What is Fourier series: Definition and 750 Discussions

In mathematics, a Fourier series () is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

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  1. D

    MATLAB Truncated Fourier series analysis in Matlab

    a) Write a Matlab function, which accepts the following inputs -a finite set of Fourier series coefficients -the fundamental period T of the signal to be reconstructed -a vector t representing the times for which the signal will be reconstructed. This function should produce an...
  2. O

    Why is there a Half Cos(x) Term in the Fourier Series for Cos(x)?

    Hello, I'm having some trouble completing a Fourier series question. We have f(x)=Cos(x), for abs(x)≤π/2 f(x)=0, for π/2≤abs(x)≤π period=2π and then show that the Fourier series f(x) reads http://img71.imageshack.us/img71/3424/84335314oi1.jpg now I can get the 1/π and the...
  3. D

    Fourier Series representation for signals

    Homework Statement a) Write a Matlab function, which accepts the following inputs -a finite set of Fourier series coefficients -the fundamental period T of the signal to be reconstructed -a vector t representing the times for which the signal will be reconstructed Homework...
  4. T

    Complex power series to calculate Fourier series

    I never took complex analysis in undergrad and always regretted it, so I'm working through the book Visual Complex Analysis on my own. Really enjoying it so far. Homework Statement Actually you can view the problem...
  5. R

    Are there steps to find the Fourier series for transformed graphs?

    Hey everybody! I am an undergrad student ,in an ee department.Ive taken "signals and systems" and two ago we got into the Fourier part. The question i have is whether or not there are some steps someone can follow in order to find the Fourier series given some specific graph.Now,our professor...
  6. R

    How can I find the Fourier coefficients for a shifted sawtooth wave?

    Homework Statement Recently in my signals class,we got to the chapter of Fourier series.We have learned that there are some basic graphs that can be described with Fourier series (sawtooth,square pulse etc) and so far i have understood their respective Fourier formulas.The problem i face...
  7. N

    Electrical engineering and Fourier series

    Homework Statement Hi all. I have used Fourier series a lot in my maths study, and now I am about to use it when analyzing electrical circuits. In my notes from a lecture, I wrote down an example done by the professor. We are looking at a RCL-circuit with an applied EMF given by the periodic...
  8. P

    Fourier Series: Exploring Java Applet Questions

    Fourier series ... Hi, everyone, I’m new to these forums. I’m just doing some work on Fourier analysis, and just have a few curious questions, I just need to clear up, and so I understand it better. I have been playing around with the java applet for Fourier series from this website...
  9. R

    Uniform convergence of Fourier Series satisfying Lipschitz condition

    Homework Statement f is integrable on the circle and satisfies the Lipschitz condition (Holder condition with a=1). Show that the series converges absolutely (and thus uniformly). i literally spent about 20 hours on this problem today but i just could not figure it out. i have a feeling...
  10. K

    Period length of product of two periodic function, Fourier Series

    Hello, I am currently dealing with Fourier Series, and a question about the period length of a product of two periodic functions is bothering me. If you have two different periodic functions with different (or the same) period and you sum them up, what happens to the period? Can this generally...
  11. T

    Fourier Series of f1(t) and f2(t): Calculating the Shifted Fourier Series

    Homework Statement Given that the Fourier series of function f1(t) = At/π + A , -π < t < 0 = -At/π +A, 0 < t < π is f(t) = 0.5A +4A/π2(cos t + 1/9 cos 3t + 1/25 cos 5t + ...) what is the Fourier series of function f2(t) = -At/π, -π < t < 0 = At/π, 0 < t < π The Attempt at a...
  12. K

    Difficult integrals (important für Fourier series)

    Hello, I couldn't figure out the following problem so far. Homework Statement m,n \in \mathbb{N}. Show the following relations: Homework Equations a) is no problem, but b) and c) seem tricky. I tried partial integration (two times) on b), which gave me a term which contained the original...
  13. K

    How Do You Calculate Fourier Series for Functions Like sin(nπ)?

    Homework Statement find Fourier series for: sinnπ=0 (n=0,+1,-1,+2,-2...) I really can't understand how Fourier series works.. Like I tried solving problems..but till now, it didn't sink in my brain.. I just want 2 know the basic way of solving a problem regarding Fourier series..then...
  14. S

    Fourier series half and full wave rectifiers

    Homework Statement (a) The negative half-cycles of a sinusoidal waveform E(t) = E0*cos(omega*t) are removed by a half- wave rectifier. Find the Fourier series representing the resulting wave in the output. (b) Find the output produced by a full-wave rectifier, which inverts...
  15. T

    What is the Fourier series for a periodic function over [-\pi,\pi]?

