Fourier Definition and 1000 Threads

I have a lot of questions, if you know something in one of them or more I will glad if you can write a replay I search after researches or others things that are correlated between optimal control and autonomous vehicles it can be things like how to calculate the shortest way, the rapid way...

I am having a hard time understanding how the Fourier sine transform works. I understand that you input a function and you get a function as an output, but I have no idea how certain inputs are even valid. Here is what the book gives for the transform: F(\omega...

Hi I'm trying to use the fft function in MATLAB to compute the discrete Fourier transform of a box signal. I'm told to assume that the signal x[n] is periodic with period N and the vector contains one period. x[n]= box [n] I'm am going to use these commands to make my vector...

Hi all, How do I compute the Fourier series coefficients for unit periods for cos(pi)x, the interval is from -1/2 to 1/2. I know the formula but I am getting a wrong answer ?

Homework Statement Using <\hat{p}n> = ∫dxψ*(x)(\hat{p})nψ(x) and \hat{p} = -ihbar∂x and the definition of the Fourier transform show that <\hat{p}> = ∫dk|\tilde{ψ}(k)|2hbar*k 2. The attempt at a solution Let n = 1 and substitute the expression for the momentum operator. Transform the...

Homework Statement How can I get to that answer? Homework EquationsThe Attempt at a Solution I'm stuck don't know how to get to the answer that I got from wolframalpha. there is no solution there unfortunately. I know how to get to that denominator w^2-1 but I don't know how to get to...

Homework Statement what values does the Fourier series for f(t) converge to if t = 0 and t = 2? Homework Equations The Attempt at a Solution My answers the red rectangles for the even function t=0 >> 1 and t=2 -->1.5 and odd function t=0 >> 0 and t=2 -->1.5 because at t=0 is continuity...

So I know that sec(x) has period 2Pi, and it's even so I don't need to figure out coefficients for bn. Let's take the limits of the integral to go from -3/2 Pi to 1/2 Pi. How do I integrate sec(x) sin(nx) dx?! Am I on the right path? PS: I know that this doesn't satisfy the Dirichlet...

I've been kicking this around for a few days. I know I'm overlooking something simple it has been a long time since I've had to do this. I'm trying to sketch the output in the time domain of a 1 kHz square wave passing through a communication channel whose bandwidth is 0 to 10 kHz. I'm trying...

Homework Statement How can I figure out the Fourier transform of the following: I'd prefer to use tables if at all possible. 1. d(z)=d_{eff}sign[\cos[2\pi z]/\Lambda]) (note this is one function inside another one.) 2. d(z)=d_{eff}(1/2)(sign[\cos[2\pi z]/\Lambda]+1) 3...

Homework Statement How did \frac{1}{2}x come from at k=1? Homework EquationsThe Attempt at a Solution because k=1 will make the first term at denominator 2(k-1) = \frac{0}{0}

Dear people, I am trying to analyze data from test bench which consists of a magnetically levitated spindle. We have a rotor/spindle which rotates and moves vertically up and down as it rotates. I measure the angle of rotation and the verticle displacement at a steady rate of 10,000 samples...

Homework Statement these two functions will give the same Fourier series? because when I write the graph they look the same? Homework Equations The Attempt at a Solution in the picture thank you

Hi I am working on a optimization problem involving vector fields. In order to define a objective function I need a measure (scalar quantity) of some properties of the vector field. The vector field comes from a finite element analysis, that is the vector field is calculated on a discretized...

Hey guys. I just started a class on Fourier Analysis and I'm having a difficult time understanding this question. Any help would be much appreciated! Homework Statement Verify that the Fourier Isometry holds on [−π, π] for f(t) = t. To do this, a) calculate the coefficients of the orthogonal...

I need more help understanding Fourier Transforms. I know that they transform a function from the time domain to the frequency domain and vice versa, but the short cuts to solve them just straight up confuse me. http://www.cse.unr.edu/~bebis/CS474/Handouts/FT_Pairs1.pdf This list of relations...

Hi all, I'm reading the following PDF about the DFT: http://www.analog.com/static/imported-files/tech_docs/dsp_book_Ch8.pdf Please see pages 152-153. So the inverse DFT (frequency to space, x[i] = ...) is given on page 152. Then it is claimed that the amplitudes for the space-domain...

