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. B

    Fourier Series for |x|: Convergence & Answers

    Homework Statement Find trigonometric Fourier series for ##f(x)=|x|##, ##x∈[−\pi, \pi]##, state points where ##F(x)## fail to converge to ##f(x)##. Homework Equations ##F(x) = \frac{a_0}{2}+\sum\limits_{n=1}^\infty a_ncos(\frac{n\pi x}{L})+b_nsin(\frac{n\pi x}{L})##...
  2. G

    Derivation of the Fourier series of a real signal

    Homework Statement Consider the Fourier series of a signal given by $$x(t)=\sum_{k=-\infty}^{\infty} a_ke^{jk\omega_0t}$$ Let's consider an approaches to this series given by the truncated series. $$x_N(t)=\sum_{k=-N}^{N} a_ke^{jk\omega_0t}$$ a- Show that if $x(t)$ is real then the series...
  3. Allan McPherson

    Using Maxima to plot error in Fourier series

    I'm trying to use Maxima to examine the error in a Fourier series as the number of terms increases. I've figured out how to produce a Fourier series and plot partial sums, but this has me stumped. If anyone experienced with the Maxima CAS has some insight into this, I would greatly appreciate...
  4. H

    Fourier series of a bandwidth limited periodic function

    Homework Statement Find Fourier coefficients of the periodic function whose template is x(t) where the Fourier Transform of x(t) is X(f) = (1-f^2)^2 where \left|f\right|<1 and period T_0= 4. Homework Equations FC=\hat x_T(k,T_0)=\sum_{k=-\infty}^\infty\frac{1}{T_0}X\left(k/T_0\right) The...
  5. Svein

    Insights Further Sums Found Through Fourier Series - Comments

    Svein submitted a new PF Insights post Further Sums Found Through Fourier Series Continue reading the Original PF Insights Post.
  6. baby_1

    Some questions about Fourier series

    Hi, First of all, I want to say that I know how can define and calculate Fourier coefficients but I have some question about the final presentation of Fourier and half-period or unknown period functions. 1)In this function how can we define T? 2)for above diagram, in a book, they define f(t)...
  7. M

    Fourier Series of a function not centered at zero

    Homework Statement f(x)=x on [0,2) Homework Equations Fourier Series is given as: f(x)=a0/2 + n=1∞∑(an*cos(nπx/L) + bn*sin(nπx/L) a0=1/L*-LL∫f(x)dxThe Attempt at a Solution Basically what I am being taught is that we take the Period, T, to be equal to 2L so, T=2L In this case T=2 and L=1. My...
  8. A

    Odd and even in complex fourier series

    Homework Statement In Complex Fourier series, how to determine the function is odd or even or neither, as in the given equation $$ I(t)= \pi + \sum_{n=-\infty}^\infty \frac j n e^{jnt} $$Homework Equations ##Co=\pi## ##\frac {ao} 2 = \pi## ##Cn=\frac j n## ##C_{-n}= \frac {-j} n ## ##an=0##...
  9. C

    What is the Definition of Period in Fourier Series?

    Homework Statement Homework Equations The Attempt at a Solution a0=4 an=8/Pi*n Heres a written solution https://gyazo.com/57e11d1e7a360914db8aec167beb6b39.png
  10. A

    Complex Fourier Series for cos(t/2)

    Homework Statement Q:/ Find the complex form of Fourier series for the following periodic function whose definition in one period is given below then convert to real trigonometry also find f(0). f(t)=cos(t/2), notes: (T=2*pi) (L=pi) Homework Equations 1) f(t)=sum from -inf to +inf (Cn...
  11. G

    Find Fourier coefficients - M. Chester text

    Homework Statement I am self studying an introductory quantum physics text by Marvin Chester Primer of Quantum Mechanics. I am stumped at a problem (1.10) on page 11. We are given f(x) = \sqrt{ \frac{8}{3L} } cos^2 \left ( \frac {\pi}{L} x \right ) and asked to find its Fourier...
  12. evinda

    MHB Sine fourier series with period 1

    Hello! (Wave) I want to find the Fourier series of $f(x)=x, 0 \leq x<1$. It is a series with period $1$. In our case, the function is odd. So in order to find the Fourier series, we would find the odd extension of $f$ and then use the following formulas: $a_n=0 , \ \ \forall n \geq 0$...
  13. evinda

    MHB How can we extend the solution of an initial value problem using Fourier series?

