Fourier Definition and 1000 Threads

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...

Hey, thanks for taking the time to look ay my post (: I have attached a file which shows the question I am stuck on, and my attempt at working it out. My problem is the answer I get, is different to what my Lecturer gets (shown in the attachment). He worked it out a different way to me, he...

Homework Statement Find the F.S.S. of f(x)=x 0≤x≤1 of period 2 Homework Equations (uploading photo with equations shown) The Attempt at a Solution (uploaded photo) I have completely worked this problem out in the provided photo but I am unsure as to whether or not I went about it in the...

Homework Statement I am looking at finding the Fourier transform of: f(t)=\exp \left[ \frac{-(t-m)^2}{2 \sigma^2}\right] Homework Equations \hat{f}(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt The Attempt at a Solution I did it a little differently that my...

This is purely conceptual and I'm just looking for opinions on whether its misguided or, indeed, plausible. From what I understand about Fourier decomposition we can break down an analog signal into component sinusoidal waves. My thinking is that the sound system at a nightclub can be...

Today in my circuits class, we were talking about Fourier transforms and my professor briefly said something about how a Fourier transform is a rotation in infinite dimensional space. I would ask him more about it but since it's beyond our course I'd rather not bug him. Where can I learn more...

I was going to post this in the learning material section but i didnt have access to it for some reason. but i guess i can post it here. its homework after all. so i have noticed that there is almost nothing learning material on fourer transform on the web. like how to transform a function to...

semi urgent Fourier series question (small) Homework Statement Hi, I have x(t) = 1/2 + cos(t) + cos(2t) so I can see that a0 = 1/2 and that it is an even function so there is no bn Also that T = 2pi so an = 2/2pi ∫02pi x(t).cos(nω0t) dt but when I integrate this I get an = 0 yet...

Homework Statement Inverse Fourier transform of X(w) = (sin(w/2).exp(-j2w))/(jw + 2) Homework Equations From table: exp(-bt).u(t) → 1/(jw+b) multiplication by sin: x(t)sin(w0t → j/2[X(w+w0)-X(w-w0] w0 being 0.5 here. shifted left or right in time: x(t-c) → X(W)exp(-jwc) The Attempt at a...

Homework Statement Please see picture attached Homework Equations The Attempt at a Solution ck = 1/T ∫ a-a x(t).e-jk2pit/T So x(t) = Ʃk=-∞∞ (sin(k.a.2pi/T).a-a e-jk2pit/T)/k.pi but is is supposed to be: So x(t) = Ʃk=-∞∞ (sin(k.a.2pi/T).a-a e-jk2pit/T)/k.pi.2.a but I...

Hi, I have x(t) = 1/2 + cos(t) + cos(2t) so I can see that a0 = 1/2 and that it is an even function so there is no bn Also that T = 2pi so an = 2/2pi ∫02pi x(t).cos(nω0t) dt but when I integrate this I get an = 0 yet I've been told that the answer is x(t) = 1/2 + Ʃn = 12 cos(nω0t) which...

Why is Fourier sine series of any function satisfying Dirichlet's theorem, not defined on the discontinuous points whereas we define it for Fourier cosine series? ex - sine series of f(x) = cosx, 0<=x<=∏ is defined on 0<x<∏ whereas cosine series of f(x) = sinx, 0<=x<=∏ is defined on 0<=x<=∏

Homework Statement For a physics problem I must take the inverse Fourier transform of 2 functions. Namely I must compute the integral ##\frac{1}{\sqrt{2\pi}}\int_{-\infty} ^\infty [A\cos (ckt)+B\sin (ckt)]e^{ikx}dk##.Homework Equations Already given. i is the complex number. t is greater or...

$$ \sum_{n = 1}^{\infty}\sum_{m = 1}^{\infty}A_{nm}\sin\frac{n\pi x}{L}\sin\frac{m\pi y}{H} = -\frac{4}{\pi}\sum_{k = 1}^{\infty}\frac{1}{(2k-1)\sinh\frac{\pi(2k-1)H}{L}}\sin\frac{\pi(2k-1)x}{L}\sinh\frac{\pi(2k-1)y}{L} $$ If I start with x on the left, can I then end up with: $$...

If you have a function with countable discontinuities on an interval, I know that the Fourier series will converge to that function without those discontinuities. But how could you explain that formally? If the basis of the Fourier series span the space L^2[a,b], that would include functions...

