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. B

    I Fourier analysis and the sinusoidal plane wave

    hey So Fourrier transform is ##f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d\omega## with ##F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt## Question 1 - The Fourier mode for the continuous case is ## \frac{1}{2 \pi} F(\omega) e^{i \omega t}##, is...
  2. 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...
  3. TheBigDig

    Need Help with Fourier Coefficients for 1-t Function?

    Homework Statement [/B] I am looking for help with part (d) of this question 2. Homework Equations The Attempt at a Solution I have attempted going through the integral taking L = 4 and t0 = -2. I was able to solve for a0 but I keep having the integrate by parts on this one. I've tried it...
  4. 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...
  5. TheSodesa

    How to Fourier-transform e^(-2|t|)?

    Homework Statement Determine the Fourier-transfroms of the functions \begin{equation*} a) f : f(t) = H(t+3) - H(t-3) \text{ and } g : g(t) = \cos(5t) f(t) \end{equation*} and \begin{equation*} b) f : f(t) = e^{-2|t|} \text{ and } g : g(t) = \cos(3t) f(t) \end{equation*}Homework Equations The...
  6. TheSodesa

    Number of subdivisions in a Riemann integral (DFT)

    Homework Statement This is a combination of two questions, one being the continuation of the other 3) Calculate the DFT of the sequence of measurements \begin{equation*} \{ g \}_{k=0}^{5} = \{ 1,0,4,-1,0,0 \} \end{equation*} 4a) Draw the DFT calculated in question 3 on the complex plane. 4b)...
  7. TheSodesa

    Amplitude and phase spectra from fundamental frequency?

    Homework Statement Let \begin{equation*} f(t) = 2 + \cos\left( 3t - \frac{\pi}{6} \right) + \frac{1}{4}\cos\left( \frac{1}{2}t + \frac{\pi}{3} \right) + \sin^2(t) \end{equation*} Determine the period ##T## and fundamental frequency ##\omega_0## of ##f## and draw images of its amplitude and...
  8. C

    Fourier Analysis: Inspect Waveforms in FIGURE 1

    Homework Statement b) state by inspection (i.e. without performing any formal analysis) all you can about each of the periodic waveforms shown in FIGURE 1 in terms of their Fourier series when analysed about t = 0 Homework Equations 3. The Attempt at a Solution Hi could someone please be...
  9. 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...
  10. FeDeX_LaTeX

    I Discrete Convolution of Continuous Fourier Coefficients

    Suppose that we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{R}, whose continuous Fourier coefficients \hat{f} are known. The convolution theorem tells us that: $$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$ where \ast denotes the...
  11. SU403RUNFAST

    Sketching a periodic function and Fourier analysis

    Homework Statement So i have a function f(x)=x^2 that is periodic -a<x<a and need to sketch this function from -3a<x<a. I know how to find the Fourier coefficients though. Homework Equations f(x)=x^2 sketch it periodically The Attempt at a Solution I know that a function is only periodic...
  12. RJLiberator

    Fourier Analysis for |x| and Proving a Series Convergence

    Homework Statement Consider a 2pi-periodic function f(x) = |x| for -pi ≤ x ≤ pi a) Compute the Fourier series of the function f. b) Prove that (from n=1 to n=infinity)∑ 1/(2k-1)^2 = pi^2/8.**note all "sums" from here on out will be defined from n = 1 to n=infinity Homework EquationsThe Attempt...
  13. ognik

    Fourier analysis of wave packet

    Homework Statement Assume ## \phi(k_x ) = \sqrt2 {\pi}## for ## \bar{k}_x - \delta \le k_x \le \bar{k}_x + \delta##, and ##= 0## for all other values of ##k_x##. Calculate ##\psi(x, 0)##, and show that ## \Delta x \Delta k_x \approx 1 ## holds if ## \Delta x## is taken as the width at half...
  14. S

