Fourier Definition and 1000 Threads

"Sketch the form of the Fourier transform" - is this right? Question ~ sketch the "form of the Fourier transform" for the function: f(k) = sin^2(ka/2) / (ka/2)^2So I'm thinking it will look like a cos [or sin] graph (shifted so that its 'above' *f(k)=0*) and that there will be some sort...

Hi I was just wondering when do we use the different variations of the General Fourier, Fourier Sine Transform, Fourier Cosine Transform, and Laplace Transforms. I missed my lecture and I overheard that apparently there needs to be specific boundary conditions or initial conditions which...

What is the statement of the mean square convergence of Fourier series?

Homework Statement Fx= sin2x for -pi<x<0 and 0 for 0<x<pi. Compute the coefficients of the Fourier series. Homework Equations The Attempt at a Solution I found Ao=0 even though I came up with a Fourier cosine series. There must be something wrong. And is that possible that I...

Homework Statement Find the waveforms x1(t) and x2(t) (expressed in a simple form) that are consistent with the sets of Fourier coefficient values provided below. Assume that the period T is equal to 1/10 seconds in both cases (a-1): a(0) = 6, a(-1) = 3 + 4j, a(1) = 3 - 4j, and a(k) = 0 for...

Hi. I just wondered why we use a 1/\sqrt{V} in the Fourier expansion of the vector potential. A regular 3 dimensional Fourier expansion is just f(\vec r) = \sum_{\vec k} c_\vec{k} e^{i \vec k \cdot \vec r} but as the solution to the equation (\frac{\partial ^2}{\partial t^2} -...

So, my friend looked at this post and told me it's beyond confusing. So let me clarify. Suppose I have a neural network connected to various sensors. How best would I process the input data from the sensor such that a neural network could learn from it best. I'm assuming my network has many...

Homework Statement S(t) = S(0)e^{-i \pi f_{o}t} e^{-t/T^{*}_{2}}, 0 \leq t < \infty S(t) = 0, t < 0 Show that the spectrum G(f) corresponding to this signal is given by: G(f) = S(0) { \frac{T^{*}_{2}}{ 1 + [2 \pi (f- f_{o} )T^{*}_{2}]^{2}} + \frac{i2 \pi (f- f_{o} )...

Plotting Laplace Functions in Matlab - Help? SORRY I MEANT LAPLACE FUNCTIONS Homework Statement so we have a function f(t) = |2t, 0≤ t < 2 |(t^2)-3, 2≤t how would you plot this over 0≤ t ≤4 in Matlab? Homework Equations - The Attempt at a Solution I guess you could...

Homework Statement I've been stuck on this for a while: Find the Fourier transform of f(t)=sin(\omega0t+\phi) Homework Equations I know that I have to use F(ω)=\intf(x)e^-iωt dt (between - and + infinity) to solve this The Attempt at a Solution So far I have...

Homework Statement h(x)=\left\{\begin{matrix} 9+2x , 0<x<\pi\\ -9+2x , -pi<x<0 \end{matrix}\right. \\ Find \ the \ sum \ of \ the \ Fourier \ series \ for \ x=\frac{3\pi}{2} and\ x=\pi \\ The \ Fourier \ series \ is: \\ h(x)=9+\pi + \sum_{n=1}^{inf}...

I really need your help - i can't work out how to do a FFT in excel. The main problem is I don't have a constant sampling rate - I recorded the time and then the corresponding magnitude of the wave. I have followed everything oneline but I can't seem to get anything to work as I can't fill the...

Homework Statement How can i find the magnitude spectra of 2sin(4000*pi*t)*sin(46000*pi*t) The Attempt at a Solution im not sure how to go about this question, can someone please give me some help on what i should do. I know that for a square wave i can find the Fourier series...

Homework Statement Any wavepacket can be obtained by the superposition of an infinite number of plane waves using the so-called Fourier integral or Fourier transform f(x,t) = \frac{1}{\sqrt{2\pi}} _{-\infty}\int^\infty A(k)e^{i(kx-\omega t)} dk Find at t=0 the representation of the...

Let $$ h(\theta) = \begin{cases} \frac{1}{2}(\theta + \pi), & 0 < \theta < \pi\\ 0, & \theta = 0, \pm\pi\\ \frac{1}{2}(\theta - \pi), & -\pi < \theta < 0 \end{cases} $$ How can I find the Fourier series without doing any integration?

What are rules for differentiating a Fourier series? For example, given $$ f = \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2n-1)\theta}{2n-1} = \begin{cases} 1, & 0 < \theta < \pi\\ 0, & \theta = 0, \pm\pi\\ -1, & -\pi < \theta < 0 \end{cases} $$ Can this be differentiating term wise? If so...

Show G(k)=\sqrt{2π}g1(k)g2(k) Given that G(k) is the Fourier transform of F(x), g1(k) is Fourier trans of f1(x), g2(k) is Fourier trans of f2(X) and F(x)=^{+∞}_{-∞}∫dyf1(y)f2(x-y) SO FAR G(k)=1/\sqrt{2π}^{+∞}_{-∞}∫F(x)e-ikxdx <-def'n of Fourier transform...

