{jnt} $$Homework Equations ##Co=\pi## ##\frac {ao} 2 = \pi## ##Cn=\frac j n## ##C_{-n}= \frac {-j} n ## ##an=0##...

Hi, this thread is an extension of this one: https://www.physicsforums.com/posts/5829265/ As I've realized that the problem is that I don't know how to properly use FFTW, from http://www.fftw.org. I am trying to calculate a derivative using FFTW. I have ##u(x)=e^{\sin(x)}##, so...

Hi. I was checking the library for the discrete Fourier transform, fftw. So, I was using a functition ##f(x)=sin(kx)##, which when transformed must give a delta function in k. When I transform, and then transform back, I effectively recover the function, so I think I am doing something right...

Homework Statement Solve the following heat Eq. using FFCT: A metal bar of length L is at constant temperature of Uo, at t=0 the end x=L is suddenly given the constant temperature U1, and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature at any...

Homework Statement Homework Equations The Attempt at a Solution a0=4 an=8/Pi*n Heres a written solution https://gyazo.com/57e11d1e7a360914db8aec167beb6b39.png

Is it possible to show that every kind of possible wave form can be decomposed into a sum of sines and cosines? If so, how is it done?

When discussing about generalized coordinates, Goldstein says the following: "All sorts of quantities may be impressed to serve as generalized coordinates. Thus, the amplitudes in a Fourier expansion of vector(rj) may be used as generalized coordinates, or we may find it convenient to employ...

Hi there - just a quick question about Fourier transforms: When learning about quantum mechanics, I found that the Fourier transform and inverse Fourier transform were both defined with constants of ##{ \left( 2\pi \right) }^{ -d/2 }## in front of the integral. This is useful, as...

Hello buddies! Please, check out these equations... Tell me, please, are they mathematically correct or not? I need a simple YES/NO answer. I have not sufficient knowledge to understand them. I just need to know whether they are correct... Thank you! P.S. Am is amplitude; I guess it is a...

Homework Statement Q/ in this inverse Fourier problem, how did he come with the results of integration of (Sinc) function and how did he come up with those results of integration with the inverse part (as in the attached picture) here is the problem: https://i.imgur.com/Ir3TQIN.png Homework...

Homework Statement Hello everyone, am trying to solve this Fourier Trans. problem, here is the original solution >> https://i.imgur.com/eJJ5FLF.pngQ/ How did he come up with this result and where is my mistake? Homework Equations All equation are in the above attached picture The Attempt at a...

Homework Statement Q:/ Find the complex form of Fourier series for the following periodic function whose definition in one period is given below then convert to real trigonometry also find f(0). f(t)=cos(t/2), notes: (T=2*pi) (L=pi) Homework Equations 1) f(t)=sum from -inf to +inf (Cn...

Could you explain a bit about the relationship between locality and uncertainty in Fourier pairs? Many pages talk about uncertainty principle stating that the precision at which we can measure time duration of signal cannot unlimitedly grow without affecting precision on bandwidth. Many other...

I'm currently reading class notes from an introductory waves course, written by the professor himself. I'm stuck in the Fourier analysis part, because he gives the formulas for the nth mode amplitude of a standing wave with fixed ends and then states some properties which I can't really make...

Homework Statement Trying to find the sum of (-1)3n+1/(2n-1)3. by using term-by-term integration on the cosine Fourier series x= L/2-4L/π2∑cos(((2n-1)πx)/L)/(2n-1)2. Homework Equations Shown below The Attempt at a Solution When integrating and substituting Lx/2 for x's sine Fourier series I...

Homework Statement [/B] I am trying to match each of the following 28-point discrete-time signals with its DFT: Set #1: Set #2: Homework EquationsThe Attempt at a Solution Set #1 We have already established (here) that: ##Signal 1 \leftrightarrow DFT3## ##Signal 4 \leftrightarrow...

Homework Statement I am self studying an introductory quantum physics text by Marvin Chester Primer of Quantum Mechanics. I am stumped at a problem (1.10) on page 11. We are given f(x) = \sqrt{ \frac{8}{3L} } cos^2 \left ( \frac {\pi}{L} x \right ) and asked to find its Fourier...

Hi! 1. Homework Statement From the website http://www1.uprh.edu/rbaretti/MomentumspaceIntegration8feb2010.htm we can see the Fourier transform of the ground state hydrogenic wave function : Φ(p) = ∫ ∫ ∫ exp(-i p r) (Z3/π )1/2 exp(-Zr) sin(θ) dθ dφ r² dr (1.1) After intregation...

Homework Statement Match each discrete-time signal with its DFT: Homework EquationsThe Attempt at a Solution I am mainly confused about Signal 7 and Signal 8. Signal 1 is the discrete equivalent to a constant function, therefore its DFT is an impulse (Dirac ##\delta##), so it corresponds...

hey So Fourrier transform is ##f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d\omega## with ##F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt## Question 1 - The Fourier mode for the continuous case is ## \frac{1}{2 \pi} F(\omega) e^{i \omega t}##, is...

Homework Statement Hi guys, I have the following transmitted power signal: $$x(t)=\alpha_m \ cos[2\pi(f_c+f_m)t+\phi_m],$$ where: ##\alpha_m=constant, \ \ f_c,f_m: frequencies, \ \ \theta_m: initial \ phase.## The multipath channel is: $$h(t)=\sum_{l=1}^L \sqrt{g_l} \ \delta(t-\tau_l).$$...

Hi! I am currently trying to derive the Fourier transform of a 2D HgTe Hamiltonian, with k_x PBC and vanishing boundary conditions in the y direction at 0 and L. Here is the Hamiltonian: H = \sum_{k}\tilde{c_k}^{\dagger}[A\sin{k_x}\sigma_x + A\sin{k_y}\sigma_y + (M-4B+2B[\cos{k_x} +...