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

    Fourier Series coefficients, orthogonal?

    Homework Statement Hello. I need help with orthogonality of the Fourier series coefficients. I know you can use the dirac delta function, (or the kronecker function) in the orthogonality relationship. I want to try and see the derivation using complex form rather than sines and cosines...
  2. P

    How do I find Fourier series coefficients for triangular and sawtooth waveforms?

    Triangular and sawtooth waveforms. I solved for the coeffiecients for the square wave, and was wondering if you do the same for triangular and sawtooth. I keep getting large/messy integrals that do not seem to simplify. The thing that is messing me up is that the waveform is linear with a...
  3. Z

    How Can Fourier Series Demonstrate the Summation Identity Ʃ1/(2m+1)^2 = pi^2/8?

    Homework Statement Consider the function: f(x) = {0 if 0<x<L/2 x-L/2 if L/2<x<L} Define a periodic extension, obtain the complex Fourier series, and show that Ʃ1/(2m+1)^2 = pi^2/8... Homework Equations complex Fourier series The Attempt at a Solution I defined it as an even function by...
  4. N

    Understanding Fourier Series for EE Homework

    Hey guys, I'm having trouble with a problem assigned for homework in an EE course on Fourier series. We have yet to have a lecture on Fourier series when the homework is due Thursday, and because of the long weekend break we don't have class Tuesday. With little knowledge on Fourier series, from...
  5. I

    Exact value of infinity sum of Fourier series coefficients

    Homework Statement Sorry for doing another thread but I can't edit the old one any longer and I found out I made some calculation error but I'm pretty sure it's right now. The problem is to find the exact value of the series. Homework Equations \sum a_{k} The summation is to be done...
  6. Z

    Is it possible to deduce a function from its fourier series ?

    Is it possible to deduce a function from its Fourier series ?
  7. K

    What is the Fourier series for f(x) = sin^2(x)?

    Homework Statement Find the Fourier series of f(x) = sin^2(x) Homework Equations bn = because f(x) is even ao = (1/(2*∏))*∫(f(x)) (from 0 to 2*∏) an = (1/(∏))*∫(f(x)*cos(x)) (from 0 to 2*∏) The Attempt at a Solution ao = (1/(2*∏))*∫(f(x)) (from 0 to 2*∏) = ao = 1/2 an =...
  8. Z

    What is the rules regarding convergence of a Fourier series?

    If you are given part of a period of a Function, what rules would you apply to draw out the full function, so that it converges as quickly as possible as a Fourier series? thanks
  9. A

    Fourier Series for real and odd signals

    Hi, I have a general question regarding the computation of Fourier Series coefficients for real and odd inputs. In this case, the following should be true: ∫x(t)*e^(-j*k*w*t)dt = ∫x(t)*sin(k*w*t)dt However, every time I compute my coefficients this way, I get the inverse sign of what it...
  10. Z

    What is the convergence rate of Fourier series?

    Okay, so I didn't really understand the professor when he talked about the speed of convergence of Fourier series. The question is what kind of functions converge faster than what kind of other functions using Fourier series representation. My guess from what I have absorbed is that functions...
  11. estro

    Why Are My Fourier Series Calculations Giving Me Nonsense Results?

    I'm trying to get Fourier series for some function however I get the same nonsense series each time: f(x)=\begin{cases} 0, & \mbox{if x $\in [-\pi,0]$}\\ x, & \mbox{if x $\in (0,\pi)$} \end{cases} \pi a_n=\int_{-\pi}^\pi f(x)cos(nx)dx=\int_{-\pi}^0 0 \cdot cos(nx)dx+\int_0^\pi...
  12. estro

    Convergence Issues with Fourier Series for Piecewise Function?

    I'm trying to find Fourier series for the following function: f(x) = \begin{cases}1, & \mbox{if x $\in (-\frac{\pi}{2}+2\pi n,\frac{\pi}{2}+2\pi n)$ } \\ -1, & \mbox{if x $\in [\frac{\pi}{2}+2\pi n,\frac{3\pi}{2} + 2\pi n]$} \end{cases} This is how I calculated a_n and b_n: [Please See...
  13. N

    Understanding Fourier Series for PDEs: Odd vs. Even Summation

    So I am in PDE's and we are deriving certain Fourier series for periodic functions. I have notice that the solution manual to the book Partial Differential Equations with Fourier Series and Boundary Value Problems by Asmar tends to emphasize infinite sums frequently with "n" equal to 2k for even...
  14. S

