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. R

    Heat conduction problem in a ring of radius a

    Homework Statement We previously solved the heat conduction problem in a ring of radius a, and the solution is c into the sum, perform the sum first (which is just a geometric series), and obtain the general solution, which should only involve one integral in ϑHomework Equations...
  2. henry wang

    I How is the constant pi/L deduced in Fourier series?

    How is pi/L part deduced in (n*pi*x)/L?
  3. M

    Complex Fourier Series into a Cosine Series

    Homework Statement a. Represent f(x)=|x| in -2<x<2 with a complex Fourier series b. Show that the complex Fourier Series can be rearranged into a cosine series c. Take the derivative of that cosine series. What function does the resulting series represent? [/B]Homework Equations...
  4. L

    Pointwise, uniform convergence of fourier series

    Hello; I'm struggling with pointwise and uniform convergence, I think that examples are going to help me understand Homework Statement Consider the Fourier sine series of each of the following functions. In this exercise de not compute the coefficients but use the general convergence theorems...
  5. F

    Engineering Circuit Analysis-Converting a square wave to a sine wave

    Homework Statement For an upcoming lab I've been asked to build a circuit to convert a square wave (vi(t))e into a sine wave (v0(t)) by selecting appropriate resistor/capacitor values for the circuit below (from what I know, it's impossible to produce an accurate sine wave with just this, I...
  6. C

    Odd or Even? - Arbritrary Period Fourier Series

    Homework Statement Hello everyone, I'm new to the great field that is Fourier analysis, and have a question about the way in which to determine if the function is a odd or even function. Given the function, of one period f(x) = { x; 0 <= x < =1, 1; 1 < x < 2, (3 -x); 2 <= x <= 3: Is...
  7. RJLiberator

    Fourier Series and deriving formulas for sums of numerical

    Homework Statement Homework EquationsThe Attempt at a Solution So I am tasked with answer #3 and #4. I have supplied the indicated parenthesis of 8 also with the image. Here is my thinking: Take the Fourier series for |sin(θ)|. Let θ = 0 and we see a perfect relationship. sin(0) = 0 and...
  8. ognik

    MHB Please help me find Fourier series mistake

    Find the Fourier sin series expansion of dirac delta function $\delta(x-a)$ in the half-interval (0,L), (0 < a < L): Now $b_n = \frac{1}{L} \int_0^L f(x)sin \frac{n \pi x}{L}dx $ - but L should be $\frac{L}{2}$ for this exercise... So I would get $ \frac{2}{L} \int_0^L f(x)sin \frac{n \pi...
  9. S

    Solving the Fourier Series of a 2π-Periodic Function

    Homework Statement The odd 2π-periodic function f(x) is defined by f(x) = x2 π > x > 0 -x2 −π<x<0 Find the coefficient bn in the Fourier series f(x) = a0/2 + ∑(an cos(nx) + bn sin(nx)). What are the values of the coefficients a0 and an and why? Homework Equations bn = 1/π ∫...
  10. ognik

    MHB Is the half interval Fourier series for f(x)=x over (0,L) correct?

    Please help me find my mistake - "find the Sine F/series of f(x)=x over the half-interval (0,L)" I get $ b_n=\frac 2L \int_{0}^{L}x Sin \frac{2n\pi x}{L} \,dx $ $ = \frac 2L \left[ x(-Cos \frac{2n\pi x}{L}. \frac{L}{2n\pi x}\right] + \frac {1}{n\pi} \int_{0}^{L} Cos \frac{2n\pi x}{L} \,dx$...
  11. ognik

    MHB Find Fourier series of Dirac delta function

    Hi - firstly should I be concerned that the dirac function is NOT periodic? Either way the problem says expand $\delta(x-t)$ as a Fourier series... I tried $\delta(x-t) = 1, x=t; \delta(x-t) =0, x \ne t , -\pi \le t \le \pi$ ... ('1' still delivers the value of a multiplied function at t)...
  12. ognik

    MHB Help with Fourier series mistake

    Hi - frustratingly I get some problems right 1st time, others just defy me (Headbang) $f(x) = -x, [-\pi,0]; = x, [0,\pi]$ I get $a_0 = \pi$ and $a_n = \frac{-4}{\pi \left(2n-1\right)^2}$ which agrees with the book - but I thought I'd check $b_n$ for practice, it should = 0 according to the...
  13. ognik

