What is Fourier analysis: Definition and 126 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. S

    Characterize Fourier coefficients

    I would try to determine whether ##p(t)## is even or odd. This would be so much easier if the values of ##\tau## and ##T## would be specified, but maybe it's possible to do without it, which I'd prefer. If for example ##\tau=1/2## and ##T=2\pi##, then ##p(t)=\sin{(2t)}## for ##0\leq t <\pi ##...
  2. stephen8686

    I Propagation of Angular Spectrum Code

    I'm making a MATLAB code to propagate a gaussian field in the angular spectrum regime (fresnel number >> 1). After Fourier transforming the field, you propagate it: $$U(k_x,k_y,z) = U(k_x,k_y,0)e^{ik_z z}$$ The thing that I am having trouble with is the propagation factor, I have looked at this...
  3. M

    I Please discuss discrete Fourier analysis

    It has been 35 years since I did the math for Fourier analysis, and I have forgotten what the subtleties are. Please be kind. So this is not a how do I calculate a DFT (though that may be my next question) but rather how do I use it, and interpret the results. All the online and software I find...
  4. P99

    Understanding Fourier Transforms

    I think that is with the Fourier transform.
  5. D

    B Fourier Analysis on musical chords in different instruments

    I wanted to do an investigation about how the same musical chord can have the same pitch but sound different on different musical instruments. Like how chord C major would sound higher played in the electric guitar than a C major played on piano. How should I approach this investigation?
  6. I

    Converting an expression of a particular k-mode to the spatial domain

    $$n_\vec{k} = \omega a^2(\vec{k})\tag{1}$$ One way is to write the inverse Fourier transforms of the terms above. So, eqn (1) becomes $$\int\mathrm{d}^3x\ n(\vec{x})e^{-i\vec{k}\cdot\vec{x}} = \omega \int\mathrm{d}^3x^\prime\ a(\vec{x^\prime})e^{-i\vec{k}\cdot\vec{x^\prime}}...
  7. arcTomato

    I Power spectrum definition

    Hello PF. I am thinking about the power spectrum when observing X-rays. We are trying to obtain the power spectrum by applying a window function ##w(t)## to a light curve ##a(t)## and then Fourier transforming it. I have seen the following definition of power spectrum ##P(\omega)##. Suppose...
  8. Mr.Husky

    Analysis Opinions on textbooks on Analysis

    What are your opinions on Barry Simon's "A Comprehensive Course in Analysis" 5 volume set. I bought them with huge discount (paperback version). But I am not sure should I go through these books? I have 4 years and can spend 12 hours a week on them. Note- I am now studying real analysis from...
  9. S

    I Why should a Fourier transform not be a change of basis?

    I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts. It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...
  10. tworitdash

    A Spatial Fourier Transform: Bessel x Sinusoidal

    I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m +...
  11. Jason-Li

    Comp Sci Fourier analysis & determination of Fourier Series

    ANY AND ALL HELP IS GREATLY APPRECIATED :smile: I have found old posts for this question however after reading through them several times I am having a hard time knowing where to start. I am happy with the sketch that the function is correctly drawn and is neither odd nor even. It's title is...
  12. Ineedhelp0

    I Parseval's theorem and Fourier Transform proof

    Given a function F(t) $$ F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega $$ I am looking for a proof of the following: $$ \int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d...
  13. arcTomato

    I How to derive the Fourier transform of a comb function

    Dear all. I'm learning about the discrete Fourier transform. ##I(\nu) \equiv \int_{-\infty}^{\infty} i(t) e^{2 \pi \nu i t} d t=\frac{N}{T} \sum_{\ell=-\infty}^{\infty} \delta\left(\nu-\ell \frac{N}{T}\right)## this ##i(t)## is comb function ##i(t)=\sum_{k=-\infty}^{\infty}...
  14. A

    A Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w))

    Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w)) Hello to my Math Fellows, Problem: I am looking for a way to calculate w-derivative of Fourier transform,d/dw (F{x(t)}), in terms of regular Fourier transform,X(w)=F{x(t)}. Definition Based Solution (not good enough): from...
  15. K

    I Fourier analysis - waves

    if I am to learn about waves at an undergraduated level, how much is it important to learn Fourier theory before I actually go into the physics?
  16. E

    Fourier series for a series of functions

    ## ## Well I start with equation 1): ## e^{b\theta }=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in\theta } ## If ## \theta =0 ## ##e^{b(0)}=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in(0) }## ##1=\frac{sinh(b\pi )}{\pi...
  17. M

    How a square or sawtooth wave can have a certain frequency?

    Hello! I know that a square or saw tooth wave consists of infinite amount of sinousoids each having different frequency and amplitude. But when I look at their plot they seem to have a well defined frequency or period. Which term in the Fourier series determines their frequency? Does a saw...
  18. K

    Fourier series of abs(sin(x))

    Homework Statement Hello, i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1...
  19. Behrouz

    A Finding a specific amplitude-frequency in the time domain

    Hello, I have a signal and got the FFT result of that. I have shown them both below along with the MATLAB code. May I ask if there is any method to find the time zone(s) in the signal that a specific frequency has(have) happened? The reason I'm asking this is that I want to specify the time...
  20. J

    What is the maximum or Nyquist frequency of a Gaussian signal?