    Solving Hard Fourier Series: Let f = e^(cos(x^2)) for 0 < x < 2pi

    Homework Statement Let f be the 2pi periodic function defined by f(x)=e^{cos(x^{2})} for 0 < x < 2pi. What is the value of the Fourier series at x=4pi Homework Equations The Attempt at a Solution I don't even know where to start. All help is much appreciated!
  15. J

    Fourier series convergence test

    Homework Statement A function f(x) is given as follows f(x) = 0, , -pi <= x <= pi/2 f(x) = x -pi/2 , pi/2 < x <= pi determine if it's Fourier series (given below) F(x)=\pi/16 + (1/\pi)\sum=[ (1/n^{2})(cos(n\pi) - cos(n\pi/2))cos(nx) -...
  16. J

    Fourier series convergence question

    Homework Statement f(x) = 5, -pi <= x <= 0 f(x) = 3, 0 < x <= pi f(x) is the function of interest Find the x-points where F(x) fails to converge to f(x) Homework Equations F(x) = f(x) if f is continuous at x\in(-L,L) F(x) = 0.5[ f(x-) + f(x+) ] if f is...
  17. mnb96

    From Fourier Series to Fourier Transforms

    Hello, I am trying to formalize the logical steps to prove that the Fourier Series of a function with period\rightarrow \infty leads to the Fourier transform. So let's have the Fourier series: f(x)=\sum_{n=-\infty}^{+\infty}c_n e^{i\cdot \frac{2\pi n}{L}x} where L is the period of the...
  18. T

    How to find the period of a Fourier Series

    Hi. My problem is that even though I can find the Fourier series, its coefficients etc. I have trouble determining the period of equation. For example let's say we have f(t) = t, t \in [-pi,pi]. I thought that the period was 2pi but in the solution it says that the period is pi. This isn't the...
  19. N

    How Do You Calculate the Fourier Series of a Piecewise Function with Period 4?

    Homework Statement The function f has period 4 and is such that f(x)=2+x, -2<x\leq0 f(x)=2, 0<x<2 Sketch the graph of f for x∈[−4, 4] and obtain its Fourier series. Homework Equations The Attempt at a Solution Okay so I've pretty much sketched the graph, but I've been...
  20. A

    Complex Fourier Series of f(x)=x from -pi to pi | Extra Credit Assignment

    What is the complex Fourier series of f(x)=x from -pi to pi? I'm in a complex variables class and we have an extra credit assignment to figure out the complex Fourier series of f(x)=x from -pi to pi. We only vaguely covered the topic in class and our book is not very good so I'm not...
  21. G

    Fourier Series: Solving f(x) = e^x Homework

    Homework Statement f(x) = e^x for -\pi/2 \leq x \leq \pi/2 Homework Equations The Attempt at a Solution Having a bit of trouble getting this out. This is a odd function right so a_n = 0 so i only have to find b_n , right?
  22. J

    Finding the signal x(t) when given properties of fourier series coefficients

    Homework Statement Suppose we are given the following information about a continuous-time periodic signal with period 3 and Fourier coefficients a_{k} 1. a_{k} = a_{k+2} 2. a_{k} = a_{-k} 3. \int_{-0.5}^{0.5}x(t)dt = 1 4. \int_{0.5}^{1.5}x(t)dt = 2 Determine x(t) Homework Equations if x(t)...
  23. D

    Fourier series problem solving for an,bn

    Homework Statement let f(x)={0;-2\leqx\leq0. x;0\leqx\leq2 find a0 an bn given the period is 4 Homework Equations a0=1/L\intf(x)dx an=1/L\intf(x)cos(n\pix/L) bn=1/L\intf(x)sin(n\pix/L) The Attempt at a Solution so I can get a0 = 1 but I run into trouble with an. so I plug...
  24. I

    Fourier series of a piecewise function

    i'm going through this question. firstly how would the graph look like? at first i thought it would be http://i.imgur.com/XfVCK.png the top graph drawn, but then i thought maybe it's the bottom one. if it was the bottom one then it'd be easier for me to do the question, since it'd be an odd...
  25. I

    Fourier series question, either i'm wrong or the answers are wrong

    Homework Statement Homework Equations The Attempt at a Solution i'm pretty much learning Fourier series from scratch today after only looking at it in lectures (and my exam is tomorrow lol @ me) and I'm sort of stuck. The solutions have something different to what i have, i think it might be an...
  26. reddvoid

    How Is x(t) Determined from Its Fourier Series Coefficients and Given Integrals?

    x(t) is periodic with T=3. X(k) is FS coeff. X(k)=X(-k) and X(k)=X(k+2) also integral from t=-0.5 to 0.5 of x(t)dt is 1 integral from 0.5 to 1.5 of x(t) dt is 2. Find x(t) ======== i found that the signal x(t) is even and X(0) is 5/3 (if I'm correct) and integral from t=-1.5 to 1.5 of...
  27. Telemachus

