What is Fourier analysis: Definition and 143 Discussions

In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.
Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis often refers to the study of both operations.
The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis.

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  1. N

    Fourier Analysis of a sound signal using Mathematica

    Homework Statement I am trying to construct a Mathematica notebook that will be able to import sound in the form of a .wmv file and then create the frequency spectrum for a given time interval. Homework Equations I managed to complete this part, though I am trying to figure out: a)...
  2. A

    A question on orthogonality relating to fourier analysis and also solutions of PDES

    a question on orthogonality relating to Fourier analysis and also solutions of PDEs by separation of variables. I've used the fact that the following expression (I chose sine, also cosine works): \int_{0}^{2\pi}\sin mx\sin nxdx equals 0 unless m=n in which case it equals pi in...
  3. B

    How can I interpret the superpostion of Fourier analysis?

    Hello guys. thanks for reading my thread first. I have been studied classical mechanics and quantum mechanics a little and after that, I got a kind of feeling "I have to study Fourier analysis(FA) again". So I have been studying FA again. Here're my question. 1. From my viewpoint...
  4. S

    Fermat's Principle, Fourier Analysis, Standing waves, Normal Modes

    Can someone provide me a link that explains and provides a proof for the following principles: 1. Fermat's Principle that light always takes the path that minimizes the time taken 2. Solution to a Fourier Series and why all periodic motion can be represented as an infinite sum of sines and...
  5. S

    What is the Sampling Period Range for Recoverable Signals?

    Homework Statement The signal y(t) is generated by convolving a band limited signal x1(t) with another band limited signal x2(t) that is y(t)=x1(t)*x2(t) where: --> X1(jω)=0 for|ω| > 1000Π --> X2(jω)=0 for|ω| >2000Π Impulse train sampling is performed on y(t) to obtain: --> yp(t)=...
  6. R

    How Does Fourier Analysis Explain Frequency Changes in Time Series Data?

    Dear friends, I generated a time series graph showing some modeled versus measured data (precipitation data)...The results show that the modeled data are sharper than the mesured data (more rounded on top)...Based on that, how can I explain the frequency content changes based on Fourier...
  7. S

    Help needed for Fourier analysis of a triangular wave

    Homework Statement Fourie analysis for a boost converter Where d = duty cycle, A = amplitude and t = time Homework Equations Fx = (2At/d) for 0<t<d/2, -A+(2A/(1-d))*(1-t-d/2) for d/2<t<(1-d/2); (2A/d)*(t-1) for 1-d/2 <t<1. Where d = duty cycle, A = amplitude and t = time The...
  8. E

    Finding x[n] from Continuous Fourier Transform with Given Properties

    Hello everyone(my first post here), I hope I have posted in the right section... Homework Statement Given x[n] is a discrete stable(absolutely summable) sequence and its continuous Fourier transform X(e^{j\omega}) having the following properties: x[n]=0, \ \ \ \forall n<1 and...
  9. N

    Fourier analysis of a Lorentzian/Cauchy/Breit–Wigner distribution

    So I'm supposed to do this but is it just me or is it too hard to do this analytically? (I put it into wolfram online integrator and he couldn't do it) I don't need it very accurate so are there any approximations to this distribution that I could use to make it easier? Anyone have any ideas of...
  10. C

    Extending an Isometry in Schwartz Space to Moderately Decreasing Functions

    Homework Statement let S(R) be the schwartz space, M(R) be the set of moderately decreasing functions, F be the Fourier transform Suppose F:S(R)->S(R) is an isometry, ie is satisfies ||F(g)|| = ||g|| for every g in S(R). Show that there exists a unique extension G: M(R)->M(R) which is an...
  11. M

    Book reccomendations on Fourier Analysis

    I was wondering if anyone has any recommendations for a Fourier Analysis textbook. I have Stein & Shakarchi's Fourier Analysis textbook, but ideally I'd like to have one that takes advantage of some of the analytic machinery that I know that Stein & Shakarchi doesn't assume. I have a basic...
  12. D

    Good intro to fourier analysis?

    Hi, I am starting in a Ph.D. program in math next fall and the prerequisites for the first-year graduate course sequences included basics of Fourier analysis. The only thing I know about it is that you calculate a projection of a function on a certain infinite dimensional subspace, so I do...
  13. T

    Fourier Analysis - uniform convergence on (-inf, inf)?

    I have a question that I just don't know how to go about. " Let Fn = x/(1+(n^2)(x^2)) where n=1,2,3,... show that Fn converges uniformly on (-infinity,infinity)" To be honest, I don't even know where to start. Is this a series? How would I solve this. Would the Abel's test apply?
  14. O

    Why Does My Fourier Transform Magnitude Plot Look Incorrect?