    Homework Statement Find the Fourier series corresponding to the following functions that are periodic over the interval [-\pi,\pi] f(x) = 1, -\pi/2 < x< \pi/2; f(x) otherwise. Homework Equations Fourier Series: f(x) = \frac{1}{2}a_0 + \sum^\infty_{n=1}a_n...
  16. M

    MATLAB Fourier Series Coefficients with MATLAB

    Hi, I'm trying to write a MATLAB Script to solve the Fourier series coefficients of a square wave via simultaneous equations in MATLAB. Any advice? I can write the expression out by hand, but cannot code it. Please advice.
  17. B

    Estimating the sum of reciprocal powers using a given fourier series

    Homework Statement Let f(x) be defined by the following Fourier series for \left|x\right|: f(x) = \frac{\pi}{2} - \frac{4}{\pi}\sum_{1,3,...}\frac{cos\left(nx\right)}{n^{2}} Show that \sum_{1,3,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{8} and \sum_{1,2,3,...}\frac{1}{n^{2}} =...
  18. Z

    Converting f(x) to MATLAB Code for Fourier Series with n=500

    The problem are as below :- f(x) = { x, for 0 < x < pi/2; x/2, for pi/2 < x < pi; pi-(x/2), for pi < x < 2*pi; how do I convert it to Matlab code and produce a graph of Fourier series with n=500? I have done analytically by hand for n=2, but the question ask to produce...
  19. N

    Parseval's Theorem and Fourier series

    Homework Statement Hi all. Please take a look at the lowest equation in this picture: http://img143.imageshack.us/img143/744/picture2ao8.png This is Parselvals Identity. Let us say that I am given a Fourier series of f(x), and I want to calculate the integral of f(x)^2 from -L to...
  20. W

    How can the Fourier series equation be used to represent the sawtooth function?

    Homework Statement Prove: f(t)=\sum_{i=1}^\infty (-1)^{n+1} \frac{Sin(n\omega t)}{n} represents the sawtooth function. Homework Equations Fourier Series Equations: f(t)=\frac{a_0}{2}+\sum_{i=0}^\infty \left(a_n Cos(nt)+b_n Sin(nt) \right) where a_n=\frac{2}{T}\int_{t}^{t+T} dt...
  21. N

    Fourier series and convergence

    Homework Statement Hi all. There is an example in my book, where we have the following Fourier series, and the author writes it as: f(x) = \sum\limits_{n = 1}^\infty {\left( {\frac{{( - 1)^n }}{{n^2 }}\cos \left( {\frac{{n\pi x}}{p}} \right) + \frac{1}{n}\sin \left( {\frac{{n\pi...
  22. N

    Calculating the Fourier Series of |sin x|

    I'm trying to find the Fourier series of |sin x| between -pi and pi. I've got it down to: a_n = \frac{1}{\pi} \int^{\pi}_{-\pi} |sin x| cos (nx) dx which i wrote as: a_n = \frac{2}{\pi}\int^{\pi}_0 sin x cos (nx) dx writing sin x cos (nx) = \frac{1}{2} (sin (n+1)x - sin (n-1)x) I...
  23. F

    Fourier series vs Integral: just one coherent notation?

    Fourier series vs Integral: just one "coherent" notation? Hi all, notations for Fourier analysis always confuse me. I understand that the "volume" normalization factor can be disposed of either in the definition of the Fourier coefficient or in the definition of the Fourier series (or...
  24. N

    Question regarding the definition of the complex fourier series

    Why is the complex Fourier series expanded from +\infty to -\infty. Yet the regular Fourier series from 0 to +\infty?
  25. W

    Fourier series (cosine series expansion?)

    Find the Fourier series of the 2Pi-periodic function f(x)= {0 , Abs(x) <= Pi/2 {Abs(x)-Pi/2, Pi/2 < Abs(x) <= Pi My attempt at a solution I have sketched the function... It equals zero between -Pi/2 and Pi/2 and it equals Pi/2 at -Pi and Pi. Then the...
  26. M

    Can Anyone Help Solve These Fourier Series and Heat Equation Problems Quickly?

