Fourier analysis

(noun)

The study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

Related Terms

Examples of Fourier analysis in the following topics:

  • Motivation

    • The representation of arbitrary functions in terms of sines and cosines is called Fourier analysis.
    • Jean Baptiste Joseph Fourier.
    • Fourier trained as a priest and nearly lost his head (literally) in the French revolution.
    • Fourier established the equation governing diā†µusion and used infinite series of trigonometric functions to solve it.
    • Fourier was also a scientific adviser to Napoleon's army in Egypt.

  • Some Basic Theorems for the Fourier Transform

    • It is very useful to be able think of the Fourier transform as an operator acting on functions.
    • This result is crucial in using Fourier analysis to study differential equations.
    • The convolution theorem is one of the most important in time series analysis.
    • Convolutions are done often and by going to the frequency domain we can take advantage of the algorithmic improvements of the fast Fourier transform algorithm (FFT).
    • Start by multiplying the two Fourier transforms.

  • Introduction to the Fourier Series

    • So, the motivation for further study of such a Fourier superposition is clear.
    • The answer is yes, and this is one of the principle aims of Fourier analysis.
    • Later we will see how to estimate the power spectrum using a Fourier transform.
    • Later on you will be given two of the basic convergence theorems for Fourier series.
    • This sort of analysis is one of the central goals of Fourier theory.

  • 1-D Separation of Variables: Summary of the Argument

    • This is our first example of a Fourier series.
    • We use Mathematica's built-in Fourier series capability to represent a "hat" function as a 6 term sine-series.
    • (Don't worry about the details of the Fourier analysis, we'll be covering that later. ) But download this notebook and run it.

  • Convergence Theorems

    • One has to be a little careful about saying that a particular function is equal to its Fourier series since there exist piecewise continuous functions whose Fourier series diverge everywhere!
    • Similarly for a left derivative) then the Fourier series for $f$ converges to
    • If $f$ is continuous with period $2\pi$ and $f'$ is piecewise continuous, then the Fourier series for $f$ converges uniformly to $f$ .
    • For more details, consult a book on analysis such as The Elements of Real Analysis by Bartle or Real Analysis by Haaser and Sullivan.

  • Introduction to the Fourier Integral

    • Looking at its Fourier series (either Equation 4.2.1 or 4.2.2) we see straight away that the frequencies present in the Fourier synthesis are
    • First, the frequencies appearing in the Fourier synthesis are now
    • In this case, our Fourier series
    • A function $f(t)$ is related to its Fourier transform $f(\omega)$ via:
    • We won't attempt to prove that the kernel function converges to a delta function and hence that the Fourier transform is invertible; you can look it up in most books on analysis.

  • X-Ray Diffraction Analysis

    • The analysis consists of indexing, merging, and phasing variations in electron density.
    • Further analysis involves structure refinement and quantitative phase using the general structure analysis system (GSAS), which ultimately leads to the identification of the amorphous or crystalline phase of a matter and helps construct its three dimensional atomic model .
    • X-ray diffraction analysis workflow.
    • The two-dimensional images taken at different rotations are converted into a three-dimensional model of the density of electrons within the crystal using the mathematical method of Fourier transforms, combined with chemical data known for the sample.
    • Summarize the methods used for x-ray diffraction analysis and the contributions they have made to science

  • The Discrete Fourier Transform

    • Now we consider the third major use of the Fourier superposition.
    • Now we write down a Fourier approximation for the unknown function (i.e., a Fourier series with coefficients to be determined):
    • This is the discrete version of the Fourier transform (DFT).
    • Implement the previous formula and compare the results with Mathematica's built in Fourier function.
    • The reason is that Mathematica uses a special algorithm called the FFT (Fast Fourier Transform).

  • Introduction to The Sampling Theorem

    • Clearly a band limited function has a finite inverse Fourier transform
    • Since we are now dealing with a function on a finite interval we can represent it as a Fourier series:
    • where the Fourier coefficients $\phi _n$ are to be determined by
    • And we know that the sinc function is also the Fourier transform of a box-shaped function.
    • In addition to his work in sampling, Nyquist also made an important theoretical analysis of thermal noise in electrical systems.

  • The Spectrum

    • A general electromagnetic wave can be expressed as a sum of the Fourier components described in the previous section.
    • The first step in obtaining the spectrum is to take a Fourier transform of the electric field of the wave
    • The table below gives a few Fourier transforms of common functions.