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. J

    Fourier Series complex coefficients

    I have been trying to follow how the complex Fourier coefficients are obtained; the reference I am using is at www.thefouriertransform.com. However I am unable to follow the author's working exactly and wondered if anyone could help me see where I am going wrong. First, I understand that the...
  2. O

    Fourier series representation for trigonometric and complex form base

    May i know to obtain Fourier series representation for trigonometric and complex form base on magnitude spectrum and phase spectrum?? what i found is that to get trigonometric form is from phase spectrum, but i don't know how.. can anyone help
  3. P

    Plotting an Exponential Fourier Series

    I'm having some problem in determining the phase of an exponential Fourier series. I know how to determine the coefficient which in turn gives me the series after I multiply by e^-(jωt) I can determine the amplitude by dividing the coefficient by 2 |Dn| = Cn/2 Now my question is how to...
  4. M

    Question on fourier series convergence

    hey pf! if we have a piecewise-smooth function ##f(x)## and we create a Fourier series ##f_n(x)## for it, will our Fourier series always have the 9% overshoot (gibbs phenomenon), and thus ##\lim_{n \rightarrow \infty} f_n(x) \neq f(x)##? thanks!
  5. A

    Relationship between Fourier series Coefficients and F Transform

    Homework Statement "Suppose x[n] is a DT (discrete time) periodic signal with fundamental period N. Let us define x_{n}[n] to be x[n] for n ε {0, 1,2, ... , N-1} and zero elsewhere. Denote the Fourier transform of x_{n}[n] with X_{n}[e^jω]. How can one find the Fourier Series coefficients...
  6. J

    MATLAB Complex Fourier Series using Matlab

    Hello, I have a problem synthesising the complex Fourier series using Matlab. The time domain periodic function is: -1, -1.0 ≤ t < -0.5 1 , -0.5≤ t <0.5 -1, 0.5 ≤ t < 1 The single non zero coefficient is: Cn = \frac{2}{\pi n}, Co is 0 (average is 0). f(t)= \sum Cn...
  7. J

    Representing a real periodic valued function with Fourier series

    Homework Statement Hey, the question i have been given reads: By a simple change of variables, show that if g(x) is a periodic real valued function with period L it can be represented as g(x)~ ∑∞n=-∞ An exp(-2\piinx/L) where the complex constants An are given by LAm =[L/2,-L/2]...
  8. T

    Extracting a fourier series of pure tones from a signal

    I believe this is an error minimization problem so I am trying to solve the following equation Min((∑ ( (S(t) - A cos(b t + C)))^2 ) Where S(t) is the input signal, t is time and I will sum over t, A is the amplitude, b is radians per second (frequency), and C is the phase angle. I...
  9. N

    Is This Fourier Series an Odd or Even Function with a Period of 4s?

    Homework Statement For he following Fourier series, which of the answers correctly describes the following function y(t) = 2 - \stackrel{1}{π}∑1inf1/nsin(n*πt/2) a) odd function, period = 2 s b) Even function, period = 2s c) Odd function, period = 4s d) Even functio, period = 4s...
  10. N

    Fourier series / calculate power over resistor

    Homework Statement V(t) = 4 for 0<t< 1 and 0 for 1<t<3 and repeats itself for all t (negative and positive) Find the first 5 harmonics of the Fourier series in cosine form and find the power if this is the voltage over 100 ohm resistor The Attempt at a Solutionpower = d_dc ^2 / R + .5sum...
  11. J

    Other definition for fourier series

    Is correct to define Fourier series like: f(t)=\sum_{k=0}^{\infty}a_k \cos \left (\frac{2 \pi k t}{T} \right ) + b_k \sin \left (\frac{2 \pi k t}{T} \right ) Where ak and bk: a_k=\frac{1}{T} \int_{-T}^{+T} f(t) \cos \left (\frac{2 \pi k t}{T} \right ) dt b_k=\frac{1}{T}...
  12. N

    How Do You Calculate Fourier Series Coefficients for a Piecewise Function?

