Fourier transforms (nonfiction)

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input, and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both the mathematical operation and to this complex-valued function. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function.[note 1] The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.

The Fourier transform relates the time domain, in red, with a function in the domain of the frequency, in blue. The component frequencies, extended for the whole frequency spectrum, are shown as peaks in the domain of the frequency. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced sine and cosine transforms (which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.

The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.[note 2] For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.[note 3]

The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional "position space" to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued.[note 4] Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or Rn, notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.

Background

History

In 1822, Fourier claimed (see Joseph Fourier § The Analytic Theory of Heat) that any function, whether continuous or discontinuous, can be expanded into a series of sines.[13] That important work was corrected and expanded upon by others to provide the foundation for the various forms of the Fourier transform used since.

Fourier transform for periodic functions

The Fourier transform of a periodic function cannot be defined using the integral formula directly. In order for integral in Eq.1 to be defined the function must be absolutely integrable. Instead it is common to use Fourier series. It is possible to extend the definition to include periodic functions by viewing them as tempered distributions.

This makes it possible to see a connection between the Fourier series and the Fourier transform for periodic functions that have a convergent Fourier series.

Alternatives

In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and standing waves are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably transients, or any signal of finite extent.

As alternatives to the Fourier transform, in time–frequency analysis, one uses time–frequency transforms or time–frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform, fractional Fourier transform, Synchrosqueezing Fourier transform,[71] or other functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform.[27]

Applications

See also: Spectral density § Applications

Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform. Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency,[note 7] so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.

Analysis of differential equations

Perhaps the most important use of the Fourier transformation is to solve partial differential equations. Many of the equations of the mathematical physics of the nineteenth century can be treated this way. Fourier studied the heat equation ...

Nonlinear Fourier transform

Main article: Inverse scattering transform

The twentieth century has seen application of these methods to all linear partial differential equations with polynomial coefficients as well as an extension to certain classes of nonlinear partial differential equations. Specifically, nonlinear evolution equations (i.e. those equations that describe how a particular quantity evolves in time from a specified initial state) that can be associated with linear eigenvalue problems whose eigenvalues are integrals of the nonlinear equations.[72][73] As it may be considered an extension of Fourier analysis to nonlinear problems, the solution method is called the nonlinear Fourier transform (or inverse scattering transform) method

Quantum mechanics

The Fourier transform is useful in quantum mechanics in at least two different ways. To begin with, the basic conceptual structure of quantum mechanics postulates the existence of pairs of complementary variables, connected by the Heisenberg uncertainty principle. For example, in one dimension, the spatial variable q of, say, a particle, can only be measured by the quantum mechanical "position operator" at the cost of losing information about the momentum p of the particle. Therefore, the physical state of the particle can either be described by a function, called "the wave function", of q or by a function of p but not by a function of both variables. The variable p is called the conjugate variable to q.

In classical mechanics, the physical state of a particle (existing in one dimension, for simplicity of exposition) would be given by assigning definite values to both p and q simultaneously. Thus, the set of all possible physical states is the two-dimensional real vector space with a p-axis and a q-axis called the phase space. In contrast, quantum mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, the q-axis alone, but instead of considering only points, takes the set of all complex-valued "wave functions" on this axis. Nevertheless, choosing the p-axis is an equally valid polarisation, yielding a different representation of the set of possible physical states of the particle. Both representations of the wavefunction are related by a Fourier transform ...

Signal processing

The Fourier transform is used for the spectral analysis of time-series. The subject of statistical signal processing does not, however, usually apply the Fourier transformation to the signal itself. Even if a real signal is indeed transient, it has been found in practice advisable to model a signal by a function (or, alternatively, a stochastic process) which is stationary in the sense that its characteristic properties are constant over all time. The Fourier transform of such a function does not exist in the usual sense, and it has been found more useful for the analysis of signals to instead take the Fourier transform of its autocorrelation function.

See also:

• Analog signal processing
• Beevers–Lipson strip
• Constant-Q transform
• Discrete Fourier transform
• DFT matrix
• Fast Fourier transform
• Fourier integral operator
• Fourier inversion theorem
• Fourier multiplier
• Fourier series
• Fourier sine transform
• Fourier–Deligne transform
• Fourier–Mukai transform
• Fractional Fourier transform
• Indirect Fourier transform
• Integral transform
• Hankel transform
• Hartley transform
• Laplace transform
• Least-squares spectral analysis
• Linear canonical transform
• List of Fourier-related transforms
• Mellin transform
• Multidimensional transform
• NGC 4622, especially the image NGC 4622 Fourier transform m = 2.
• Nonlocal operator
• Quantum Fourier transform
• Quadratic Fourier transform
• Short-time Fourier transform
• Spectral density
• Spectral density estimation
• Symbolic integration
• Time stretch dispersive Fourier transform
• Transform (mathematics)

External links

• Fourier transform @ Wikipedia