    Solving phi-fourth theory using Fourier analysis

    The equation of motion of ##\phi^4## theory is ##(\partial^{2}+m^{2})\phi = -\frac{\lambda}{3!}\phi^{3}##. Why can't this equation be solved using Fourier analysis? Can't we simply write the equation in Fourier space and take it from there?
  15. T

    Electrodynamics Fourier Analysis (Fouriers Trick)

    Homework Statement Two infinitely grounded metal plates at y=0 and y=a are connected at x=b and x=-b by metal strips maintained at a constant potential V. Find the potential inside the rectangular pipe.Homework Equations Laplaces EquationThe Attempt at a Solution I posted a photo of what I've...
  16. 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...
  17. D

    What are the best textbooks for learning Fourier analysis and its applications?

    Hello all, I'm a third year university physics major. I haven't read much on Fourier analysis however I have had been introduced to it through an oscillations and waves class. My professor was saying that it can be applied to many different areas and is extremely helpful tool to have under your...
  18. R

    Wavelets vs disadvantages of Fourier Analysis and Synthesis?

    I am a beginner. The Fourier series, Fourier Transform and it's inverse play very important role in Fourier Analysis and Fourier Synthesis. I have read that Fourier transform is localised in only frequency domain.Also,it contains information about the signal in phase and frequency spectrum...
  19. Z

    Fourier Analysis vs Topology: Which is More Useful for Physics?

    Hey guys, long story short. I am completing my Math minor this semester and need to decide on whether Topology or Fourier Analysis. I am an undergraduate physics major and neither one of those classes is required for my B.S. in physics. So what do you guys think, Topology or Fourier Analysis?
  20. J

    Understanding the Significance of Fourier Analysis in Signal Processing

    Ok so this isn't a homework question per se, but I'm currently writing a report on Fourier Analysis but a bit stuck as to what the results can actually help with. I realized that I don't grasp how a Fourier Transform can be used. In the experiment we recorded the signal created by a remote...
  21. T

    Fourier analysis and continuous spectra

    So I've been self-studying from Griffiths Intro to QM to get back in shape for graduate school this fall, and I guess I'd just like some confirmation that I'm on the right track... So while I am sure there are many other applications, the one I am dealing with is eigenfunctions of an operator...
  22. A

    Engine Test Bed Fourier Analysis

    I have done a test on a 4 piston test engine which is expected to exhibit torsional resonance at 800RPM and a vertical translational resonance at 1200RPM. The data we gathered from the test bed machine was as follows: Theta | Signal 0 | -5 60 | -1 120 | 7 180...
  23. F

    Fourier Analysis of Sawtooth Signal with To = 1

    Homework Statement Sawtooth signal with To = 1, at T=0, x = 0, at T=1, x =1 verify: a_{k} = \left\{\begin{matrix} \frac{1}{2}, for k=0; & \\\frac{j}{2\pi k}, for k \neq 0; & \end{matrix}\right. Homework Equations \frac{1}{T_{0}} \int_{0}^{T_{0}} te^{-j(2\pi/T_{0}))kt}dt The Attempt...
  24. S

    What is Fourier Analysis and its Applications?

    Is this college or graduate math? Is it pure or applied math? Is it useful for physics and electrical engineering?
  25. PsychonautQQ

    Deriving of the constants in Fourier Analysis

    Homework Statement Wow LaTex fail... anyone know how i make this look like not ****? I had it looking all pretty on http://www.codecogs.com/latex/eqneditor.php and it gave me these codes as my latex markup but it didn't come out right... why do they look like source code and not the pretty...
  26. H

    Help with Eulers relation in Fourier analysis

    Hi I'm doing Fourier analysis in my signals and system course and I'm looking at the solution to one basic problem but I'm having trouble understanding one step Can anyone explain to me why becomes From Eulers formula: http://i.imgur.com/1LtTiKX.png for example the Cosine in my problem. I...
  27. G