    Homework Statement x(t) = 4 for 0 <x<1 and 0 otherwise and this process repeats for all values including negative. find X_0 and X_n and find the first 6th harmonics of the Fourier series in cosine form Homework Equations The Attempt at a Solution x_0 = 4/3 x_n =...
  13. M

    Exploring Alternating Solutions in Fourier Series

    Homework Statement This is a general question, no real problem statement and is connected to solving Fourier series. You know that to solve it, you need to find a_{n}, a_{0} and b_{n}. Homework Equations When solving the above mentioned ''coefficients'' you can get a solution with sin or...
  14. dexterdev

    Which of the signals is not the result of fourier series expansion?

    Homework Statement Which of the signals is not the result of Fourier series expansion? options : (a) 2cos(t) + 3 cos(3t) (b) 2cos(\pit) + 7cos(t) (c) cos(t) + 0.5 Homework Equations Dirichlet conditionsThe Attempt at a Solution From observation, I thought all are periodic and so must be...
  15. K

    Fourier Series for a Square-wave Function

    Homework Statement Consider the square wave function defined by y(t) = h (constant) when 0 ≤ (t + nT) ≤1, y(t) = 0 elsewhere, where T = 2 is the period of the function. Determine the Fourier series expansion for y(t). Homework Equations Fourier Analysis Coefficients The Attempt...
  16. D

    Fourier Series from m=1 to infinity

    Simple question; Why isn't it \sum am (from m=1 to infinity) Thanks in advance.
  17. P

    Find the Fourier series solution to the differential equation

    Find the Fourier series solution to the differential equation x"+x=t It's given that x(0)=x(1)=0 So, I'm trying to find a Fourier serie to x(t) and f(t)=t, and I'm know it must a serie of sin... So here's my question...the limits of integration to the Bn, how do I define them? Will...
  18. N

    Fourier series of a periodic function not starting at x=-L

    Homework Statement In "oppgave 4" http://www.math.ntnu.no/emner/TMA4120/2011h/xoppgaver/tma4120-2010h.pdf you have a periodic function which is NOT periodic from ##x=-L=-\pi## to ##x=L=\pi##, but at ##x=0## and ends at ##x=2 \pi=2L##. The formulas I have (like these...
  19. N

    Why Is the Coefficient Doubled in the Fourier Sine Series Calculation?

    Homework Statement https://wiki.math.ntnu.no/_media/tma4120/2013h/tma4120_h11.pdf Check out the solution to problem 4b) My question is: Why do they set ##b_n = \frac{2}{\pi} \int_{0}^{\pi}(...)dx## instead of ##b_n = \frac{1}{\pi} \int_{0}^{\pi} (...)dx##? Ie, why did they multiply the...
  20. N

    Fourier series: complex vs ordinary method

    Hi! Which is the better method for finding Fourier expansions of a function? The ordinary one (find a_0, b_n and a_n with separate integrals), or the one which uses complex numbers (just find c_n)?
  21. G

    Fourier series of complex function

    Homework Statement Hello guys, I have problem with the Fourier series, since we had only one lecture about it and I cannot find anything similar to my problem in internet. should we consider for the first f(x+1) integrated from -1 to 0 ? http://img819.imageshack.us/img819/3508/wbve.jpg when...
  22. U

    Complex Fourier series of sin (t)

    In finding Cn, I arrived at a different answer. I got an extra factor of (1/i) instead, which came when you do the integral of each exponential with respect to t; so you get a factor of 1/i(1-n) and 1/i(1+n) respectively.. Did they intentionally leave that out?
  23. D

    How Do You Compute the Fourier Series for |sin(x)| Over the Interval (-1,1)?

    Hello, Find the Fourier serie of f(x)=|sin(x)| on the interval (-1,1) I'm just a little confused, does that mean that I have to integrate from -1 to 1 to find the coefficients ? Because the formula of the coefficients is in terms of the period T, for this function the period is pi. Or do I...
  24. U

    Continuous Periodic Fourier Series - Coefficients

    Homework Statement In the dirac notation, inner product of <f|g> is given by ∫f(x)*g(x) dx. Why is there a 1/∏ attached to each coefficient an, which is simply the inner product of f and that particular basis vector: <cn|f>? Homework Equations The Attempt at a Solution
  25. N

    What is the point of Fourier Series and what is it used for?