    Hallo all, i have a problem with Fourier analysis and i really hope yoou can help me in this forum. i have been trying to find the reason for my problem since many weeks but i could not. Well, i have this Equation:s(t) = (1/(4*(Pi*t)^(3/2))) * Exp[-3)/(4*t)] and i need to find the...
  15. W

    Fourier Analysis: Finding Coefficients for f(t)=1

    Homework Statement I need to find the Fourier coefficients for a function f(t)=1 projected onto trigonometric polynomials of infinite order Homework Equations Equation finding the first coefficient, the constant term: The Attempt at a Solution So I feel quite stupid because this...
  16. N

    Fourier Analysis, definition of convolution

    I having a hard time understanding an aspect of the definition of the convolution of two functions. Here is the lead up to its definition... It goes on to discuss what the observed distribution h(z) will be if we try to measure f(x) with an apparatus with resolution function g(y). And tries...
  17. X

    Questions related to Fourier Analysis

    Dear all I have recently taken up the study of Fourier analysis. My background knowledge is limited - some basic notions of analysis, including the Riemann integral, as well as uniform and pointwise convergence of series of functions. These are not exactly homework problems, but questions...
  18. O

    How important is fourier analysis to Mechanical Engineering?

    I'm an Engineering Physics Major senior and I'm going to get my Masters in Mechanical Engineering. I have an internship in the Polymer and Coatings Department Analyzing Data. I am mostly going to be doing Fast Fourier Transforms and studying the data. The problem is I don't really know...
  19. O

    Fourier Analysis: Discrete Series & Transform

    Hello all, I'm looking for a good textbook, tutorial or anything like that about Fourier analysis: the discrete series and the Transform (very important - for a project about waves). I don't know a lot about Fourier analysis, so this textbook should start quite from the beginning. What I do...
  20. H

    Fourier Analysis: What is the Difference with (w-wo)?

    Hi all. I am learning a numerical method that involves Fourier transform. As far as I know, I think Fourier transform is tool to find the frequency spectrum of a signal. And the usual form shall be "Integrate from negtive infinity to positive infinity, f(x)*exp(i*w*x)dx" However, when i...
  21. C

    Why is the Fourier Transform in QFT Divided by (2pi)?

    is Fourier analysis in qft just used for going from a position wavefunction to a wavefunction described by the wave vector (k)? also why is the integral divided by [2(pi)]^n where n is the number of dimensions and how do you know when to divide the integral by the 2(pi) factor or not?
  22. E

    Expanding Wavefunction in Infinite Well: Fourier Analysis

    I have an infinite well from -a to with a particle in its ground. The initial wavefunction is then \psi(x) = u_1^+(x;a) = cos(\pi x/ 2a)/\sqrt{a} for |x| < a. In order to get the wavefunction for this particle when box that is instantaneously expanded to [-b,b] should I apply Fourier...
  23. S

    Will studying fourier analysis prepare one for string theory and QP?

    hey does anyone know if studying Fourier analysis is going to aid in physics, particularly in string theory or quantum physics?
  24. H

    Fourier analysis of my experimental data

    hi Im doing a project about chaos theory. Over the past few weeks I've built a Lorenzian waterwheel (its a waterwheel with buckets with holes in them, it shows chaotic behaviour.) One of the aims of the experiment was to try and plot the Lorenz attractor. The lorenz attractor is a strange...
  25. T

    A question on Fourier Analysis

    it's a curiosity more than a HOmework, given the integral: \int_{-\infty}^{\infty}dx e^{if(x)+iwx}=g(w) Where g(w) can be viewed as the Fourier transform of exp(if(x)) then my question is if we can prove g(w) satisfies the ODE: -if'(x)\frac{\partial g(w)}{\partial w}+wg(w)=0 for...
  26. R

    Fourier Analysis: Pointwise vs. L^2 Convergence

    I'm just taking Calculus 4 this semester, where part of it is also Fourier analysis. When I was browsing a little bit about the subject I found out that there are several different approaches and so I'm a bit confused now. So this is how I understand it, correct me if I'm wrong: There...
  27. M

    Text on Discrete Fourier Analysis & FFT

    Does anyone know of any good texts that cover Discrete Fourier Analysis and the Fast Fourier Transform? *I don't know if this belongs here in the HW help or in General Math section.
  28. E

    Discrete-time Fourier Analysis

    I am trying to compute the DTFT of x[n]=n(.5)^n u[n]. This is my work so far. Is this correct?
  29. C

    Experiment utilizing Fourier Analysis

    Can anyone help me devise some kind of physics related question (ie. How does resistance varry with area?) involving light or sound where the frequency spectum varries when Fourier Analysis is performed? Would "How does the frequency spectrum of a light bulb varry with the voltage across it."...
  30. C

    Fourier Analysis: Signal Representation & Non-Monochromatic Light/Sound

    From what I understand, I can use Fourier Analysis to represent a periodic signal using a sum of sine waves. However, isn't this just a mathematical tool? Can I take any non-monochromatic light source and use Fourier Analysis to break it into a sum of the physically meaningfuly frequencies it's...
  31. C

    Using Fourier Analysis in a Senior Year Project: Any Experiment Ideas?