    Integrating Fourier Series for a Cubic Function

    Homework Statement Hi there. I wanted to intagrate the Fourier series for g(t)=t^2 to get the Fourier series for f(x)=t^3 So I thought making something like: f(t)=3\int_0^t x^2 dx I know that g(t)=t^2\sim \frac{p^2}{3}+\sum_{n=1}^{\infty}\frac{4p^2(-1)^n}{n^2\pi^2}\cos\left (\frac{n\pi...
  28. H

    Finding Fourier series coefficient a0

    Homework Statement We're given this 'interactive page' that gives us the values, so T=2.8066, W=0.9542 and A=8.5988 and then told to find a0, b0, a1, b1, a3, b3, Total Power and 3rd harmonic power. Homework Equations Cn given above and: a0=1/T \int s(t) dt integrating from 0 to T. Also...
  29. Y

    Find the Fourier Series for f(x)=x^2 evaluate f(0) and show that the summation

    Find the Fourier Series for f(x)=x^2 evaluate f(0) and show that the summation \sum^{\infty}_{n=1}\frac{1}{(2n-1)^2}=\frac{\pi^2}{8} The first part of this problem asked that I find a_{n} and b_{n} Since x^2 is an even function b_n=0 and for a_n I got \frac{4(-1)^2}{n^2} for a_0 I got...
  30. C

    Find the Fourier Series of the Function

    Homework Statement For the function f(t)={t+2, -2<t<0 {2, 0<t<2 {f(t+4) all t Homework Equations a(n)=2/T*Integration[f(t)cos(n*w*t)dt] b(n)=2/T*Integration[f(t)sin(n*w*t)dt] xcos(ax)=...
  31. P

    Sample Exam Fourier Series & reverse engineering the question

    Homework Statement Hey, This is a question from a sample exam we are given for our engineering maths exam. Firstly given that the Fourier series contains only "sin(x)" doesn't this mean that it is an "odd" function? Can the period be calculated from the given function easily? Secondly is...
  32. Telemachus

    Proving the Sum of a Fourier Series Using Parseval's Relation

    Hi. Well, I have to demonstrate that \sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}=\frac{\pi^2}{8} Using a previous result of a Fourier series for f(t)=1 if 0<t<1, f(t)=0 if -1<t<0. I've found that: f(t)\sim{}1/2+\sum_{n=1}^{\infty}\frac{2}{(2n-1)\pi}\sin([2k-1]\pi t) I think this result, the series...
  33. Telemachus

    Troubleshooting Fourier Series Expansion for Piecewise Function

    Hi there. I have some trouble with this problem, it asks me to find the Fourier expansion series for the function f(t)=0 if -pi<t<0, f(t)=t^2 if 0<t<pi So I've found the coefficients a_0=\displaystyle\frac{1}{\pi}\displaystyle\int_{0}^{\pi}t^2dt=\displaystyle\frac{\pi^2}{3}...
  34. Telemachus

    How to be sure that a series isn't a fourier series of a derivable function

    Hi there. I have this interesting problem which I don't know how to solve. I'll post it here because I think more people will se it, but I'm not sure if this is the proper subforum. The problem says: How can be sure that \sum_{n = 1}^\infty \frac{1}{n}\sin (nx) isn't the Fourier series of...
  35. C

    Troubleshooting Fourier Series Analysis for Periodic Functions | Help Needed

    Hi All, I'm having a lot of trouble determining the a and b coefficients of the Fourier Series. I am given the following periodic function below and am asked to determine the coefficients. I am usually off by a factor of 2 or 1/2, if that means anything. It could be my bounds or my value for...
  36. A

    Transforming Complex Fourier Series to Real - Help Guide

    How can I transform a complex Fourier series into a real one in general? If for example I have the complex Fourier series for K=2m+1 (\frac{1}{2}+\frac{2}{i\pi})\sum\frac{1}{2m+1}e^{i(2m+1)t} what shoudl I do to transform it into a series with real coeff?
  37. J

    How Is the Fourier Series Derived for Odd and Even Functions?

    I am so stuck on my revision and i really need someones help! I am using the definition of Fourier series as My lecturer has told us that if f is odd. Could someone please tell me how he has derived this because i can't understand how he's got to it, iv tried using trig identities and...
  38. Z

    Inverse function and fourier series

    let be the Fourier expansion of the function f(x) = \sum_{m=-\infty}^{m=\infty}c_{m} exp(imx) valid on the interval (-1,1) , from this can we obtain the inverse function f^{-1} (x) by reflection of the Fourier series through the line y=x ??
  39. B

    Fourier Series as (Generalized)Least Squares?