    Orthogonality Problem (From Fourier Analysis Text)

    Hello all, I'm working through a majority of the problems in "A First Course in Wavelets with Fourier Analysis" and have stumbled upon a problem I'm having difficulty with. Please view the PDF attachment, it shows the problem and what I have done with it so far. Once you have seen the...
  28. L

    Use Fourier analysis to construct an amplitude spectrum

    Homework Statement A sensor yields a signal y(t) = |sin(120\pit)| a. Using Fourier analysis please construct an amplitude spectrum for this signal. Homework Equations A0 = \frac{1}{T}\int ^{-T/2}_{T/2}y(t) dt An =\frac{2}{T}\int^{-T/2}_{T/2}y(t)cos\frac{2n\pi t}{T}dt The Attempt at...
  29. H

    Basic questions about signal processing and fourier analysis

    1)A note consists of a fundamental frequency and the multiples of that frequency called harmonics. Peak frequency means that one that contributes most to the note. Is the fundamental frequency always the peak frequency? Since the frequencies die out very quickly as the value of n increases...
  30. H

    How to Choose Values for N, N1, and N2 in MATLAB for Fourier Analysis?

    Hi I'm trying to use the fft function in MATLAB to compute the discrete Fourier transform of a box signal. I'm told to assume that the signal x[n] is periodic with period N and the vector contains one period. x[n]= box [n] I'm am going to use these commands to make my vector...
  31. K

    Resonant Frequencies from Fourier Analysis

    I did Fourier analysis on a set of force data from a vibrating string. In my graph of magnitute and frequency, I'm getting major peaks at 62.1 Hz and 249.0 Hz. There is a tiny blip in the data at 125 Hz and nothing at 186 Hz. I have two questions. Do the peaks at 62.1 and 249 mean that those...
  32. W

    Basic Fourier analysis proof

    Hi. Just going through my notes from the last lecture I remember having some troubles understanding the proof the lecturer gave for the following theorem: Suppose that f is Riemann integrable and that all its Fourier coefficients are equal to 0, then f(x)=0 at all points of continuity. The...
  33. R

    Fourier analysis: Impulse Symbol(dirac Delta Function)

    1. what is the even part of δ(x+3)+δ(x+2) -δ(x+1) +1/2δ(x) +δ(x-1) -δ(x-2) -δ(x-3)? 2. δ= 0 x≠0; ∞ x = 0 1/2 (f(x) + f(-x)) 1/2 (f(x) - f(-x)) Knowing the piecewise definition of the delta function, and knowing 1/2 (f(x) + f(-x)) for even parts of a function. I plug this in...
  34. X

    Potential Fourier Analysis Metrics?

    So, my friend looked at this post and told me it's beyond confusing. So let me clarify. Suppose I have a neural network connected to various sensors. How best would I process the input data from the sensor such that a neural network could learn from it best. I'm assuming my network has many...
  35. D

    MHB Trig identities Fourier Analysis

    Prove the identities $$ \frac{\sin\left(\frac{n + 1}{2}\theta\right)}{\sin\frac{\theta}{2}}\cos\frac{n}{2}\theta = \frac{1}{2} + \frac{\sin\left(n + \frac{1}{2}\right)\theta}{2\sin\frac{\theta}{2}} $$ By using the identity $\sin\alpha + beta$, I was able to obtain the $1/2$ but now I am not to...
  36. A

    Fourier Analysis of Real Sound Waves

    If a basic sin sound wave is analysed with a Fourier transform, the result is just a spike at a certain frequency. My maths isn't the best so bare with me... if we take a real sound file and take Fourier transforms at regular intervals (I assume that's what's being done when calculating a...
  37. G

    Learning Elementary Fourier Analysis

    Hey guys, Is it possible to learn, (at least) elementary Fourier analysis, after completing Spivak's "Calculus"?. If not, what more is there to learn before one can begin Fourier analysis?
  38. M

    What is a good book on Fourier Analysis for understanding Dirac Delta functions?

