Fourier coefficients Definition and 67 Threads

Hello! How do I prove ? Thank you! (it can be proven by using the convergence of the Fourier series in L_p-norm, but I want to use the above result to prove the convergence in L_2-norm, so I want to avoid that)

letsa say i have an ak = cos ( k*Pi/4) + sin(3*k*Pi/4), the signal is discrete time, fundamental period N=12. the way i would derive its x[n] is.. Sum(k=0, to 11 of: 0.5*exp(j*k*Pi/4) *exp(j*k*w*n) + 0.5*exp(-j*k*Pi/4) *exp(j*k*w*n) + (1/2*j)*exp(j*k*3*Pi/4) *exp(j*k*w*n) -...

Homework Statement Let f be a C1 function on [-pi,pi]. Prove the Fourier coefficients of f satisfy |an| <= K/n and |bn| <= L/n n=1,2,... Homework Equations an = 1/pi * int[-pi..pi] (f(x)*cos(nx)) dx bn = 1/pi * int[-pi..pi] (f(x)*sin(nx)) dx Sorry if my form is slightly...

Homework Statement Given the first cycle of a waveform: f(t)=2u(t)-2u(t-1)+u(t-2)-u(t-3) -- Plot the first cycle of the wave form -- Find the Fourier Coefficients Homework Equations Given above The Attempt at a Solution No idea yet. Will appreciate any help.

Homework Statement Hi i would just like some fast hints, I'm doing the integrals wrong, I am splitting up the integral below and get the wrong answer. well it's about finding the Fourier series for f(t)={0 for -π<t<0 and sint for 0≤t≤π} Homework Equations a_{n} =...

Homework Statement So I'm supposed to show that a finite Fourier approximation is the optimal approximation for a given function. I am to suppose we have a given set of functions \phi _k(x),k=1,2,\text{...}N defined on a\leq x\leq b. They are orthogonal \int _a^b\phi _m(x)\phi _n(x)dx=0...

Homework Statement f(t) = 1 0<=t<T/2 -1 T/2 <=t<=T ie. step function.frequency w_0 = 2pi/T Homework Equations The Attempt at a Solution What's the definition for the Fourier coefficients a_n and b_n again? Not the one in wikipedia.

Homework Statement f (x) = 0 -pi<x<0 x^2 0<x<pi Find the Fourier series and use it to show that (pi^2)/6=1+1/2^2+1/3^2+... Homework Equations N/A The Attempt at a Solution I was able to find the Fourier series and my answer matched with the back of the...

Hi, I was wondering if it is possible to express the norm of a function in terms of Fourier coefficient. If so, how do you go through it if given a particular function. Thanks

Homework Statement 1. f is a function defined on the interval -a<x<a and has Fourier coefficients an=0 bn=1/n^(1/2) what can you say about the integral from -a to a of f^2(x)dx? 2. Show that as n goes to infinity the Fourier sine coefficients of the function f(x)=1/x -pi<x<pi tend to a...

[SOLVED] Fourier coefficients Homework Statement For f \in C^{2\pi}\cap C^1[-\pi,\pi] , I have to show that \sum_{n\in\mathbb{Z}}|c_n(f)| < \infty where c_n(f) is the Fourier coefficient of f; c_n(f) = (f, e_n) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)e^{-int}\,dt f \in...

There are two functions f(t) and g(t); t is the independent variable. The distance between the two functions will be given by [1/2pi integral{f(t)-g(t)}^2 dt]^1/2 between -pi and +pi. Apparently, this distance also is the Fourier coefficient of each term in the Fourier expansion of a...

A function f(t) can be represented by the expansion f(t) = \frac{1}{2}A_{0} + A_{1}cos(\omega t) + A_{2}cos(2 \omega t) + A_{3}cos(3 \omega t) + ... B_{1}sin(\omega t) + B_{2}sin(2 \omega t) + B_{3}sin(3 \omega t) + ... Do the constants A_{n} and B_{n} the same thing as the real and...

Im just looking through examples of finding the Fourier coefficients. in one particular example bn is found to be = (-1/npi) (cosnpi - cos 0) then it says this is 0 when n is even and 2/npi when n is odd are we just substituting values of n e.g. 1,2,3... to find this result? i...

Yes, another thread... lab due tomorrow :-p We take the integral of the function f(t) times one of the components: integral(0->T) of [a0 + sigma(n=1->N) acos(nwt) + bsin(nwt)]sin(nwt) Now, in order to evaluate this is it correct to say we multiple sin(nwt) through then take the integral of...

I need to find the Fourier series for the function f(x)=x. I have come across trying to find the integral from -pi to pi of -ixSin(nx). How do I go about evaluating this integral when n is infinity? I seem to only be able to find integrals in an integral table where n is an integer, but not...

Hi! I have to calculate the Fourier coefficients c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx and the Fourier series for the following function: f(x)= \begin{cases} \frac{2}{\pi}x + 2 & \text{for } x\in \left[-\pi,-\pi/2\right]\\ -\frac{2}{\pi}x & \text{for } x\in...