    Hi, All: Given a normed vector space (X,||.||), and an inconsistent system Ax=b, the generalized least squares solution x^ to Ax=b is the point in the span of Ax that is closest to b, i.e., given a fixed matrix A, we define AX={Ax: x in X}, and then: x^:={ x in AX...
  40. B

    How to expand f(x)=x^2, -pi<x<pi as fourier series

    Homework Statement how to expand f(x)=x^2, -pi<x<pi as Fourier series answer at back of the book says: f(x)=((pi^2)/3)+4sigma(from 1 to infinity) [((-1)^n)/(n^2)]cos nx i tried the a(n) b(n) stuff but i don't see where they get the n^2 in the denominator from when i integrated to try to...
  41. S

    Am I missing something with exponential Fourier series?

    Hi, I'm in a signal processing class, and I'm having some trouble with complex numbers. As an example, I've attached a pretty simple question about an exponential Fourier series. I don't find these questions particularly hard, it just takes me ridiculously long to do them. I mean, after I...
  42. S

    Fourier series odd and even functions

    Hello I'am a little confused. In my textbook it is written that all odd function can be described by a sine series. I have this following equation from an exercise: A_{0}+\sum\limits_{n=1}^\infty (A_{n} cos(n \phi) + B_{n} sin(n \phi))c^{n} = sin(\dfrac{\phi}{2}) It's a standard...
  43. jegues

    Sketching Full Fourier Series of a Function

    Homework Statement See figure attached. Homework Equations The Attempt at a Solution See figure attached for my attempt. Is this the correct way to sketch the full Fourier series of the given function? If not, what am I doing wrong/misunderstanding? Thanks again!
  44. D

    Fourier Series Problem: Finding Coefficients for Piecewise Function f(t)

    Homework Statement one period of a function f(t) is given by the piecewise function f(t)= 2t if -pi/2 <t <0 and -1 if 0< t < pi/2 I need to find a sub 0, a sub n, a sub 1,2,3,& 4, and b sub 1,2,3 &4 If I could just get a sub 0 the rest are easy, however, my answer does not look correct...
  45. G

    Therefore D_n=\frac{2}{\pi(1-4n^2)}

    D_n=\frac{1}{\pi}\int_0^\pi\sin{(t)}\cdot e^{-i2nt}dt=\frac{2}{\pi(1-4n^2)} I have no idea on how they get from one side of the equation symbol to the other, can i get some tips and tricks ? I have try ed writing sint as an exp function, but i don't feel it gets me anywhere close.
  46. H

    Fourier Series: Is f(x) Even or Piecewise Continuous?

    Hello. I have to find the Fourier series for f(x) = 1 + cos(pi x / L). My question is about f(x) Is this function even? I plotted it out and it looks even. The question I am completing starts off by saying: assume that any function f for which f and its derivative are piecewise continuous...
  47. jegues

    How Diff. Ranges Affect Fourier Series

    Homework Statement I just have a general question about a set of questions given in my textbook. Homework Equations The Attempt at a Solution The questions are given in sequential order, and they both ask the same thing, Find the Fourier series of the function f(x). One...
  48. L

    How to Determine the Fourier Series for f(x)=cos3x

    Homework Statement determine the Fourier Series for f(x)=cos3x Homework Equations f(x)=ao/2+(sum) an cos(nx)+ (sum) bncos(nx) ao= (integral) f(x)dx (from -\pi to\pi) an= (integral) f(x)cos(nx)dx (from -\pi to\pi) bn= (integral) f(x)sin(nx)dx (from -\pi to\pi) The Attempt at a Solution i...
  49. N

    Differential Equations and Fourier Series

    Homework Statement Q1) (dy/dx)= 2x(y2+9); y(0)=0 Q2) (x4+y2) dx - xy dy =0; y(2)=1 Q3) (dy/dt)= 4y+t Q4) y"+2y'+y=0; y(0)=4 and y'(0)=-6 Q5) y"+3y'+2y= 30e2t 3. Solutions found A1) y= tan (x2/3) A2) this is not an exact differential and so cannot be solved A3) couldn't solve this...
  50. L

    Exploring Fourier Series and Gibbs Phenomenon

    Homework Statement http://img684.imageshack.us/img684/4496/fourier.png Homework Equations http://en.wikipedia.org/wiki/Small-angle_approximation The Attempt at a Solution This is part of a larger question, I have underlined in Red the areas I am struggling with and have cut out...
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