    Hello! (Wave) The following problem shall show the way with which the Fourier series can be used for the solution of initial value problems.Find the solution of the initial value problem $$y''+ \omega^2 y=\sin{nt}, y(0)=0, y'(0)=0$$ where $n$ is a natural number and $\omega^2 \neq n^2$. What...
  14. dumbdumNotSmart

    Heat equation integral - Fourier Series coefficient is zero

    Homework Statement WE have a thermally insulated metallic bar (from enviroment/surroundings) . It has a temperature of 0 ºC. At t=0 two thermal sources are applied at either end, the first being -10 ºC and the second being 10 ºC. Find the equation for the temperature along the bar T(x,t), in...
  15. R

    Fourier Series of Sawtooth Wave from Inverse FT

    Homework Statement I want to find the Fourier series of the sawtooth function in terms of real sine and cosine functions by using the formula: $$f_p (t)=\sum^\infty_{k=-\infty} c_k \exp \left(j2\pi \frac{k}{T}t \right) \tag{1}$$ This gives the Fourier series of a periodic function, with the...
  16. chikou24i

    B Fourier series of a step function

    Hello, can we make a Fourier series expansion of a (increasing or decreasing) step function ? like the one that I attached here. I just want to know the idea of that if it is possible.
  17. Marcus95

    Fourier Series Coefficient Symmetries

    Homework Statement Let ## f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx) ## What can be said about the coefficients ##a_n## and ##b_n## in the following cases? a) f(x) = f(-x) b) f(x) = - f(-x) c) f(x) = f(π/2+x) d) f(x) = f(π/2-x) e) f(x) = f(2x) f) f(x) = f(-x) =...
  18. T

    Fourier Series of a Piecewise Function

    Homework Statement f(x) = -1, -π ≤ x ≤ 0 2, 0 ≤ x ≤ π Given this find the Fourier series using both $$a) \sum_{n=-∞}^\infty a_n e^{inx}$$ $$b) \sum_{n=0}^\infty [A_n cos(nx) + B_n sin(nx)]$$ Homework Equations $$a_o = \frac {1} {2L} \int_{-L}^L f(t) \, dt $$ $$a_n = \frac {1} {L}...
  19. D

    Fourier/heat problem involving hyperbolic sine

    Homework Statement A rectangular box measuring a x b x c has all its walls at temperature T1 except for the one at z=c which is held at temperature T2. When the box comes to equilibrium, the temperature function T(x,y,z) satisfies ∂T/∂t =D∇2T with the time derivative on the left equal to zero...
  20. G

    Evaluate Fourier series coefficients and power of a signal

    Homework Statement Derive the expression for coefficients of Fourier series in exponential form for the sequence of rectangular pulses (with amplitude A, period T and duration θ) shown in this image: Derive the expression for signal power depending on the coefficients of Fourier series...
  21. D

    How Does Time Affect the Displacement of a Plucked Violin String?

    Homework Statement A violin string is plucked to the shape of a triangle with initial displacement: y(x,0) = { 0.04x if 0 < x < L/4 (0.04/3)(L-x) if L/4 < x < L Find the displacement of the string at later times. Plot your result up to the n = 10...
  22. bradzyc

    Fourier Series: Stamping Machine Positioning Function

    Homework Statement Homework Equations All Fourier series trigonometric equations. I think we are required to use sigma function, integrals, etc.[/B]The Attempt at a Solution We are currently working through our Fourier series revision studying integrals of periodic functions within K.A...
  23. N

    Change of variables in Heat Equation (and Fourier Series)

    Q: Suppose ##u(x,t)## satisfies the heat equation for ##0<x<a## with the usual initial condition ##u(x,0)=f(x)##, and the temperature given to be a non-zero constant C on the surfaces ##x=0## and ##x=a##. We have BCs ##u(0,t) = u(a,t) = C.## Our standard method for finding u doesn't work here...
  24. R

    Mathematica How to plot several terms in a Fourier series

    I was given a function that is periodic about 2π and I need to plot it. I was wondering if there is a way to input a value and have mathematica generate a new graph with the number of iterations. The function is: $$\sum_{n=1}^{N}\frac{sin(nx)}{n}$$ where n is an odd integer. I guess a better...
  25. J

    Fourier series expansion. Find value of a term in expansion

    Homework Statement Fourier series expansion of a signal f(t) is given as f(t) = summation (n = -inf to n = +inf) [3/(4+(3n pi)2) ) * e j pi n t A term in expansion is A0cos(6 pi ) find the value of A0 Repeat above question for A0 sin (6 pi t) Homework Equations Fourier expansion is summation...
  26. E