Hi, I am taking a random process class and I came across a problem that has stumped me. I believe I know the end result but I would like to know how it is solved. I have been out of college for a while and I am a little rusty with integration. Homework Statement What I need is to find out...

When is a function "equal to" its Fourier series? First of all - a bit unsure where this post fits in, there seems to be no immediately appropriate subforum. So I'm a physics student and currently looking at what it takes for a Fourier series to converge. I've looked at wiki...

I am trying to prove a theorem related to the convergence of Fourier series. I will post my proof below, so first check it and then my question will make sense. Is there any flaw in my proof? Also, here I proved it for integrable functions monotonic on an interval on the left of 0. But what if...

Where is the fallacy in this "proof" that the Fourier series of f(x) converges to f(x) if f is continuous at x and has period 2π? (I read in Wikipedia that a counterexample had been provided). Start with the Dirichlet integral for the N-th partial sum of the (trigonometric) Fourier series...

how to get the Fourier transform of (1+at^2)^-n ? n is a natural number such that (n>1) and a is any positive number. i.e. ∫((1+at^2)^-n)*exp(-jωt)dt; limits of integration goes from -∞ to ∞

Supposing $f$ is bounded and $A_n$ is given by 1-8, prove that $\sup_n|A_n|$ is finite. $$ f(\theta) = \sum_{n = -\infty}^{\infty}A_ne^{in\theta} $$ Since $f$ is bounded, $|f| < M = |z|\in\mathbb{C}$. Since it could be $\mathbb{C}$, $M$ would be the modulus correct? We know that the modulus of...

Suppose $f(\theta) = |\theta|$ for $-\pi < \theta < \pi$. Find the formal series solution of the corresponding heat problem in the disk. How many terms of the series will give $u(r,\theta)$ with an error $< 0.1$ throughout the disk? Evaluate $u\left(\frac{1}{2},\pi\right)$ to two decimals. Show...

Homework Statement Find the Fourier transform of x(t) = e-t sin(t), t >=0. We're barely 3 weeks into my signals course, and my professor has already introduced the Fourier transform. I barely understand what it means, but I just want to get through this problem set.Homework Equations I...

Homework Statement show that xf(x)=integral from 0 to infinity of [B*(w)sin(wx)]dw , // B* is a function not B * w where B* = -dA/dw A(w) = 2/pi integral from 0 to infinity [f(v) cos(wv)] dv Homework Equationsf(x)=integral from 0 to infinity [A(w)cos(wx)] dw The Attempt at a Solution...

Find the Fourier series for $$ f(\theta) = \begin{cases} \theta, & 0\leq \theta \leq\pi\\ \pi + \theta, & -\pi\leq \theta < 0 \end{cases}. $$ $$ a_0 = \frac{1}{\pi}\int_0^{\pi}\theta d\theta + \frac{1}{\pi}\int_{-\pi}^0\theta d\theta + \int_{-\pi}^0 d\theta $$ The first and second integral...

Hi all, I have a seemingly simple problem which is I'd like to efficiently evaluate the following sums: Y_k = \sum_{j=0}^{n-1} c_j e^{\frac{i j k \alpha}{n}} for k=0...n-1. Now if \alpha = 2\pi, then this reduces to a standard DFT and I can use a standard FFT library to compute the...

Apply Theorem 1.4 to evaluate various series of constants. Theorem 1.4: Let $f$ be periodic and piecewise differentiable. Then at each point $\theta$ the symmetric partial sums $$ S_N(\theta) = \sum_{n=-N}^Na_ne^{in\theta} $$ converge to $\frac{1}{2}\left[f(\theta)+f(-\theta)\right]$; if $f$ is...

We have $f(\theta) = \theta^2$ for $-\pi < \theta\leq \pi$. $$ a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi}\theta^2 d\theta = \frac{2\pi^2}{3} $$ $$ a_n = \frac{1}{\pi}\left[\left.\frac{\theta^2}{n}\sin n\theta\right|_{-\pi}^{\pi}-\frac{1}{n}\left[\left.\frac{-\theta}{n}\cos...

$f(\theta) = \theta$ for $-\pi < \theta \leq \pi$ $f$ is odd so $\sum\limits_{n = 1}^{\infty}a_n\sin n\theta$. $$ \frac{1}{\pi}\int_{-\pi}^{\pi}\theta\sin n\theta d\theta $$ So I have that $$ a_n = \begin{cases} \frac{-2\pi}{n}, & \text{if n is even}\\ \frac{2\pi}{n}, & \text{if n is odd}...

Calculate the Fourier series of the function $f$ defined on the interval [\pi, -\pi] by $$ f(\theta) = \begin{cases} 1 & \text{if} \ 0\leq\theta\leq\pi\\ -1 & \text{if} \ -\pi < \theta < 0 \end{cases}. $$ f is periodic with period 2\pi and odd since f is symmetric about the origin. So f(-\theta)...