    Hello. I'm studying Fourier analysis. If we look at attached graph where Gaussian functions are transformed by Fourier analysis, we can find Gaussian functions in frequency domain have maximum value at 0 hertz. So I confused what is the Nyquist frequency at Gaussian signal. I need to know...
  21. N

    I Understanding what the complex cosine spectrum is showing

    The complex exponential form of cosine cos(k omega t) = 1/2 * e^(i k omega t) + 1/2 * e^(-i k omega t) The trigonometric spectrum of cos(k omega t) is single amplitude of the cosine function at a single frequency of k on the real axis which is using the basis function of cosine, right? The...
  22. C

    Fourier Analysis and the Significance of Odd and Even Functions

    Homework Statement Q1. a) In relation to Fourier analysis state the meaning and significance of 4 i) odd and even functions ii) half-wave symmetry {i.e. f(t+π)= −f(t)}. Illustrate each answer with a suitable waveform sketch. b) State by inspection (i.e. without performing any formal analysis)...
  23. D

    Finding the fourier spectrum of a function

    Homework Statement Find the Fourier spectrum ##C_k## of the following function and draw it's graph: Homework Equations 3. The Attempt at a Solution [/B] I know that the complex Fourier coefficient of a rectangular impulse ##U## on an interval ##[-\frac{\tau}{2}, \frac{\tau}{2}]## is ##C_k =...
  24. R

    I 2D Fourier transform orientation angle

    The orientation of frequency components in the 2-D Fourier spectrum of an image reflect the orientation of the features they represent in the original image. In techniques such as nonlinear microscopy, they use this idea to determine the preferred (i.e. average) orientation of certain features...
  25. D

    Question regarding Fourier Transform duality

    Homework Statement Given the Fourier transformation pair ##f(t) \implies F(jw)## where ##f(t) = e^{-|t|}## and ##F(jw)=\frac{2}{w^2+1}## find and make a graph of the Fourier transform of the following functions: a) ##g(t)=\frac{2}{t^2+1}## b) ##h(t) = \frac{2}{t^2+1}\cos (w_ot)## Homework...
  26. A

    I Complex Fourier Series: Even/Odd Half Range Expansion

    Does the complex form of Fourier series assume even or odd half range expansion?
  27. S

    Fourier Series Expansion

    Homework Statement There is a sawtooth function with u(t)=t-π. Find the Fourier Series expansion in the form of a0 + ∑αkcos(kt) + βksin(kt) Homework Equations a0 = ... αk = ... βk = ... The Attempt at a Solution After solving for a0, ak, and bk, I found that a0=0, ak=0, and bk=-2/k...
  28. DeathbyGreen

    I Frequency contributions

    I'm trying to relate some different frequencies together in an experiment. Say I have 3 different frequencies, \omega_1,\omega_2, \omega_3. Omega 3 is the large envelope, and the other two must fit inside of it, and so they are integer multiples of each other. Is there some way to express...
  29. J6204

    Calculating the Fourier integral representation of f(x)

    Homework Statement Considering the function $$f(x) = e^{-x}, x>0$$ and $$f(-x) = f(x)$$. I am trying to find the Fourier integral representation of f(x). Homework Equations $$f(x) = \int_0^\infty \left( A(\alpha)\cos\alpha x +B(\alpha) \sin\alpha x\right) d\alpha$$ $$A(\alpha) =...
  30. J6204

    Extending function to determine Fourier series

    In the following question I need to find the Fourier cosine series of the triangular wave formed by extending the function f(x) as a periodic function of period 2 $$f(x) = \begin{cases} 1+x,& -1\leq x \leq 0\\ 1-x, & 0\leq x \leq 1\\\end{cases}$$ I just have a few questions then I will be able...
  31. A

    A Fourier Transform for 3rd kind of boundary conditions?

    I am studying online course notes from University of Waterloo on 'Analytical mathematics in geology' in which the author describes a 'modified Fourier transform' which can be used to incorporate 3rd kind of boundary conditions. The formula is ## \Gamma \small[ f(x) \small] = \bar{f}(a) =...
  32. A

    Odd and even in complex fourier series

    Homework Statement In Complex Fourier series, how to determine the function is odd or even or neither, as in the given equation $$ I(t)= \pi + \sum_{n=-\infty}^\infty \frac j n e^{jnt} $$ Homework Equations ##Co=\pi## ##\frac {ao} 2 = \pi## ##Cn=\frac j n## ##C_{-n}= \frac {-j} n ##...
  33. G

    Find Fourier coefficients - M. Chester text

    Homework Statement I am self studying an introductory quantum physics text by Marvin Chester Primer of Quantum Mechanics. I am stumped at a problem (1.10) on page 11. We are given f(x) = \sqrt{ \frac{8}{3L} } cos^2 \left ( \frac {\pi}{L} x \right ) and asked to find its Fourier...
  34. 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...
  35. 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...
  36. TheBigDig

    Finding Fourier Coefficient

    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...
  37. 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...
  38. 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...
  39. 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)...
  40. 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...
  41. 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...
  42. 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...
  43. 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...
  44. 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...
  45. RJLiberator

    Fourier Analysis Question

    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 Equations The...
  46. 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...
  47. 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?
  48. 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 Equation The Attempt at a Solution I posted a photo of what...
  49. Titan97

    Learning Fourier Analysis

    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...
  50. D

    Textbooks on Fourier analysis

    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...