    MHB Understanding Fourier Series: Solving a Problem with Sinusoidal Functions

    Hi, appreciate some help with this FS problem - $f(t)= 0$ on $[-\pi, 0]$ and $f(t)=sin\omega t$ on $[0,\pi]$ I get $a_0=\frac{2}{\pi}$ and $b_1 = \frac{1}{2}$, which agree with the book; all other $b_n = 0$ because Sin(mx)Sin(nx) orthogonal for $m \ne n$ But $a_n...
  14. ognik

    MHB How to decompose a fourier series

    Hi, in a section on FS, if I were given $\sum_{n=1}^{\infty} \frac{Sin nx}{n} $ I can recognize that as the Sin component of a Fourier Series, with $b_n = \frac{1}{n} = \frac{1}{\pi} \int_{0}^{2 \pi}f(x) Sin nx \,dx$ Can I find the original f(x) from this? Differentiating both sides doesn't...
  15. ognik

    MHB Fourier series coefficient problem

    Hi - an example in my book shows that FS coefficiants can be arrived at by minimizing the integrated square of the deviation, i.e. $ \Delta_p = \int_0^{2\pi}\left[ f(x) - \frac{a_0}{2}-\sum_{n=1}^{p}\left( a_nCosnx + b_nSinnx \right) \right]^2dx $ So we're looking for $ \pd{\Delta_p}{a_n}...
  16. J

    Fourier series of square wave on Matlab?

    Homework Statement How Can i do this on matlap the question in Attached files Homework Equations The Attempt at a Solution i try a lot but i failed
  17. S

    Finding Fourier Series for (-π, π): Sketch Sum of Periods

    Homework Statement Find the Fourier series defined in the interval (-π,π) and sketch its sum over several periods. i) f(x) = 0 (-π < x < 1/2π) f(x) = 1 (1/2π < x < π) 2. Homework Equations ao/2 + ∑(ancos(nx) + bnsin(nx)) a0= 1/π∫f(x)dx an = 1/π ∫f(x)cos(nx) dx bn = 1/π ∫f(x) sin(nx) The...
  18. K

    Discrete Fourier series derivation

    Hello,*please refer to the table above. I started from x(n)=x(n*Ts)=x(t)*delta(t-nTs), how can we have finite terms for discrete time F.S can anyone provide me a derivation or proof for Discrete F.S.?
  19. M

    Fourier series of periodic function

    Homework Statement Periodic function P=3 f(t) = 0 if 0<t<1 1 if 1<t<2 0 if 2<t<3 a) Draw the graph of the function in the interval of [-3,6] b) Calculate the Fourier series of f(x) by calculating the coefficient. Homework EquationsThe Attempt at a Solution a) in attached...
  20. Aristotle

    Can somebody check my work on this Fourier Series problem?

    Homework Statement Homework Equations The Attempt at a Solution Since P=2L, L=1 ? a_o = 1/2 [ ∫(from -1 to 0) -dx + ∫(from 0 to 1) dx ] = 1/2 [ (0-1) + (1-0) ] = 1/2(0) = 0 a_n = - ∫ (from -1 to 0) cosnπx dx + ∫ (from 0 to 1) cosnπx dx = 0 b_n = - ∫ (from -1 to 0) sinnπx dx...
  21. kostoglotov

    Fourier, square sign wave, f(x)sin(kx) integration

    I'm not sure whether to put this here or in Linear Algebra, if any Mod feels it should go in Linear Algebra I won't mind. I've just been introduced to Fourier Series decompositions in my Linear Algebra text, and I understand all the core concepts so far from the Linear Algebra side of it (a...
  22. kostoglotov

    Verifying the Fourier Series is in Hilbert Space

    The text does it thusly: imgur link: http://i.imgur.com/Xj2z1Cr.jpg But, before I got to here, I attempted it in a different way and want to know if it is still valid. Check that f^{*}f is finite, by checking that it converges. f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x +...
  23. B

    Dirac Delta Function - Fourier Series

    1. Homework Statement Find the Fourier series of ##f(x) = \delta (x) - \delta (x - \frac{1}{2})## , ## - \frac{1}{4} < x < \frac{3}{4}## periodic outside. Homework Equations [/B] ##\int dx \delta (x) f(x) = f(0)## ##\int dx \delta (x - x_0) f(x) = f(x_0)##The Attempt at a Solution...
  24. Remixex

    Should i be worried? (DE Course)

    Hello all, my question is if i should or not be worried about the apparent "missing" (or alternative) content about my DE course. My DE course (and the only one in the list of courses i must take to get my degree) consisted on First order ODEs (Separation of variables, homogeneous, Bernoulli's...
  25. O

    Continuous Time Fourier Series of cosine equation

    Homework Statement Using the CTFS table of transforms and the CTFS properties, find the CTFS harmonic function of the signal 2*cos(100*pi(t - 0.005)) T = 1/50 Homework Equations To = fundamental period T = mTo cos(2*pi*k/To) ----F.S./mTo---- (1/2)(delta[k-m] + delta[k+m]) The Attempt at...
  26. A

    Relationship between Fourier transform and Fourier series?

