Fourier transform

The term "Fourier transform", without any further qualification, is often taken to refer specifically to the continuous Fourier transform, which is described briefly below, and treated at greater length in the Wikipedia article of that title. The article also includes a table of Fourier transforms.

The continuous Fourier transform can be thought as a continuous generalization of Fourier series to non-periodic functions. Other Fourier transforms than the "continuous" one are also described below, and at greater length in other articles. 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. The book by Dym and McKean cited at the end of this article treats many such applications.

Definition of the Fourier transform

The Fourier transform is an integral transform (and thus, a linear operator) that maps one complex function of a real variable into another; the original function and its transform are sometimes called a transform pair. As the transform and its inverse are unique (by the Fourier inversion theorem), there is exactly one transform pair for each function for which the transform is defined.

The Fourier transform of a function f is defined by an integral,

denoting the transformed function as F, and the transformation operator itself by the script F. Note that the definition is sometimes written with different scaling factors in the exponent or in front of the integral.

The inverse Fourier transform, given the above definition for the transform, is a similar integral,

There is also a discrete version of the Fourier transform.

Properties of the Fourier transform

Parseval's theorem states that

Fourier transforms also possess the property that if F(s) and G(s) are the Fourier Transforms of f(t) and g(t) respectively then:

where * denotes convolution. Likewise,

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. 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 real parts of the resulting complex-valued function F represent the amplitudes of their respective frequencies (s), while the imaginary parts represent the phase shifts.

Discrete Fourier transforms and Fourier series

There are also discrete Fourier transforms and Fourier series. These two, along with the continuous transform, can all be considered as special cases of a general theory.

In general, the Fourier transform taking functions with domain A into functions with domain B may be:

• a Fourier series, in which A is an interval, such as [−π, π] and B is the set of all integers)

• a discrete Fourier transform in which A and B are segments of natural numbers - usually 0, ..., N − 1)

• a continuous Fourier transform, in which A and B are the whole real line; that is the concept treated above.

Both the continous and discrete Fourier transforms, and also the Fourier series, are generalized by the Fourier transform on locally compact abelian topological groups, which is studied in harmonic analysis; here, A is the group and B is its dual group. This treatment also allows a general formulation of the convolution theorem, which relates Fourier transforms and convolutions.

The Fourier transform can be viewed as a special case of the Z-transform: the Fourier transform is the Z-transform evaluated at the unit circle in the complex space.

Computational implementations

Implementations of Fourier transforms of arbitrary signals are computationally intensive, but the fast Fourier transform can greatly reduce the computation required.

Such transforms are used in some types of RF modulation.

The free software library FFTW is a C library for computing the discrete Fourier transform, which claims to be especially fast.

See the Fourier transform in action on the SETI at home project.

See also

• Continuous Fourier transform
• Fourier series
• Discrete Fourier transform
• Integral transform
• Laplace transform
• Wavelet
• Orthogonal functions

External links

• Kevin Cowtan's Book of Fourier
  • Offers an introduction to the Fourier transform, especially regarding its application to X-ray crystallography.
• Dym & McKean's Fourier Series and Integrals
  • This book includes a very extensive collection of examples from physics, geometry, number theory, etc. It also gives an excellent exposition of the theory for those who have sufficient analysis background. For those who do not, it introduces some of that prerequisite material, but that is best learned from other books.

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