Calculate the Fourier series of the function $f$ defined on the interval $[\pi, -\pi]$ by $$ f(\theta) = \begin{cases} 1 & \text{if} \ 0\leq\theta\leq\pi\\ -1 & \text{if} \ -\pi < \theta < 0 \end{cases}. $$ $f$ is periodic with period $2\pi$ and odd since $f$ is symmetric about the origin. So...

Homework Statement Hello, Check each function to see whether it is piecewise smooth. If it is, state the value to which its Fourier series converges at each point x in the given interval and the end points (a.) f(x)=|x|+x, -1<x<1 (it would be very helpful to see if i did this right, as the...

This is for an assignment, (not sure if its in the right section) but anyway I'm considering the system response to H(w) = 10/(jw + 10) when the input is x(t) = 2 + 2*cos(50*t + pi/2) so I know that Y(w) = X(w).H(w) but I'm not sure what to do about the '2 + ' in the input. I know that...

greetings, why do we use Euler equation that is e ^(jωt)=cos(ωt)+i sin(ωt) in Fourier series and what does it represent? advanced thanks.

Homework Statement The argument of the kernel of the Fourier transform has a different sign for the forward and inverse transform. For a general function, show how the original function isn’t recovered upon inverse transformation if the sign of the argument is the same for both the forward and...

Prove the identities $$ \frac{\sin\left(\frac{n + 1}{2}\theta\right)}{\sin\frac{\theta}{2}}\cos\frac{n}{2}\theta = \frac{1}{2} + \frac{\sin\left(n + \frac{1}{2}\right)\theta}{2\sin\frac{\theta}{2}} $$ By using the identity $\sin\alpha + beta$, I was able to obtain the $1/2$ but now I am not to...

If you write $$ e^{ik\theta} = \cos k\theta + i\sin k\theta, $$ then $\sum\limits_{k = 0}^ne^{ik\theta} = \frac{1 - e^{i(n + 1)\theta}}{1 - e^{i\theta}}$ yields two real sums $$ \sum\limits_{k = 0}^n\cos k\theta = \text{Re}\left(\frac{1 - e^{i(n + 1)\theta}}{1 - e^{i\theta}}\right) $$ and $$...

Hello, Please could someone explain to me about Boltzmann distribution and Fourier transformation? or point me in the direction of some really really easy-to-understand guide? I need to understand it for biology - to understand NMR and mass spectrometry. Thanks.

Hi, I was wondering if anyone might know what the analytic Fourier transform of a Super-Gaussian is? cheers

Homework Statement Use the Fourier series of f(x) = { 1 |x|<a { 0 a<|x|<\pi for 0<a<\pi extended as a 2-Pi periodic function for x \inR to find \sum Sin2(na)/n2 [b]2. Homework Equations [/b I got that the Fourier series of f(x) was a/\pi+\sum (2/(m\pi) sin(ma) sin(mx)...

Since I lack the understand of real world applications of Fourier Transform in the real world I decided to buy a signals and systems book (Lathi) do some Fourier Transform problems and them do the same problem in Matlab. The question in the book wants me to find the Fourier Transform of...

Homework Statement Determine Fourier Transform of f(t) = cos^2 ω_p t ... for |t|<T also, for |t|>T, f(x) = 0, although i don't think you need to do anything with that. The Attempt at a Solution okay so: f(t) = cos^2 ω_p t ... for |t|<T becomes f(t) =...

Homework Statement I am using the time differentiation property to find the Fourier transform of the following function: Homework Equations f(t)=2r(t)-2r(t-1)-2u(t-2) The Attempt at a Solution f'(t)=2u(t)-2u(t-1)-2δ(t-2) f''(t)=2δ(t)-2δ(t-1)-?? Can somebody explain what the...

hi I know the Fourier transform of a lorentzian function is a lorentzian but i was wondering if the Fourier transform of the second derivation of a lorentzian function is also a second derivative of a lorentzian function Thanks

Homework Statement Given the function f(x) = Acos(∏x/L), find its Fourier series Homework Equations Okay so, f(x) is even, so the Fourier series is given by: f(x) = a0 + \sumancos(nx) where a0 = 1/∏\int f(x).dx with bounds ∏ and -∏ and an = 1/∏\int f(x)cos(nx).dx with bounds ∏...

Hi all, I've been trying to solve the following I = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{x}{(x^2+y^2+d^2)^{\frac{5}{2}}} e^{-i(kx+\ell y)} \ dx \ dy where d,k,\ell are constants. I haven't been able to put this into a tractable analytic form and I figured I'd consult all...

Hey Physics Forums, Grading an assignment, the current topic is continuous Fourier Transforms. They're trying to prove the convenient property: \mathcal{F} \left[ \frac{d^n}{dx^n} f(x) \right] = (i \omega)^n \mathcal{F} \left[ f(x) \right] So there's a simple way to get it: Let f(x) be...

\dfrac{\mathcal{F}^{-1} \Big( \sqrt{\mathcal{F}(f(x))} \Big)}{f(x)} = g(x) g(x) is known, and for an example let's say g(x) is something simple like g(x) = x so we have \mathcal{F}^{-1} \Big( \sqrt{\mathcal{F}(f(x))} \Big) = x \cdot f(x) my question is, how do i find f(x)? it's...