    Any one Solution will be appreciated , just solve as soon as possible! 1) Fourier Series 2) Heat Equation
  27. M

    Book on laplace transforms & fourier series

    Hi hows it going, Im currently doing a linear maths course, i suppose it'd be introductory. I am using Elementary Linear algebra by Howard anton and find that very good. However the course goes on to deal with laplace transforms and Fourier series, can anyone recommend a good (introductory)...
  28. R

    Find Fourier Series Coefficients

    [b]1. Find Fourier Series Coefficient for a signal. (alternanate sawtooth pulse from t= -3 to 3 amplitude 1 on positivwe side and-1 on negative side. 3 pulse on positive side. 3 pulse on negative side. pulse width is 1.) as given in attachment file. [b]2. Compute the Fourier series...
  29. G01

    Fourier Series from old Physics GRE

    Homework Statement Hi everyone. I have a Fourier series problem from an old Physics GRE that I would like to discuss. Here is a link to the problem and a solution. The answer they get is correct, but their explanation is not: http://grephysics.net/ans/9277/39Homework Equations Fourier...
  30. M

    Find the coefficients in the Fourier series

    Homework Statement The odd periodic function f(x) is defined by f(x) = -4-x for -4 <= x <= 0 and 4-x for 0 <= x <= 4 f(x + 8) = f(x) Sketch f(x) for -12 <= x <= 8 Find the coefficients in the Fourier series for the function defined by equation (1) and write out the series...
  31. L

    Why do I have a Fourier Serie in my heat equation solution?

    I'm trying to solve a basic heat equation \frac{\partial T}{\partial t}=\alpha \frac{\partial^2 T}{\partial x^2} I manage to get T=X(x)\tau(t) Then \tau(t)=A*e^{-\alpha*\lambda^2*t} and X(x)=C*sin(\lambda x) where \lambda=\pi/Ln n=1,2,3,... From here I don't know how or why I get to a...
  32. T

    Fourier series expansion of Sin(x)

    May be simple, but I'm getting problem with doing Fourier series expansion of Sin(x) for -pi\leqx\leqpi
  33. M

    Understanding Fourier Series: Period and Sketching for a Unit Step Function

    I can do the Fourier series but this question is really confusing F(x) is 1 for 0<x< pi , 0 for pi < x < 2pi and f(x+2pi) = f(x) for all x I know that it means that the period is 2pi and L = 2pi/2 = pi 1) Sketch f(x) for -4pi <x < 4pi I sketched the graph so that is alrite 2) Find...
  34. C

    Discover the Fourier Series of (sin x)^2 with Helpful Tips!

    fourier series, please help! Homework Statement Find the Fourier series of f(x)=(sin x)^2. Homework Equations The Attempt at a Solution I know that I need to use the double angle formulas for this problem: (sin x)^2=1/2-1/2(cos 2x) but I do not know where to go from here.
  35. D

    Fourier series and the dirchlet integral

    Use a Fourier series to prove that: \int_{0}^{ \infty} \frac{\sin(x)}{x}\ \mbox{d}x = \frac{ \pi}{2} What function do I need to Fourier transform?
  36. A

    Understanding Fourier Series: Solving Problems and Finding Coefficients

    fourier series pleeeease :( Hi every one, I want your help please. I have a difficulty in answering these questions. Actually it seems simple but I could not answer them because I did not understand the lesson very well. This homework worths 5% and I have to hand it as soon as...
  37. G

    Proving Fourier Series of f(x) for 0<\lambda<1

    Homework Statement if 0<\lambda<1 and f(x) = x for 0<x<\lambda\pi and f(x) = (\lambda/(1-\lambda))(\pi-x) for \lambda\pi<x<\pi show that f(x)= 2/(\pi(1-\lambda))\Sigma(sin( n\lambda\pi)sin(nx))/n^{}2 Homework Equations The Attempt at a Solution am i right in saying that...
  38. G

    How can I use Fourier series to solve for f(x) when given specific conditions?

    Homework Statement if 0<\lambda<1 and f(x) = x for 0<x<\lambda\pi and f(x) = (\lambda/(1-\lambda))(\pi-x) for \lambda\pi<x<\pi show that f(x)= 2/(\pi(1-\lambda))\Sigma(sin( n\lambda\pi)sin(nx)(/n^{}2 Homework Equations The Attempt at a Solution am i right in saying that...
  39. G

    Finding Fourier Series for Piecewise Function with Parameter

    Homework Statement if 0<\lambda<1 and f(x) = x for 0<x<\lambda\pi and f(x) = (\lambda/(1-\lambda))(\pi-x) for \lambda\pi<x<\pi show that f(x)= 2/(\pi(1-\lambda))\Sigma(sin( n\lambda\pi)sin(nx)(/n^{}2 Homework Equations The Attempt at a Solution am i right in saying that...
  40. G

    Is this the correct way to calculate a Fourier series with given conditions?