    What is the relationship between the Fourier transform of a periodic function and the coefficients of its Fourier series? I was thinking Fourier series a special version of Fourier transform, as in it can only be used for periodic function and only produces discrete waves. By this logic, aren't...
  27. R

    Exploring Fourier Series: An=An*sin() & bn=An*cos()

    Consider the following article: https://en.wikipedia.org/wiki/Fourier_series At definition, they say that an = An*sin() and bn = An*cos() So with these notations you can go from a sum having sin and cos to a sum having only sin but with initial phases. Why can I write an = An*sin() and bn =...
  28. ognik

    MHB How do we find A0 in Fourier series for f(x)=x?

    My book says the expansion of $f(x)=x, -\pi \lt x \lt \pi = \sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{n}$, I get double that so please tell me where this is wrong: f(x) is odd, so $a_n=0$ $ b_n=\frac{1}{\pi} \int_{-\pi}^{\pi}x Sin(nx) \,dx = \frac{1}{\pi} [\frac{1}{n^2}Sin(nx) - \frac{x}{n}...
  29. davidbenari

    Is a function really equal to its Fourier series?

    Suppose all Dirichlet conditions are met and we have a function that has jump discontinuities. Dirichlet's theorem says that the series converges to the midpoint of the values at the jump discontinuity. What bothers me then is: Dirichlet's theorem is basically telling us the series isn't the...
  30. B

    Can You Confirm My Fourier Series Calculation for a Square Wave?

    Hello, I think that I have done this correctly, but this is the first problem I have done on my own and would appreciate confirmation. 1. Homework Statement Find the Fourier series corresponding to the following functions that are periodic over the interval (−π, π) with: (a) f(x) = 1 for...
  31. I

    How Can Fourier Series Aid in Solving the Sinc Function Integral?

    Homework Statement Compute ##\int_0^\infty \frac{\sin x}{x}dx## using that ##\frac{\sin x}{x} = \frac{b_0}{2} +\sum_1^\infty b_n \cos nx \; \; , \; \; 0 < x < \pi## with ##b_n = \frac{1}{\pi} \int_{(n-1)\pi}^{(n+1)\pi} \frac{\sin x}{x}dx##. Homework Equations Perhaps the following...
  32. Terocamo

    Fourier Transform of Dirac Comb/Impulse Train

    With Dirac Comb is defined as follow: $$III(t)=\sum_{n=-\infty}^\infty\delta(t-nT)$$ Fourier Transform from t domain to frequency domain can be obtained by: $$F(f)=\int_{-\infty}^{\infty}f(t)\cdot e^{-i2\pi ft}dt$$ I wonder why directly apply the above equation does not work for the Dirac Comb...
  33. Titan97

    Is Fourier Analysis Useful for High School Students?

    Is learning Fourier analysis useful for a high school student? If so, which book should I refer for learning the basics of Fourier analysis? This topic is not in my syllabus. But will it be useful for solving problems? (even if its not, it seems interesting to me). I have learned single variable...
  34. D

    Fourier series solution of wave equation

    Homework Statement Suppose a horizontally stretched string is heavy enough for the effects of gravity to be significant, so that the wave equation must be replaced by ##u_{tt} = c^2u_{xx} - g## where ##g## is the acceleration due to gravity. The boundary conditions are ##u(0,t) = u(l,t) = 0##...
  35. B

    Fourier series expansion problem

    < Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown > hi I've got a problem that I've partially worked but don't understand the next part/have made a mistake? f(x)=0 for -π<x<0 and f(x)=x for 0≤x≤π i got a0=π/4 and an=0 and bn=0 if n is even and 2/n if n...
  36. E

    Calculus Calculus books dealing with Fourier series

    Hey all, I am looking for **calculus**(and not all these books of Advanced Engineerigng Math or etc...) books dealing with Fourier Series ,its expansions , half reange extensions etc... I have found that "Stewart'c calculus" includes a chapter dealing generally with Fourier Series but *not *...
  37. G

    Using a Spreadsheet to determine the Fourier Series

    Homework Statement Use a spreadsheet to determine the F.S. of the data given in Fig 6 See attached for Fig 6 Homework Equations N/A - Use the Fourier Series tool of MS Excel. Tools > Data Analysis > Fourier Series. If you don't have the Data Analysis tool loaded you can load it by going...
  38. R

    How are Fourier Series and Fourier transform are related?

