Fourier analysis

The Fourier transform, named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i.e. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes"). There are many closely-related variations of this transform, summarized below, depending upon the type of function being transformed. See also: List of Fourier-related transforms.

Fourier transforms have many scientific applications — in physics, number theory, combinatorics, signal processing, probability theory, statistics, cryptography, acoustics, oceanography, optics, geometry, and other areas. (In signal processing and related fields, the Fourier transform is typically thought of as decomposing a signal into its component frequencies and their amplitudes.) This wide applicability stems from several useful properties of the transforms:

The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality).

The transforms are invertible, and in fact the inverse transform has almost the same form as the forward transform.

The sinusoidal basis functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones. (For example, in a linear time-invariant physical system, frequency is a conserved quantity, so the behavior at each frequency can be solved independently.)

By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.

The discrete version of the Fourier transform (see below) can be evaluated quickly on computers, using algorithms based on the fast Fourier transform (FFT).

Variants of the Fourier transform

Most often, the unqualified term "Fourier transform" refers to the continuous Fourier transform, representing any square-integrable function f(t) as a sum of complex exponentials with angular frequencies ω and complex amplitudes F(ω):

f(t)={\mathcal {F}}^{-1}(F)(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }F(\omega )e^{i\omega t}\,d\omega .

This is actually the inverse continuous Fourier transform, whereas the Fourier transform expresses F(ω) in terms of f(t); the original function and its transform are sometimes called a transform pair. See continuous Fourier transform for more information, including a table of transforms, discussion of the transform properties, and the various conventions.

The continuous transform is actually a generalization of an earlier concept, a Fourier series, which was specific to periodic (or finite-___domain) functions f(x) (with period 2π), and represents these functions as a series of sinusoids:

f(x)=\sum _{n=-\infty }^{\infty }F_{n}\,e^{inx},

where $F_{n}$ is the (complex) amplitude. Or, for real-valued functions, the Fourier series is often written:

f(x)={\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }\left[a_{n}\cos(nx)+b_{n}\sin(nx)\right],

where a_n and b_n are the (real) Fourier series amplitudes.

For use on computers, both for scientific computation and digital signal processing, one must have functions x_k that are defined over discrete instead of continuous domains, again finite or periodic. In this case, one uses the discrete Fourier transform (DFT), which represents x_k as the sum of sinusoids:

x_{k}={\frac {1}{n}}\sum _{j=0}^{n-1}f_{j}e^{2\pi ijk/n}\quad \quad k=0,\dots ,n-1

where f_j are the Fourier amplitudes. Although applying this formula directly would require O(n²) operations, it can be computed in only O(n log n) operations using a fast Fourier transform (FFT) algorithm (see Big O notation), which makes Fourier transformation a practical and important operation on computers.

These Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact abelian topological groups, which are studied in harmonic analysis; there, one transforms from a group to its dual group. This treatment also allows a general formulation of the convolution theorem, which relates Fourier transforms and convolutions. See also the Pontryagin duality for the generalized underpinnings of the Fourier transform.

Interpretation in terms of time and frequency

In terms of signal processing, the transform takes a time series representation of a signal function and maps it into a frequency spectrum, where ω is angular frequency. That is, it takes a function in the time ___domain into the frequency ___domain; it is a decomposition of a function into harmonics of different frequencies.

When the function f is a function of time and represents a physical signal, the transform has a standard interpretation as the spectrum of the signal. The magnitude of the resulting complex-valued function F represents the amplitudes of the respective frequencies (ω), while the phase shifts are given by arctan(imaginary parts/real parts).

However, it is important to realize that Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze spatial frequencies, and indeed for nearly any function ___domain.

References

Smith, Steven W. The Scientist and Engineer's Guide to Digital Signal Processing, 2nd edition. San Diego: California Technical Publishing, 1999. ISBN 0-9660176-3-3. (also available online: [1])