    Finding Fourier Series of f(x)=√(x2) -pi/2<x<pi/2

    Homework Statement Find the Fourier series of the function f(x) =√(x2) -pi/2<x<pi/2 , with period pi Homework EquationsThe Attempt at a Solution I have tried attempting the question, but couldn't get the answer. uploaded my...
  27. Gopal Mailpalli

    Fourier Series for Periodic Functions - Self Study Problem

    Self Study 1. Homework Statement Consider a periodic function f (x), with periodicity 2π, Homework Equations ##A_{0} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)dx## ##A_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)cos\frac{2\pi rx}{L}dx## ##B_{n} =...
  28. Pouyan

    Fourier series and differential equations

    Homework Statement Find the values of the constant a for which the problem y''(t)+ay(t)=y(t+π), t∈ℝ, has a solution with period 2π which is not identically zero. Also determine all such solutions Homework Equations With help of Fourier series I know that : Cn(y''(t))= -n2*Cn(y(t)) Cn(y(t+π)) =...
  29. Svein

    Insights Using the Fourier Series To Find Some Interesting Sums - Comments

    Svein submitted a new PF Insights post Using the Fourier Series To Find Some Interesting Sums Continue reading the Original PF Insights Post.
  30. CricK0es

    Find the Fourier series for the periodic function

    < Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown > Hi all. I'm completely new to these forums so sorry if I'm doing anything wrong. Anyway, I have this question... Find the Fourier series for the periodic function f(x) = x^2 (-pi < x < pi)...
  31. MAGNIBORO

    Complex Fourier Series Problem

    Hi, I'm starting to studying Fourier series and I have troubles with one exercises of complex Fourier series with f(t) = t: $$t=\sum_{n=-\infty }^{\infty } \frac{e^{itn}}{2\pi }\int_{-\pi}^{\pi}t\: e^{-itn} dt$$ $$t=\sum_{n=-\infty }^{\infty } \frac{cos(tn)+i\, sin(tn)}{2\pi...
  32. W

    I Spectral representation of an incompressible flow

    Hi PH. Let ##u_i(\mathbf{x},t)## be the velocity field in a periodic box of linear size ##2\pi##. The spectral representation of ##u_i(\mathbf{x},t)## is then $$u_i(\mathbf{x},t) = \sum_{\mathbf{k}\in\mathbb{Z}^3}\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}$$ where ι denotes the usual imaginary...
  33. LLT71

    I Can Fourier Analysis Represent Any Function Using Sin and Cos?

    has Fourier used sin(x) and cos(x) in his series because "there must be such interval [a,b] where integral of "some function"*sin(x) on that interval will be zero?" so based on that he concluded that any function can be represented by infinite sum of sin(x) and cos(x) cause they are "orthogonal"...
  34. J

    MHB Need help on Fourier Series (badly)

    Need help on Fourier series! Been stuck on this questions, it is too tough for me!
  35. D

    I Inverse Laplace to Fourier series

    I have the following laplace function F(s) = (A/(s + C)) * (1/s - exp(-sα)/s)/(1 - exp(-sT)) I think that the inverse laplace will be- f(t) = ((A/C)*u(t) - (A/C)*exp(-Ct)*u(t)) - ((A/C)*u(t-α) - (A/C)*exp(-C(t-α))*u(t-α)) and f(t+T)=f(t) Now I want to find the Fourier series expansion of f(t)...
  36. Captain1024

    Fourier Series Coefficients of an Even Square Wave

    Homework Statement Link: http://i.imgur.com/klFmtTH.png Homework Equations a_0=\frac{1}{T_0}\int ^{T_0}_{0}x(t)dt a_n=\frac{2}{T_0}\int ^{\frac{T_0}{2}}_{\frac{-T_0}{2}}x(t)cos(n\omega t)dt \omega =2\pi f=\frac{2\pi}{T_0} The Attempt at a Solution Firstly, x(t) is an even function because...
  37. MexChemE

    I Motivation for Fourier series/transform

    Hello, PF! I am currently learning Fourier series (and then we'll move on to the Fourier transform) in one of my courses, and I'm having a hard time finding motivation for its uses. Or, in other words, I can't seem to find its usefulness yet. I know one of its uses is to solve the heat...
  38. mr.tea

    How Does the Poisson Summation Formula Apply to Uniformly Convergent Series?

    Homework Statement let ##g## be a ##C^1## function such that the two series ##\sum_{-\infty}^{\infty} g(x+2n\pi)## and ##\sum_{n=-\infty}^{\infty} g'(x+2n\pi)## are uniformly convergent in the interval ##0\leq x \leq 2\pi ##. Show the Poisson summation formula: ##\sum_{n=-\infty}^{\infty}...
  39. G

    I Understanding the Intuition Behind Fourier Series?