    We know that Fourier series is used for periodic sinusoidal signals and Fourier transform is used for aperiodic sinusoidal signals. But i want to know that Is there any relation present between Fourier Series and Fourier transform ? Also,Can we derive mathematical formula of Fourier...
  39. M

    MHB Where does the Fourier series converge?

    Hey! :o I want to find the Fourier series of the following function : $$g: [-\pi, \pi]\rightarrow \mathbb{R} \\ g(x)=\left\{\begin{matrix} -\frac{\pi+x}{2} & , -\pi \leq x \leq 0\\ \frac{\pi-x}{2} & , 0<x\leq \pi \end{matrix}\right.$$ I have done the following: $$g \sim...
  40. P

    How to define the Integral limits for fourier series

    Homework Statement hello in the college we have Fourier series and i have a problem with the integral limits i add a pdf ( 2 pages only) my question is: how did he get the integral limits from the question the limits are from ##-\pi## to ##-\frac{\pi}{2}## for f(x)=-2 as shown in the first...
  41. R

    What are the limitations/ disadvantages of the Fourier Tran

    I am fond of Fourier series & Fourier transform. In Fourier domain, we can come to know what frequency components are present and the contribution of each component in forming the given signal.But every approach has some advantages and disadvantages.Here, I want to know what are the limitations/...
  42. R

    Algorithm to compute Basis images of an image

    I know from the Fourier Analysis that any signal can be represented as summation of elementary signals i.e. basis functions .Likewise,any image can be represented as summation of Basis images. Is there any available code, or even an algorithm, that would allow me to compute Basis images of an...
  43. Zaent

    Find the value of x of the fourier series defined by

    Homework Statement Homework Equations The Attempt at a Solution http://imgur.com/7TRWjBg I don't really get what it's asking. I don't know how to define a Fourier series when the boundaries for X are between non-multiples of Pi. On top of that, it has one boundary that has 4<x<2Pi. How can...
  44. R

    What Does the Fourier Decomposition of an Image Represent?

    Hello everyone have a look at this video of Fourier Decomposition of an image.also we know that Fourier series is given in the image as...
  45. R

    How Does Fourier Math Apply to Image Processing?

    I am beginer in image processing. Any signal whether it is 1D,2D or any multidimensional signal can be represented using combination of number of sine and cosine waves.Similerly any image can be termed as a sinusoidal function. Fourier series and transform plays vital role in image processing...
  46. T

    Fourier series, is this valid?

    Hi, I have a Fourier problem that i do not know if it is valid to do the calculations like this. The Fourier transform looks like this ## \hat{v}(x,\omega) = \frac{\hat{F}(\omega)}{4(EI)^{\frac{1}{4}}i \omega^{\frac{3}{2}}(\rho A)^{\frac{3}{4}}}\left[ e^{-i\left[\omega^2 \frac{\rho A}{EI}...
  47. T

    Find the actual sum of a fourier series at a given point

    Homework Statement You have series expansions of the function f(x) = 0 from 0 to .5, and 1 from .5 to 1 : the halfrange cosine series, the half-range sine series, and the Fourier series. For each of these series, find the actual sum of the series at x = 0, and x =1/2, and x =1 Homework...
  48. B

    Comp Sci Output format of a matrix in Fortran 90

    Hello, I'm new here and I'm also new in programming. I never did it before and now I have a problem with one of the programs in fortran 90 and I can't figure out how to solve it. Maybe some of you can help me. Many thanks in advance. 1. Homework Statement I need to plot the results of a...
  49. I

    PDE: How to use Fourier Series to express a real function?

    actually have two questions: here we have a Fourier series.. $$f(t) = \sum c_k e^{2\pi ikt}$$ (c is complex) if we're trying to express a real function via Fourier series, and we do it the following way.. Impose condition: $$\overline{c_k} = c_{-k}$$ $$f(t) = \sum\limits_{k= -n}^n c_k e^{2\pi...
  50. Coffee_

    Fourier series, Hermitian operators

    (First of all I never saw Hilbert spaces in a mathematical class, only used it in intro QM so far, so please don't assume I know that much when answering.) Let's consider the Hilbert space on the interval [a,b] and the operator ##\textbf{L} = \frac{d^{2}}{dx^{2}} ##. Then ##\textbf{L}## is...
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