Evaluating a "Fourier Transform" Integral Homework Statement Evaluate I = ∫[0,∞] e-ktw2 cos(wx) dw in the following way: Determine ∂I/∂x, then integrate by parts. Homework Equations Possibly? The Attempt at a Solution Since integral limits do not depend on x, the partial with respect...

Hey, I was wondering if you guys could offer any course guidance on independent studies I could try to take my senior year. I have some ideas, but I was wondering whether you guys could give me any recommendations/book recommendations. My background: I initially wanted to go into a more...

Hi All, I'm just trying to practice graphing signals in frequency domain and I came across a stiuation I wasn't familiar with. If the exp() has a constant*t in it I'm not sure how to graph it, I remember that just cos it like a double sided exp(jwt) but with half the magnitude. I've attached a...

I have calculated a k-space function to be f(k) = \frac{1}{2k} I want to Fourier transform this to find f(x), I have found many different Fourier transform equations...can I use this one? f(x) = \frac{1}{\sqrt{2π}}\int\frac{1}{2k}e-ikxdk Limits fo integration -Infinity to Infinity...

Homework Statement Find the Fourier Transform of: f(t)=\frac{cos(\alpha t)}{t^2+\beta^2} Homework Equations F(\omega)=\frac{1}{2\pi}\int^{∞}_{-∞}\frac{cos(\alpha t)exp(i \omega t)}{t^2+\beta^2} The Attempt at a Solution I start with: cos(\alpha t)=\frac{exp(i \alpha t)+exp(-i...

Homework Statement Ok I know Fourier transform pair for u(t) is pi*del(w)+1/(j*w) Am I right to say the transform pair of u(t)-u(t-1) is [pi*del(w)+1/(j*w)]-[pi*del(w-1)+1/(j*(w-1)] If not what is it? thanks

Hi I am dealing with problem that says f(t)=ƩFn.exp(inwt) . f(t)=f(t+T) show that if f(t)=f[t+T/2) then Fn is zero for odd n? Attempt: I wrote formula for Fn=1/T∫f(t).exp(-inwt) and then just replace f(t) by f(t+T). but I do not get anything, I do not know how I should approach this problem...

Say you have some function that is periodic in a parameter k. The discrete Fourier transform from a sampling may be found in the usual way, giving the frequency spectrum in k. But what if I want to find the frequency spectrum in 1/k ? I'm not really sure what this is called, and so I've had a...

can we simply truncate a Fourier series if it is divergent?? given a Fourier series of the form \sum_{n=0}^{\infty}\frac{cos(nx)}{\sqrt{n}} can i simply truncate this series up to some number finite N so i can get finite results ?? thanks.

Homework Statement I'm trying to Solve for an impulse response h(t) Given the excitation signal x(t) and the output signal y(t) x(t) = 4rect(t/2) y(t) = 10[(1-e-(t+1))u(t+1) - (1-e-(t-1))u(t-1)] h(t) = ? y(t) = h(t)*x(t) --> '*' meaning convolution! I am unsure how to take the Fourier...

Hi Does anyone know if there is a relation between the Fourier transform of a function and the Fourier transform of the inverse function in summary FT[f(x)] ?= FT[f-1(x)] Thanks!

Homework Statement The entries of the time-domain vector: x(1) = [2 1 -1 -2 -1 1 2 1 -1 -2 -1 1] ; N = 12 are given by 2cos(ωn) where n = 0:11. what is the value of ω? express x(1) as the sum of two Fourier sinusoids. By considering the appropriate columns of the Fourier matrix V...

Homework Statement Let f(x)=x, 0≤x≤p (a.) Compute the half-range sine series (b.) Use the series to show that 1-(1/3)+(1/5)+...=π/4 Homework Equations bn=(2/L)*int(from 0 to L) f(x)*sin(nπx/L) dx The Attempt at a Solution bn=(2/p)*int(from 0 to p) x*sin(nπx/p) dx Using...

Homework Statement Use the Fourier series technique to show that the following series sums to : 1+\frac{1}{3^2}+\frac{1}{5^2}+...=\frac{\pi^2}{8} Homework Equations The Attempt at a Solution Don't know what the first few steps are...but I assume that I need to first express the sum as...