    I'm wondering if anyone could give me the intuition behind Fourier series. In class we have approximated functions over the interval ##[-\pi,\pi]## using either ##1, sin(nx), cos(nx)## or ##e^{inx}##. An example of an even function approximated could be: ## f(x) = \frac {(1,f(x))}{||1||^{2}}*1...
  40. C

    Sum of sinosoids that can be a Fourier Series expansion

    Homework Statement I was given a problem with a list of sums of sinusoidal signals, such as Example that I made up: x(t)=cos(t)+5sin(5*t). The problem asks if a given expression could be a Fourier expansion. Homework Equations [/B]The Attempt at a Solution My guess is that it has something to...
  41. T

    Calculating Coefficients of Fourier Series Homework

    Homework Statement I'm calculating the coefficients for the Fourier series and I got to part where I can't simplify an any further but I know I have to. a_n = \frac{1}{2π}\Big[\frac{cos(n-1)π}{n-1}-\frac{cos(n+1)π}{n+1}-\frac{1}{n-1}+\frac{1}{n+1}\Big]Homework EquationsThe Attempt at a...
  42. P

    Generalised Fourier Series

    Homework Statement By applying the Gram–Schmidt procedure to the list of monomials 1, x, x2, ..., show that the first three elements of an orthonormal basis for the space L2 (−∞, ∞) with weight function ##w(x) = \frac{1}{\sqrt{\pi}} e^{-x^2} ## are ##e_0(x)=1## , ##e_1(x)= 2x## ,##e_2(x)=...
  43. V

    I Solving u_x=(sin(x))*(u) in Fourier space

    Does anyone know if it is possible to solve an equation of the type u_x=(sin(x))*(u) on a periodic domain using the fft. I have tried methods using convolutions but have had no success thanks in advance
  44. thegreengineer

    I Fourier Series: I don't understand where I am wrong --

    Good afternoon people. Recently I started taking a course at my college about Fourier series but I got extremely confused. Here's what's going on. In school we were asigned to use the symmetry formulas to find the Fourier series of the following: f\left ( t \right )=\begin{cases} 1 & \text{ if...
  45. baby_1

    Fourier Series in cylindrical coordinate

    Homework Statement Here is my question Homework Equations I don't know with what formula does the book find Fourier series? The Attempt at a Solution
  46. sa1988

    I Is this even possible? Question about Fourier Series....

    Today I had a maths exam with a question which was worded something like: Write ##sin(3x-x_0)## as its Fourier representation. By doing a suitable integral or otherwise, find the possible values of its Fourier coefficients. You may find the following useful: ##sin(\alpha-\beta) =...
  47. P

    How's Fourier series modified for function f(t)= f(2Pi t)?

    Homework Statement How are the coefficients of the Fourier series modified for a function with a period 2πT? Homework Equations a0 = 1/π ∫π-π f(x) dx an = 1/π ∫π-π f(x) cos(nx) dx bn = 1/π ∫π-π f(x) sin(nx) dx The Attempt at a Solution I tried letting x= t/T so dx = dt/T and the limits x = ±...
  48. P

    Fourier Series: Solving Homework Equations for f(x)

    Homework Statement The following function is periodic between -π and π: f(x) = |x| Find the Coefficients of the Fourier series and, by examining the Fourier series at x=π or otherwise, determine: 1 + 1/32 + 1/52 + 1/72 ... = Σ∞j=1 1/(2j - 1)2 Homework Equations f(x) = a0/2 + ∑∞n=1 ancos(nx) +...
  49. Alana02011114

    Solving Coefficient not using Fourier Series coefficient

    Given the Laplace's equation with several boundary conditions. finally i got the general solution u(x,t). One of the condition is that: u(1,y)=y(1-y) After working on this I finally got: ∑An sin(π n y )sinh (π n) = y(1-y) However, i was asked to find An, by not using Fourier series...
  50. Nemo's

    Fourier series neither odd nor even

    Homework Statement I'm trying to calculate the Fourier Series for a periodic signal defined as: y = x 0<x<2Π y = 0 2Π≤x<3Π Homework Equations Fn = 1/T ∫T f(t)cos(kwοt + θk)[/B] cn/2 + ∑k=1k=∞(cn)cos(kwοt+θk) cn= 2|Fn| θk=∠Fn The Attempt at a Solution I got Cn =...
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