In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions. The attempt to understand functions (or other objects) by breaking them into basic pieces that are easier to understand is one of the central themes in Fourier analysis. Fourier analysis is named after Joseph Fourier who showed that representing a function by a trigonometric series greatly simplified the study of heat propagation.
Today the subject of Fourier analysis encompasses a vast spectrum of mathematics with parts that, at first glance, may appear quite different. In the sciences and engineering the process of decomposing a function into simpler pieces is often called an analysis. The corresponding operation of rebuilding the function from these pieces is known as synthesis. In this context the term Fourier synthesis describes the act of rebuilding and the term Fourier analysis describes the process of breaking the function into a sum of simpler pieces. In mathematics, the term Fourier analysis often refers to the study of both operations.
In Fourier analysis, the term Fourier transform often refers to the process that decomposes a given function into the basic pieces. This process results in another function that describes how much of each basic piece are in the original function. It is common practice to also use the term Fourier transform to refer to this function. However, the transform is often given a more specific name depending upon the domain and other properties of the function being transformed, as elaborated below. 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.
This wide applicability stems from many useful properties of the transforms:
Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses Fourier transformation of small square pieces of a digital image. The Fourier components of each square are rounded to lower arithmetic precision, and weak components are eliminated entirely, so that the remaining components can be stored very compactly. In image reconstruction, each Fourier-transformed image square is reassembled from the preserved approximate components, and then inverse-transformed to produce an approximation of the original image.
When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection and/or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.
Some examples include:
Fourier analysis has different forms, some of which have different names. Below are given several of the most common variants. Variations with different names usually reflect different properties of the function or data being analyzed. The resultant transforms can be seen as special cases or generalizations of each other.
Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, such as time (t). In this case the Fourier transform describes a function ƒ(t) in terms of basic complex exponentials of various frequencies. In terms of ordinary frequency ν, the Fourier transform is given by the complex number:
Evaluating this quantity for all values of ν produces the frequency-domain function.
Also see How it works, below. See Fourier transform for even more information, including:
for all integers n. And the Fourier series of ƒ(x) is given by:
Equality may not always hold in the equation above and the study of the convergence of Fourier series is a central part of Fourier analysis of the circle.
See Fourier series for more information, including:
For functions of an integer index, the discrete-time Fourier transform (DTFT) provides a useful frequency-domain transform.
A useful "discrete-time" function can be obtained by sampling a "continuous-time" function, s(t), which produces a sequence, s(nT), for integer values of n and some time-interval T. If information is lost, then only an approximation to the original transform, S(f), can be obtained by looking at one period of the periodic function:
which is the DTFT. The identity above is a result of the Poisson summation formula. The DTFT is also equivalent to the Fourier transform of a "continuous" function that is constructed by using the s[n] sequence to modulate a Dirac comb.
Applications of the DTFT are not limited to sampled functions. It can be applied to any discrete sequence. See Discrete-time Fourier transform for more information on this and other topics, including:
Since the DTFT is also a continuous Fourier transform (of a comb function), the Fourier series also applies to it. Thus, when s[n] is periodic, with period N, S_{T}(ƒ) is another Dirac comb function, modulated by the coefficients of a Fourier series. And the integral formula for the coefficients simplifies to:
Since the DTFT is periodic, so is S[k]. And it has the same period (N) as the input function. This transform is also called DFT, particularly when only one period of the output sequence is computed from one period of the input sequence.
When s[n] is not periodic, but its non-zero portion has finite duration (N), S_{T}(ƒ) is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the (finite) portion of s[n] that was analyzed. The same discrete set is obtained by treating N as if it is the period of a periodic function and computing the Fourier series coefficients / DFT.
The DFT can be computed using a fast Fourier transform (FFT) algorithm, which makes it a practical and important transformation on computers.
See Discrete Fourier transform for much more information, including:
The following table recaps the four basic forms discussed above, highlighting the duality of the properties of discreteness and periodicity. I.e., if the signal representation in one domain has either (or both) of those properties, then its transform representation to the other domain has the other property (or both).
Name | Time domain | Frequency domain | ||
---|---|---|---|---|
Domain property | Function property | Domain property | Function property | |
(Continuous) Fourier transform | Continuous | Aperiodic | Continuous | Aperiodic |
Discrete-time Fourier transform | Discrete | Aperiodic | Continuous | Periodic (ƒ_{s}) |
Fourier series | Continuous | Periodic ($tau$) | Discrete | Aperiodic |
Discrete Fourier transform | Discrete | Periodic (N) | Discrete | Periodic (N) |
The Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact abelian topological groups, which are studied in harmonic analysis; there, the Fourier transform takes functions on a group to functions on the 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.
In signal processing, the Fourier transform often takes a time series or a function of continuous time, and maps it into a frequency spectrum. That is, it takes a function from the time domain into the frequency domain; it is a decomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.
When the function ƒ is a function of time and represents a physical signal, the transform has a standard interpretation as the frequency spectrum of the signal. The magnitude of the resulting complex-valued function F at frequency ω represents the amplitude of a frequency component whose initial phase is given by the phase of F.
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.
To measure the amplitude and phase of a particular frequency component, the transform process multiplies the original function (the one being analyzed) by a sinusoid with the same frequency (called a basis function). If the original function contains a component with the same shape (i.e. same frequency), its shape (but not its amplitude) is effectively squared.
The complex numbers produced by the product of the original function and the basis function are subsequently summed into a single result.
The contributions from the component that matches the basis function all have the same sign (or vector direction). The other components contribute values that alternate in sign (or vectors that rotate in direction) and tend to cancel out of the summation. The final value is therefore dominated by the component that matches the basis function. The stronger it is, the larger is the measurement. Repeating this measurement for all the basis functions produces the frequency-domain representation.