    Hey all, Looking for a good recommendation on a supplemental book on Fourier Analysis (in particular Dirac Delta functions). I'm taking an electrodynamics class and want more than the two pages the book is devoting to it (Dirac's Func.). Thanks. ...Links would also be appreciated.
  39. C

    Use Fourier Analysis to solve

    Homework Statement let f[k]=1/k!, then let fsub2[k]=f[k] convoluted with f[k] what is a simple formula for fsubm[k]? Homework Equations f[k] convoluted with f[k] = summation from negative infinity to infinity of 1/m! * 1/(n-m)! The Attempt at a Solution I tried a base case, but...
  40. C

    MATLAB Non linear ODE system matlab fourier analysis problem

    Hello all. I am studying a system and want to investigate how the frequency of y(3) varies under different conditions. However, my the fft I perform on it tells me the frequency is zero, which must be incorrect. I have tried a stack of things but can't see what the problem is. I'm relatively...
  41. I_am_learning

    Application and Limitation of Fourier Analysis

    Suppose I have a non-linear load in my home (A half wave rectifier supplied DC load, say). Since it will consume current from the source only during +ve cycle of the Voltage, the current will be half-wave too. The current isn't sinusoidal. We can mathematically say that Distored Current =...
  42. C

    Book recommendation on intro into fourier analysis

    I'm starting my first math class that involves Fourier analysis. The book we are using is Boyce and Diprima's Elementary Differential Equations and Boundary Value Problems. I find the examples helpful but I have a hard time grasping the conceptual sections. Any recommendations for good intro...
  43. G

    Fourier analysis (Using Audacity)

    Homework Statement I'm doing a project for my school, deconstructing musical instruments into their various sinusoidal waves. I understand the outlying theory behind Fourier analysis and found what seems like a simple way to do it in the newest version of Audacity. Go to "Analyze", then...
  44. C

    Fourier analysis - prerequisites

    I was wondering at what point most students encounter Fourier analysis and what the prerequisites are considered to be?
  45. P

    THz-TDS Fast Fourier Analysis in MATLAB

    Hello, Basically I have some Terahertz time-domain-spectroscopy which I'm trying to analyse. The data that I have has Voltage and "Delay length" within the array. Apparently I can convert the delay length to a time using t=(delay length)/c. Then I can do a fast Fourier transform in...
  46. W

    Wave packets & Fourier analysis: minimum pulse duration w/ given frequency

    Homework Statement I: A telephone line can transmit a range of frequencies \Delta f = 2500 Hz. Roughly what is the duration of the shortest pulse that can be sent over this line? II: A space probe sends a picture containing 500 by 500 elements, each containing a brightness scale with 256...
  47. G

    Fourier Analysis: Calculating 2nd Harmonic

    All, I'm having some problems with a Fourier analysis I am doing as a coursework, we have been given waves with multiple functions to try and work out the harmonics of them. My three functions are: f1 = -5sin (2*pi*t) - sin wave part f2 = 21.74t-32.61 - straight line part f3 = 0 - horizontal...
  48. J

    Intro Quantum Course Fourier Analysis

    Homework Statement Using ultra-short laser pulses, it is possible to probe the reaction dynamics of many chemical reactions, such as A+BC->(ABC)*->AB+C. This worksheet estimates some of the parameters necessary to do this. (a) If the energy available to each final states species is roughly...
  49. H

    What Is the Method for Conducting 2D Fourier Analysis of Complex Shapes?

    I'm studying for a re-examn in a physical measuring techniques course that includes some Fourier analysis. On old exams there are always a question about a graphical 2d Fourier transform. It can be an ellipse containing a grid of circels, or something like that. In the solutions he takes the...
  50. R

    A question about fourier analysis

    Hi. see the image. In the integral there's no dx,dy,dz thing. Does this make any sense? Isn't it errata?
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