    I recently had an asignment where i calculated the Fourier series coefficients for f= 1+t for t= -1 to 0 f= 1-t for t=0-1 basically triangle looking. And as i summed more and more coefficients my function started looking more like this triangle (which was really cool). My question...
  26. T

    Fourier Series of Full Wave Rectifier

    Homework Statement Determine the Fourier series for the full-wave rectifier defined as f(t) = sinωt for 0 < ωt < pi -sinωt for -pi < ωt < 0Homework Equations The Attempt at a Solution This looks like an even function, so bm = 0 Ao = 1/pi∫sinωt from 0 to pi = 1/pi(-cos(ωt))/ω) from 0 to...
  27. D

    Fourier Series - Asymmetric Square Wave

    Good morning everyone, I am taking a signals and systems course where we are now studying the Fourier series. I understand that this is for signals that are periodic. But I get hung up when determining the Fourier coefficients. In the video by Alan Oppenheim, he derives the equation for...
  28. P

    Proof of Fourier Series Coeffecients

    Homework Statement #35 on this page Homework Equations Integral of a series can be assumed to be the sum of integrals The Attempt at a Solution Picture of Work I am not sure where to proceed from here, advice?
  29. S

    Why Doesn't the Fourier Series of a Dirac Comb Match Pointwise Values?

    http://en.wikipedia.org/wiki/Dirac_comb Please have a look at the Fourier Series section, and its last equation. Let T = 1. After expanding the Equation x(t) = 1 + 2cos(2∏t) + 2cos(4∏t) + 2cos(6∏t) ... Now this does not give the original Dirac Comb. Eg: at t = 1/2 x(1/2) = 0 But RHS =...
  30. N

    MHB What Properties of Fourier Series Are Revealed by This Equation Transformation?

    Is there some properties I should be aware of? after making the relevant substitutions, I ended up with $2 = 1 + \sum\nolimits_{m=0}^\infty \frac{4}{(2m+1)\pi}\sin(\frac{(2m+1)\pi}{2})$ but I can't get past this
  31. M

    What Do Fourier Series Actually Reveal About Signals?

    I've just started learning Fourier series and I'm having trouble understanding it. What do they actually do? And what does the amplitude-frequency show me? I'm asking as a rookie in signal analysis, so if you could explain it to me as simple as you can it will be of great help. Thanks!
  32. A

    Fourier Series and Absorption Questions?

    Sorry if I am posting in the wrong place. I'm really interested in the Fourier series, but I'm not an expert on it yet. I am very well aware yoy can do it with sound waves, but can you manipulate any other waves? What about light waves? And for absorption, how can you measure the...
  33. A

    Fourier Series and Absorbtion questions?

    I have to do a science fair, and I am really interested in physics. I spent forever trying to think of a topic, and I got myself stuck doing something with waves. After weeks of thinking of an expirement that will compete well without being extremely difficult, I came up with one. Does using...
  34. B

    Saw-tooth Wave and Fourier Series amplitude of oscillation

    Homework Statement An oscillator with free period \tau is critically damped and subjected to a force with the saw-tooth form \F(t)=c(t-n\tau) for (n-0.5)\tau<t<(n+0.5)\tau for each integer n. Find the amplitudes a_n of oscillation at the angular frequencies 2\pi n/\tau if c is a...
  35. E

    Justify an equality involving hyperbolic cosine and Fourier series

    Homework Statement The problem: Justify the following equalities: \cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2} I am trying to figure out how to start this. When I insert the Euler identity of \coth (using...
  36. B

    MATLAB Need help plotting Fourier series in matlab

    Hi all my first post as I need to seek help! I have just learned some simple Fourier series stuff and would like to be able to plot my answers in matlab. Assuming this is correct I was wondering if someone would be able to walk me through plotting this equation in Matlab...
  37. A

    Finding the Fourier Series of a periodic rectangular wave

    Homework Statement The problem/question is attached in the file called "homework". In the third signal (the peridic rectangular wave), I am requested (sub-question b) to find the Fourier series of the wave. Homework Equations The file called "solution" presents a detailed solution to the...
  38. E

    Odd & Even Functions (was thread Fourier Series )

    Odd & Even Functions (was thread "Fourier Series") Homework Statement determine if the functions below are odd even or neither: a) f(x)=x^2+2 b) f(x)=(x^2+2)tan(x^2) c) f(x) = (x^2+2)sin(x)tan(x^2)Homework Equations even - f(x) = f(-x) odd - f(-x)=-f(x) The Attempt at a Solution I've managed...
  39. A

