# 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. ### 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. ### 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. ### 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. ### Understanding Fourier Transforms

I think that is with the Fourier transform.
5. ### 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?

13. ### 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 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### I Complex Fourier Series: Even/Odd Half Range Expansion

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

44. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...