Fourier Definition and 1000 Threads

Homework Statement In "oppgave 4" http://www.math.ntnu.no/emner/TMA4120/2011h/xoppgaver/tma4120-2010h.pdf you have a periodic function which is NOT periodic from ##x=-L=-\pi## to ##x=L=\pi##, but at ##x=0## and ends at ##x=2 \pi=2L##. The formulas I have (like these...

Homework Statement https://wiki.math.ntnu.no/_media/tma4120/2013h/tma4120_h11.pdf Check out the solution to problem 4b) My question is: Why do they set ##b_n = \frac{2}{\pi} \int_{0}^{\pi}(...)dx## instead of ##b_n = \frac{1}{\pi} \int_{0}^{\pi} (...)dx##? Ie, why did they multiply the...

Hi! Which is the better method for finding Fourier expansions of a function? The ordinary one (find a_0, b_n and a_n with separate integrals), or the one which uses complex numbers (just find c_n)?

Homework Statement What is the Fourier transform of a single short pulse and of a sequence of pulses? The Attempt at a Solution In class we haven't dealt with the mathematics of a Fourier transform, however my professor has simple stated that a Fourier transform is simply a equation...

I have a tutorial question for maths involving the heat equation and Fourier transform. {\frac{∂u}{∂t}} = {\frac{∂^2u}{∂x^2}} you are given the initial condition: u(x,0) = 70e^{-{\frac{1}{2}}{x^2}} the answer is: u(x,t) = {\frac{70}{\sqrt{1+2t}}}{e^{-{\frac{x^2}{2+4t}}}} In this course...

I learned how to integrate it using the complex plane and semi circle contours but I was wondering if there is a way using Fourier transforms. I know that the Fourier transform of the rectangle wave form is the sinc function so I was thinking maybe i could do an inverse Fourier on sinc x and get...

Hello, Recently I've learned about Fourier Transform, and the uncertainty principle that is arose from it. According to Fourier Transform, if there is only one pulse in a signal, then it is composed from a lot more frequencies, compared to the number of frequencies that are building a...

Homework Statement Hello guys, I have problem with the Fourier series, since we had only one lecture about it and I cannot find anything similar to my problem in internet. should we consider for the first f(x+1) integrated from -1 to 0 ? http://img819.imageshack.us/img819/3508/wbve.jpg when...

In finding Cn, I arrived at a different answer. I got an extra factor of (1/i) instead, which came when you do the integral of each exponential with respect to t; so you get a factor of 1/i(1-n) and 1/i(1+n) respectively.. Did they intentionally leave that out?

Fourier transform of RF signal with a "prism"? We can use a prism to decompose visible light into components of different frequencies. This is a Fourier transform by nature. For an ideal prism, the energy is conserved in the process. How about RF signals? There is no fundamental difference...

Hi guys, I'm not sure how they got from first step to the second. Did they use integration by parts? I tried but I didn't arrive at the same result..

\mathcal{F}\{f(r)\}=\int e^{i\vec{k}\cdot \vec{r}}f(r)d\vec{r} in spherical polar coordinates \mathcal{F}\{f(r)\}=\int^{\infty}_0r^2dr\int^{\pi}_0\sin\theta d\theta\int^{\pi}_0d\varphi e^{ikr\cos \theta}f(r) Why could I take ##e^{ikr\cos \theta}## and to take that ##\theta## is angle which goes...

Homework Statement Finding the Fourier Transform using Transform Pair and Properties x(t) = 2[u(t+1)- t^{3}e^{6t}u(t)] Homework Equations The Attempt at a Solution For the first problem, I got u(t) \leftrightarrow ∏δ(ω)+\frac{1}{jω} F(at-t_{0}) \leftrightarrow...

Find the Fourier SIne Series for f(x) = x on -L < x < L (Full Fourier) Ok, so my issue is in calculating the coefficients for the sine and cosine parts, more so an interpretation. So I have calulated the sine and cosine series to this point: let An: Cosine series Bn: sine series...

The windowed Fourier transform on R Definition-Proposition-Theorems (Plancherel formula-Parseval formula-inversion formula-Calderon's formula) http://www.4shared.com/office/b2Ho5n7H/The_windowed_Fourier_transform.html

Hello, Find the Fourier serie of f(x)=|sin(x)| on the interval (-1,1) I'm just a little confused, does that mean that I have to integrate from -1 to 1 to find the coefficients ? Because the formula of the coefficients is in terms of the period T, for this function the period is pi. Or do I...