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

    Homework Statement Determine the Fourier series for the periodic function of period 2∏ defined by: -2 when (-∏ ) ∠ x ∠ (-∏/2) f(x)= 2 when ( -∏/2) ∠ x ∠ (∏ /2) -2 when (∏/2) ∠ x ∠ (∏) how to start i?. I have already drawn it but what next. thank you...
  40. H

    What is the Fourier Series for (sin(x))^2 on the interval [-π, π]?

    Hey there! I'm trying to calculate the Fourier Series for sin2x on [-π, π] For a0 I found 1/2. (By determining the average value of the function on the interval) Since sin2x is even, I know that bn = 0. Now, for an.. The following link shows the integral I used to try to evaluate an...
  41. A

    Square Pulse Train Fourier Series help?

    Square Pulse Train Fourier Series help?? Homework Statement problem+directions below: Homework Equations ω=2\pif β=\frac{2\pi}{\lambda} The Attempt at a Solution Since the problem asks to make all time-dependent sinusoidal functions deal with x-direction, i don't think i need to...
  42. S

    Understanding the Convergence of Fourier Series for Periodic Functions

    Hey. I'm looking for a proof of: Theorem: If f \in C^1(\mathbb{T}), then the Fourier series converges to f uniformly (and hence also pointwise.) I have looked around for it, googled, etc, but I only found proofs which used theorem they did not prove. (Or I misunderstood what they said.) I'd...
  43. T

    Fourier Series: A_0=0, a_n=0, b_n=2/(n∏); Period p=2

    Homework Statement Fourier coefficients: A_0=0, a_n=0, b_n=2/(n∏) ; period p=2 Homework Equations Fourier series The Attempt at a Solution Attempt was not good enough!
  44. D

    Issue with Fourier Series of an even function.

    Hi, I wish to obtain the Fourier series of the signal in red (please see attached figure fig1_sine_plots.png). Basically, it is a full-wave rectified 3f sinusoid, where f = 50Hz. The blue signal represents a sinusoid with frequency f = 50Hz. In the following equations (please see attached...
  45. FeDeX_LaTeX

    Obtaining One Fourier Series from Another

    Homework Statement (a) On (-π,π), find the Fourier series of f(x) = x. (b) Hence, or otherwise, find the Fourier series of g(x) = x2 (c) Hence, show that \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90} Homework Equations f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n...
  46. Q

    Graphing a Fourier Series on a TI-84 Plus not working so well

    Hi! I was wondering how you would graph a Fourier series for a square wave that periodic. My textbook gave something proper that I would expect (just google 'Square Wave Fourier Series Expansion') while my TI - 84 gave a image like the one somewhere in this thread. It would be nice if anybody...
  47. D

    The Fourier Series: Exploring Its Use, Purpose & Benefits

    The Fourier series can be used to represent an arbitrary function within the interval from - π to + π even though function does not continue or repeat outside this interval. Outside this interval the Fourier series expression will repeat faithfully from period to period irrespective of whether...
  48. D

    Exploring Fourier Series: Uncovering the Mysteries of an Arbitrary Function

    The Fourier series can be used to represent an arbitrary function within the interval from - π to + π even though function does not continue or repeat outside this interval. Outside this interval the Fourier series expression will repeat faithfully from period to period irrespective of whether...
  49. phosgene

    Finding the value of pi/4 using Fourier series

    Homework Statement The function f(x) is defined by: f(x) = -1 when \pi < x <0 and 0 when 0<x<\pi Show that \sum^{∞}_{0}\frac{(-1)^n}{2n+1}=\frac{\pi}{4} Homework Equations Fourier series for a function of period 2\pi = a_{0} + \sum^{∞}_{1}a_{n}cos(nx) + b_{n}sin(nx)...
  50. A

    Writing Fourier Series for Open and Closed Intervals

    In a rigorous mathematical course I am talking, it seems to make a difference when I am given a function f and need to write its Fourier series, whether it is defined on [0,2∏] or [0,2∏). What difference does it make for my series whether it is an open or a closed interval?
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