    I'm supposed to do an "investigation into a physics-related question." for my senior year project. I want to do something realted to Fourier Analysis but I need some sort of experiment. Any ideas? How hard would a computer model be? Also, would this topic fall more under math?
  32. M

    Finding a PC Application for Audio Recording & Fourier Analysis

    Can anyone recommend an application for PC that can take an audio recording, perform a Fourier transform, and return the audio recordings' frequency spectrum AND phase information? I can find plenty of programs that return the frequency spectrum, but I've had no luck finding one that...
  33. C

    Fourier Analysis: Independent Study for Senior in HS

    I have a chance to do an Independent Study in place of a regular class (senior, high school), and with my physics teacher as an advisor I've thought of the possibility of doing some work with Fourier Analysis. My problem is I don't know at what level this is normally taught at and what...
  34. quasar987

    Proving Continuity of a Fourier Series Function

    I'm puzzled and don't know where to begin with this question; it goes like "Consider the function f:R²-->R defined by f(x,y) = \sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}sin(nx)sin(ny) Show that f is continuous." Any hint? .
  35. quasar987

    Unraveling the Mystery: Solving a Tricky Fourier Analysis Problem

    This seemingly not-so-harsh math problem has me stumped. I tried solving it every free minute I had this weekend but no trails or any combination of them led me anywhere happy. The little ba$tard goes as follow: "Consider f: [-\pi,\pi)\rightarrow \mathbb{R} a function (n-1) times...
  36. Z

    Fourier analysis and prob. distributions?

    Ok, this might seem like either a really idiotic question or a really profound one. Consider a probability distribution. I'm picturing a normal distribution, is it meaningful to be able to build up a final probability distribution from a set of narrower probability distributions? Ok...
  37. J

    Using Fourier analysis to find frequency-amplitude spectrum?

    The signal is from a voltage supply. I see lots of pages on the internet about this, such as this one, which shows what the magnitude spectrum looks like for a square wave with an arbitrary number of co-efficients. But how would I actually create that graph myself?
  38. Oxymoron

    Finding Embedded Harmonics w/ Fourier Series & Integrals

    1) Why can't f(x) = 1, -&inf; < x < &inf; be represented as a Fourier integral? Is it because it must be defined on a finite interval? 2) Could someone tell me how you find the embedded harmonics in a given periodic function using Fourier series and integrals? Either a quick demonstration or...
  39. B

    Exercise from basic Fourier Analysis

    I really need help with this exercise (it's from a course in basic Fourier analysis). It consists of two parts: (i) Let s_0 = 1/2 and s_n = 1/2 + \sum_{j=1}^{n}\cos(jx) for n \geq 1 . By writing s_n = \left(\sum_{j=-n}^{n}e^{ijx}\right)/2 and summing geometric series show that...
  40. H

    Fourier Analysis in QM: Practical Applications & Potential Pitfalls

    Myself and a user on another board have come to the following hurdle we can't overcome: Mathematically we represent an arbiarty matter wave as a superposition of plane waves; using the theory of Fourier analysis. We can write an arbitary wavepacked as a Fourier integral of the form: $\psi...
  41. S

    Is fourier analysis unavoidable?

    From the beginning of quantum mechanics the main tool for mathematical analysis was Fourier anaysis. (Heisenberg intoduced his famous matrices using Fourier analysis). Fourier analysis has a non-local character-that the basis functions extend infinitely in space and time. What would be the...
  42. Loren Booda

    Fourier analysis of the prime distribution

    Does it make sense to Fourier-analyse pi(n) for finding patterns toward a comprehensive prime-predictive formula?
  43. E

    Fourier Analysis of Angular Momentum Operator

    Okay, if I want to do a Fourier Analysis of a wavefunction, I can use the following transform pairs for real space and momentum space. &Psi;(x) = (2&pi; hbar)^(-1/2) * &int; dp &Phi;(p) exp(ipx/hbar) &Phi;(p) = (2&pi; hbar)^(-1/2) * &int; dx &Psi;(x) exp(-ipx/hbar) So, what I want to...
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