Homework Statement Two similar problems, but once I find out how to do the first one, I can figure out how to do the second. My signals book tells me the answers to the following "Dn"s are: First problem: Dn = (1/∏) ∫ sin(t) * e^(-j2nt) dt = 2/(∏ (1-4n^2) ) if x(t) = rectified sin(t)...

Homework Statement Homework Equations Not sure The Attempt at a Solution No idea how to even begin. I don't even know how to start this equation. My textbook has no examples of this type. Do I need to transform x(t)? If someone could simply steer me in the right direction...

Hiya. I have to solve this bad boy under the assumptions that f, f' and f'' tend to 0 as |x| tends to infinity: 1/2(f')^2 = f^3 + (c/2)f^2 + af + b where a,b,c are constants. My thoughts are use Fourier Transforms to use the assumptions given, but not sure how to do them on these terms...

I am SO annoyed with this problem. Ready to jump out a window. Homework Statement Find the first three terms of the Fourier series that approximates f(θ) = tan(θ) from θ = -π/2 to π/2. The Attempt at a Solution So, I know that for an equation on [\frac{-b}{2}, \frac{b}{2}], to...

I am SO annoyed with this problem. Ready to jump out a window. Homework Statement Find the first three terms of the Fourier series that approximates f(θ) = tan(θ) from θ = -π/2 to π/2. The Attempt at a Solution So, I know that for an equation on [\frac{-b}{2}, \frac{b}{2}], to define the...

1. what is the even part of δ(x+3)+δ(x+2) -δ(x+1) +1/2δ(x) +δ(x-1) -δ(x-2) -δ(x-3)? 2. δ= 0 x≠0; ∞ x = 0 1/2 (f(x) + f(-x)) 1/2 (f(x) - f(-x)) Knowing the piecewise definition of the delta function, and knowing 1/2 (f(x) + f(-x)) for even parts of a function. I plug this in...

Homework Statement Given the function 10sin^2(10t) Find the fundamental frequency and period. Find the exponential and trigonometric coefficients of the Fourier Series. Homework Equations The Attempt at a Solution I really have no idea how to start this problem. The sin^2...

Is there a name for a transformation using the orthonormal base s_k(x)=\lceil \sin kx \rceil,\: c_k(x) = \lceil \cos kx \rceil \quad ? So basically a Fourier transform or Fourier series using periodic rectangles. What are the properties? Is there some kind of convolution theorem?

I am having trouble with this homework problem, I know how to get started but I just don't know how to carry through the completion of the problem: Question: Given the Fourier transform of an aperiodic signal X(ω) = 2*sin(3(ω-2π))/ω-2π (a)find its inverse Fourier transform x(t) using...

Hello, First post. I will attempt to use latex, something that involves me jabbing my keyboard with a pen since my \ key is missing. We have an assignment question which I have solved, but there is a deeper issue I don't understand. We are asked to find the complex Fourier series...

dealing with absolute functions that are limited always throws me off so let's consider this f(x)=|x| for -∏ ≤ x < ∏ f(t)= f(t+2∏) it's not too bad however finding the energy density is throwing me off a little.. the questions tend to be generally phrased as below: Find the energy...

Homework Statement Suppose, in turn, that the periodic function is symmetric or antisymmetric about the point x=a. Show that the Fourier series contains, respectively, only cos(k_{n}(x-a)) (including the a_0) or sin(k_{n}(x-a)) terms. 2. Homework Equations The Fourier expansion for the...

##\hat{c}_{i\sigma}=\frac{1}{\sqrt{N}}\sum_{\bf{q}}e^{i\bf{q}\cdot \bf{R}_i}\hat{c}_{\bf{k}\sigma}## ##\hat{c}^+_{i\sigma}=\frac{1}{\sqrt{N}}\sum_{\bf{q}}e^{-i\bf{q}\cdot \bf{R}_i}\hat{c}^+_{\bf{k}\sigma}## Then ##-J\sum_{i}\hat{S}_i^z\hat{c}^+_{i\sigma}\hat{c}_{i \sigma}## in...

Suppose $f$ is continuous and periodic with period $2\pi$ on $(-\infty,\infty)$, and $f'$ exist and is in $\mathcal{P}\mathcal{C}[-\pi,\pi]$. Then $\sum\limits_{k = -\infty}^{\infty}\lvert A_k\rvert < \infty$.$f'$ has a Fourier series so let's call the coefficients $A_n'$. Then $f' =...