Homework Statement In the dirac notation, inner product of <f|g> is given by ∫f(x)*g(x) dx. Why is there a 1/∏ attached to each coefficient an, which is simply the inner product of f and that particular basis vector: <cn|f>? Homework Equations The Attempt at a Solution

I recently had an asignment where i calculated the Fourier series coefficients for f= 1+t for t= -1 to 0 f= 1-t for t=0-1 basically triangle looking. And as i summed more and more coefficients my function started looking more like this triangle (which was really cool). My question...

Homework Statement Determine the Fourier series for the full-wave rectifier defined as f(t) = sinωt for 0 < ωt < pi -sinωt for -pi < ωt < 0Homework Equations The Attempt at a Solution This looks like an even function, so bm = 0 Ao = 1/pi∫sinωt from 0 to pi = 1/pi(-cos(ωt))/ω) from 0 to...

Good morning everyone, I am taking a signals and systems course where we are now studying the Fourier series. I understand that this is for signals that are periodic. But I get hung up when determining the Fourier coefficients. In the video by Alan Oppenheim, he derives the equation for...

I have done several exercises concering periodic potentials in crystal. Especially I did one, where I had to show that the Fourier component of the shortest reciprocal lattice vector (call this vector a) in the z-direction was zero. Now solving the problem was just about writing up the right...

Homework Statement #35 on this page Homework Equations Integral of a series can be assumed to be the sum of integrals The Attempt at a Solution Picture of Work I am not sure where to proceed from here, advice?

Homework Statement A function F(x) = x(L-x) between zero and L. Use the basis of the preceding problem to write this vector in terms of its components: F(x)= \sum_{n=1}^{\infty}\alpha _{n}\vec{e_{n}} If you take the result of using this basis and write the resulting function outside the...

I'm working on some research with a professor, and we're looking at data collected by an x-band radar array looking at ocean waves as they approach the coast (the radar is on land, and we can see about 3 miles out). What we're trying to do is perform an fft on the signal using Matlab, and...

I'm given a Gaussian function to apply a Fourier transform to. $$f(x)=\frac{1}{\sqrt{a\sqrt{\pi}}}e^{ik_ox}e^{-\frac{x^2}{2a^2}}$$ Not the most appetizing integral... $$g(k)=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{a\sqrt{\pi}}}\int_{-\infty}^{\infty}e^{ik_ox}e^{-\frac{x^2}{2a^2}}e^{-ikx}dx$$...

can someone spot my mistake, I'm stuck thanks, jenny

So the other day in class my teacher gave a proof for the completeness of \phi_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx} in L^2([-\pi,\pi]) . And I'm trying to convince my self I understand it at least a little. He defined Frejer's Kernel K_n(x) = \frac{1}{2\pi(n+1)}...

I understand the Fourier transform conceptually, but I am unable to reproduce it mathematically; I am very familiar with calculus and integration, but I am taking a QM course and I need to know how to apply it. No websites or videos are able to give me a good explanation as to how I can use it...

http://en.wikipedia.org/wiki/Dirac_comb Please have a look at the Fourier Series section, and its last equation. Let T = 1. After expanding the Equation x(t) = 1 + 2cos(2∏t) + 2cos(4∏t) + 2cos(6∏t) ... Now this does not give the original Dirac Comb. Eg: at t = 1/2 x(1/2) = 0 But RHS =...

When the rod is infinite or semi-infinite, I was taught to use Fourier transform. But I don't know when should the full Fourier transform or sine/cosine transform be used. how's the B.C. related to the choice of the transform ?

Homework Statement f(t) a continuously differentiable function twice over the circle T1 cr its Fourier coefficients and σn(f,t) partial sum of Fejer. a.Demonstrate that http://imageshack.us/a/img94/5992/ds35.png b. Consider k as -n≤k≤n , using cr coefficients calculate...

Is there some properties I should be aware of? after making the relevant substitutions, I ended up with $2 = 1 + \sum\nolimits_{m=0}^\infty \frac{4}{(2m+1)\pi}\sin(\frac{(2m+1)\pi}{2})$ but I can't get past this

I've just started learning Fourier series and I'm having trouble understanding it. What do they actually do? And what does the amplitude-frequency show me? I'm asking as a rookie in signal analysis, so if you could explain it to me as simple as you can it will be of great help. Thanks!

Sorry if I am posting in the wrong place. I'm really interested in the Fourier series, but I'm not an expert on it yet. I am very well aware yoy can do it with sound waves, but can you manipulate any other waves? What about light waves? And for absorption, how can you measure the...