    Homework Statement if 0<\lambda<1 and f(x) = x for 0<x<\lambda\pi and f(x) = (\lambda/(1-\lambda))(\pi-x) for \lambda\pi<x<\pi show that f(x)= 2/(\pi(1-\lambda))\Sigma(sin( n\lambda\pi)sin(nx)(/n^{}2 Homework Equations The Attempt at a Solution am i right in saying that...
  41. O

    Exponential Fourier Series for Pulse Train

    Homework Statement I need help finding the exponential Fourier series for the pulse train below. http://img329.imageshack.us/img329/1184/20080427091409ev5.jpg Homework Equations p_T(t)=\sum_{n=-\infty}^{\infty}{D_n e^{j2nt} D_n=\frac{1}{T_0}\int_{T_0}^{}{x(t)e^{-j2nt}dt}The Attempt at a...
  42. R

    Solving a Fourier Series for a Saw-Tooth Wave

    Hey, I am trying to solve this question: obtain a Fourier series for a saw-tooth wave, a periodic signal, with period T, defined such that x(t)=At -T/2<= t >= T/2 where A has a value of 1 at the maximum value of x(t) i) obtain the Fourier series for this periodic signal in form...
  43. S

    Solving a Two-Equation Fourier Series

    [SOLVED] Fourier Series Homework Statement I know this may be a simple problem but I am just beginning to understand this subject and this question has confused me. Expand the following as a whole-range Fourier series: f(x) = 1 -pi < x < 0 f(x) = x 0 < x < pi I...
  44. W

    Complex Fourier Series Problem

    Homework Statement Having shown that: f(x) = e^{ax} = \frac{sinh(a\pi)}{\pi}\sum^{\infty}_{n=-\infty}\frac{(-1)^{n}(a+in)}{(a^{2}+n^{2})}e^{inx} on (-\pi,\pi) with 'a' a real constant Deduce that: \pi coth(a\pi) = \frac{1}{a} + \sum^{\infty}_{n=1}\frac{2a}{(a^{2}+n^{2})}...
  45. M

    Fourier series via complex analysis

    Homework Statement Show that f is 2-pi periodic and analytic on the strip \vert Im(z) \vert < \eta, iff it has a Fourier expansion f(z) = \sum_{n = -\infty}^{\infty} a_{n}z^{n}, and that a_n = \frac{1}{2 \pi i} \int_{0}^{2\pi} e^{-inx}f(x) dx. Also, there's something about the lim sup of...
  46. R

    Fourier Series Integration by Parts Solution

    Homework Statement Solve \frac{1}{1} \int^{0}_{-1} -t e^{-j2\pi*nt}dtHomework Equations So I use integration by parts u = -t and dv = e^{-j2\pi*nt} , du= -1 and v = \frac{1}{-j2\pi*n}e^{-j2\pi*nt}The Attempt at a Solution after integration by parts I get: \frac{e^-j2\pi*nt}{j2pi*n} +...
  47. N

    Fourier series and even/odd functions

    [SOLVED] Fourier series and even/odd functions Homework Statement I found the solution to a PDE in this thread: https://www.physicsforums.com/showthread.php?t=224902 (not important) The solution is the sum of u(rho, phi) = [A_n*cos(n*phi)+B_n*sin(n*phi)]*rho^n. I have to find the general...
  48. M

    Fourier series technique to show that the following series sum to the quantities

    Use the Fourier series technique to show that the following series sum to the quantities shown: 1+1/3^2+1/5^2+...+1/n^2=pi^2/8 for n going to infinity I foudn the series to be: sum(1/(2n-1)^2,n,1,infinity) but I don't know how to prove the idenity. I don't know how to go about...
  49. M

    Proving the Sum of a Series using Fourier Series Technique

    Use the Fourier series technique to show that the following series sum to the quantities shown: 1+1/3^2+1/5^2+...+1/n^2=pi^2/8 for n going to infinity I foudn the series to be: sum(1/(2n-1)^2,n,1,infinity) but I don't know how to prove the idenity. I don't know how to go about...
  50. C

    The Fourier Series of Sin^5(x)

    Homework Statement So I have to find the Fourier series for sin^{5}(x). Homework Equations I know the a_{n} in: \frac{a_{0}}{2} + \sum^{\infty}_{n=1}a_{n}cos_{n}x + \sum^{\infty}_{n=1}b_{n}sin_{n}x goes to zero, which leaves me with taking the b_{n}. The Attempt at a Solution So what I got...
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