Homework Statement Consider a Gaussian pulse exp[-(t/Δt)^2/2]exp(i*w*t), where Δt is its approximate pulse width in time. Use the Fourier transform to find its spectrum. Homework Equations The Fourier transform of a Gaussian is a Gaussian. If a Gaussian is given by f(t) = exp(-t^2/2)...

Hi All, Usually the Fourier transform is defined as the one in the Wiki page here (http://en.wikipedia.org/wiki/Fourier_transform), see the definition. My question is can I define Fourier transform as \intf(x)e^{2\pi ix \varsigma}dx instead, i.e., with the minus sign removed, as the...

I did Fourier analysis on a set of force data from a vibrating string. In my graph of magnitute and frequency, I'm getting major peaks at 62.1 Hz and 249.0 Hz. There is a tiny blip in the data at 125 Hz and nothing at 186 Hz. I have two questions. Do the peaks at 62.1 and 249 mean that those...

Somebody posted a question about Fourier series yesterday that got me thinking about an argument I heard some time before. If we have a (complex-valued) analytic function f, then any closed loop in the complex plane will be mapped by f to another closed loop. (If the loop doesn't enclose any...

Homework Statement I'm trying to derive the result on slide 1 of this link: http://www.physics.ucf.edu/~schellin/teaching/phz3113/lec13-3.pdf Unfortunately, I'm not sure how to integrate the Fourier transform when my u(x,t) function is undefined. Could someone help me get the...

Homework Statement I have been given the following: A(r', ω) = μ/4∏*∫ J(r', ω)*exp(-j*k*R)/R dV' And am being asked to find the inverse FT of A(r', ω) Homework Equations Given that k = ω/c and R = |r - r'| The Attempt at a Solution I know what the inverse FT transform is, but...

Hi guys, I was studying the proof below and just can't figure out the the first highlighted step leads to the second and I was wondering if you guys can help me to fill that in. (: Thank you so much for your help in advance guys!

The DCT of an even function is comprised of just cosine coefficients, correct? I'm playing around in MATLAB and I came up with a simple even function 1.0000 0.7500 0.5000 0.2500 0 0.2500 0.5000 0.7500 1.0000 0.7500 0.5000 0.2500 0 0 0 0...

Just ahead of time, no this is not related to homework or coursework in my case. I am a TA for an optics lab and need to know it so I can help the students in my class. I am trying to find an expression for the x-spacing for a grating imaged using the 4f method, which is a grating that is 1...

Homework Statement Third question of the day because this assignment is driving me crazy: Suppose that \left\{ f_{k} \right\} ^{k=1}_{\infty} is a sequence of Riemann integrable functions on the interval [0, 1] such that \int ^{0}_{1} |f_{k}(x) - f(x)|dx \rightarrow 0 as k \rightarrow...

I know the result: \widehat{H(f)}=i\textrm{sgn}\hspace{1mm}(k)\hat{f} I thought I could use fft, and ifft to compute the transform easily, is there a MATLAB command for sgn? Mat

Hi. Just going through my notes from the last lecture I remember having some troubles understanding the proof the lecturer gave for the following theorem: Suppose that f is Riemann integrable and that all its Fourier coefficients are equal to 0, then f(x)=0 at all points of continuity. The...

Hi, The Fourier series can (among others) expressed in terms of sines and cosines with coefficients a_n and b_n and solely by sines using amplitudes A_n and phase \phi_n. I want to express the latter using a_n and b_n. Using a_n = A_n \sin(\phi_n) \\ b_n = A_n \cos(\phi_n) I...

Homework Statement See figure attached. Homework Equations The Attempt at a Solution See pdf attached for my attempt at the solution. I'm a little confused as to how to draw the phase spectrum for y(t). Would it simply be a line equation of, -\frac{\pi}{6000}f \pm...

Homework Statement Is the Fourier transform of a even/odd function also even/odd ? Homework Equations The Attempt at a Solution So far this result seems to be true. I can't find a confirmation however... Thanks ahead. Daniel.

Homework Statement Hi there! I'm just trying to figure out the Fourier transform of the hyperbolic secant function... I already know the outcome: 4\sum\ ((-1)^n*(1+2n))/(ω^2*(2n+1)^2) But sadly, I cannot figure out how to work round to it! :( maybe one of you could help me... Homework...

Why does a discrete Fourier transform seems to produce two peaks for a single sine wave? It seems to be the case that the spectrum ends halfway through the transform and then reappears as a mirror image; why is that? And what is the use of this mirror image? If I want to recover the frequency...