I have to do a science fair, and I am really interested in physics. I spent forever trying to think of a topic, and I got myself stuck doing something with waves. After weeks of thinking of an expirement that will compete well without being extremely difficult, I came up with one. Does using...

Hi all, Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.) Now I have a doubt on CTFT (continuous time Fourier transform). This is a concept I got from my...

I've been assigned the following homework: I have to compute the spectral density of a QFT and in order to do so I have to compute Fourier tranform of the following quantity (in Minkowsky signature, mostly minus) \rho\left(p\right) = \int \frac{1}{\left(-x^2 + i \epsilon...

Somehow I have really hard time wrapping my head around the concept.I mean,I get it,but I can't seem to solve any questions regarding it. Here are some examples ,and I just get stuck.Its a part of test,so I think it shouldn't be that hard to solve,and if it looks hard,I know there are some...

Homework Statement Sawtooth signal with To = 1, at T=0, x = 0, at T=1, x =1 verify: a_{k} = \left\{\begin{matrix} \frac{1}{2}, for k=0; & \\\frac{j}{2\pi k}, for k \neq 0; & \end{matrix}\right. Homework Equations \frac{1}{T_{0}} \int_{0}^{T_{0}} te^{-j(2\pi/T_{0}))kt}dt The Attempt...

Homework Statement An oscillator with free period \tau is critically damped and subjected to a force with the saw-tooth form \F(t)=c(t-n\tau) for (n-0.5)\tau<t<(n+0.5)\tau for each integer n. Find the amplitudes a_n of oscillation at the angular frequencies 2\pi n/\tau if c is a...

I'm having trouble finding a definite answer to this question: When finding the Fourier series of a function is it always possible to find ##a_0## by first finding ##a_n## and just plugging in ##n=0##?

Homework Statement The problem: Justify the following equalities: \cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2} I am trying to figure out how to start this. When I insert the Euler identity of \coth (using...

Hi all my first post as I need to seek help! I have just learned some simple Fourier series stuff and would like to be able to plot my answers in matlab. Assuming this is correct I was wondering if someone would be able to walk me through plotting this equation in Matlab...

Hi, I'm following the proof of the "Scaling Property of the Fourier Transform" from here: http://www.thefouriertransform.com/transform/properties.php ...but don't understand how they went from the integral to the right hand term here: The definition of the Fourier Trasform they...

Solving a "simple" second order PDE, do I need the Fourier? Homework Statement The problem as given: y'' + 2y' + 5y = 10\cos t We want to find the general solution and the steady-state solution. We're using \mu y'' + c y' + k y = F(t) as our general form. OK, so I first want the general...

Consider \(u_t = -u_{nxxx} - 3(u^2)_{nx}\). The Fourier Transform is linear so taking the Inverse Fourier transform of the Fourier Transform on the RHS we have \begin{align} -\mathcal{F}^{-1}\left[\mathcal{F}\left[u_{nxxx} - 3(u^2)_{nx}\right]\right] &= -\mathcal{F}^{-1}...

3. Fourier sin series for f(x) = 1, 0 < x < pi is given by 1 = 4/n E 1/ (2n-1) times sin (2n-1) x, (0 < x < n). Using this, find the Fourier sinc series for f(x)= 1, on 0 < x < c where c > 0. Then find the Fourier series for g(x), x > 0 where g(x) = 1, 0 < x < c, -1, c < x < 2c, g (x + 2c)...

2. Fourier cosine series correspondence for f(x)= x, o < x < pi given by x ~ pi / 2 - 4/n, E infinity on top and n=1 on bottom. cos (an-1)/x / (2n-1)squared, (0 < x < pi). Explain why this correspondence is actually an equality for 0 is less than or equal to x and x is less than or equal to...

Does every FFT have \(i\) in it? Given \(u_t = -(u_{xxx} + 6uu_x)\). \(f'''(x) = \mathcal{F}^{-1}\left[(ik)^3\mathcal{F}(f(x))\right]\) \(f'(x) = \mathcal{F}^{-1}\left[(ik)\mathcal{F}(f(x))\right]\) The only equation I have used the pseudo-spectral method on was the NLS which is \(u_t =...

When a Fourier series contains only sine and cosine terms, evaluating the series isn't too difficult. However, I want to show a Fourier series with sine and sinh converges to \(\frac{\pi}{16}\). \[ T(50, 50) = \sum_{n = 1}^{\infty}...