Homework Statement Find the cosine Fourier transform of the function f(t)=e-at Homework Equations The Attempt at a Solution F(w)=(2/π)0.5∫dt e-atcos(wt) The integral is from 0 to +∞ Using euler's formula I got the result F(w)=(2/π)0.5( eit(w-a)/i(w-a) - e-it(w+a)/i(w+a)...

Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform. My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where...

The Fourier series of a delta train is supposedly (1/T) + (2/T ) Ʃcos(nωt) ... where T is period and ω=2*Pi/T ...but when I plot this, it doesn't give me just a spike towards positive infinity, but towards negative infinity as well (see attached pic), so this does not seem to converge to the...

$$ u_y(x,\pi) = \frac{x}{\pi} + \sum_{n = 1}^{\infty}nB_n\sin xn\cosh\pi n = 0. $$ How can I solve for $A_n$ here?

Is there a way to "crudely" approximate PDEs with Fourier series? By saying crudely, I meant this way: Assuming I want a crude value for a differential equation using Taylor series; y' = x + y, y(0) = 1 i'd take a = 0 (since initially x = 0), y(a) = 1, y'(x) = x + y; y'(a)...

Homework Statement Consider the function φ(x) ≡ x on (0, l). Find the sum of the first three (nonzero) terms of its Fourier sine series. Reference: Strauss PDE exercise 5.1.3 Homework Equations The Attempt at a Solution I have found the coefficient without difficulty, it is...

The problem statement: Obtain a Fourier Series Expression Form from the above graph: I can't post the graph, so I will describe it. It's a periodic function with period 1 and magnitude 5. The equation is the following: f(x) = -x, -1/2<x<1/2 I'm really stuck at trying to obtain a series...

Homework Statement a) Find the Fourier transform of the function f(x) defined as: f(x) = 1-3|x| , |x|<2 and 0 for |x|>2 b) Find the values of the inverse Fourier transform of the function F(k) obtained in a) Homework Equations F(k) = \frac{1}{\sqrt{2π}}\int f(t) eikx dx f(x) =...

Consider continuous function x(t), which has zero time average: \lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2} x(t)\,dt = 0 and exponentially decaying autocorrelation function: \lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2} x(t)x(t-\tau)\,dt = C_0e^{-\gamma |\tau|}...

Homework Statement The field E(r,t) can be written as a Fourier expansion of plane waves E(r,t)=∫E(k,w)e^{i(kr-wt)}d^{3}kdw with similar expansions for other fields. Need to show the derivation of kXE(k,w)=wB(k,w) from Faraday's law ∇XE(r,t)=-∂B(r,t)/∂t and also the derivation of...

I've been trying to figure out why it's standard to use complex discrete Fourier transforms instead of just the real version. It's discussed a bit here. http://dsp.stackexchange.com/questions/1406/real-discrete-fourier-transform As far as I can tell there's a hypothetical efficiency...

Homework Statement Show that integral from 0 - > infinity (cos(alpha*x)/(alpha^2 + 1))dalpha = (pi/2)exp(-x) Homework Equations The Attempt at a Solution cos(alpha*x) = (1/2)(exp(i*alpha*x)+exp(-i*alpha*x)) Really don't know where to go from here.

Homework Statement I've reached a relation but then I need to obtain the coefficients ##A_m## and ##B_m##'s, those are the only unknowns. Here's the expression: ##\sum _{m=0}^\infty a^m [A_m \cos (m \theta ) + B_m \sin (m \theta )]=T_0\sin ^3 \theta##. Homework Equations Fourier...

In calculating some basic Fourier transform I seem stumble on the proble that I don't know how to take the limit in infinity of an exponentialfunction with imaginary exponent. In the attached example it just seems to give zero but I don't know what asserts this property. I would have thought...

Homework Statement A function f(x) has the following series expansion: ##f(x)=\sum _{n=0}^\infty \frac{c_n x^n}{n!}##. Write down the function ##g(y)=\sum _{n=0}^\infty c_n y^n## under a closed form in function of f(x). Homework Equations Not sure at all. The Attempt at a Solution...

Homework Statement Solve: PDE: 9Uxx=Utt BC: U(0, t) = U(∏, t) = 0 IC: U(x, 0) = sin4x + 7sin5x Ut = x 0 < x < ∏/2 = ∏ - x ∏/2 < x < ∏ Homework Equations Fourier Sine Series and Cosine Series Equations The Attempt at a Solution...