Mathematical transform that expresses a function of time as a function of frequency
Fourier transforms
Fourier transform
Fourier series
Discrete-time Fourier transform
Discrete Fourier transform
Discrete Fourier transform over a ring
Fourier transform on finite groups
Fourier analysis
Related transforms
An example application of the Fourier transform is determining the constituent pitches in a musical waveform. This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed.
In physics, engineering and 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 this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function. 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.
The red sinusoid can be described by peak amplitude (1), peak-to-peak (2), RMS (3), and wavelength (4). The red and blue sinusoids have a phase difference of θ.
The top row shows a unit pulse as a function of time (f(t)) and its Fourier transform as a function of frequency (f̂(ω)). The bottom row shows a delayed unit pulse as a function of time (g(t)) and its Fourier transform as a function of frequency (ĝ(ω)). Translation (i.e. delay) in the time domain is interpreted as complex phase shifts in the frequency domain. The Fourier transform decomposes a function into eigenfunctions for the group of translations. The imaginary part of ĝ(ω) is negated because a negative sign exponent has been used in the Fourier transform, which is the default as derived from the Fourier series, but the sign does not matter for a transform that is not going to be reversed.
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 the 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 1] 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 2]
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 3] 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.
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In physics, engineering and mathematics, the Fouriertransform (FT) is an integral transform that takes a function as input and outputs another function...
A Fast FourierTransform (FFT) is an algorithm that computes the Discrete FourierTransform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis...
In mathematics, the discrete Fouriertransform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of...
the quantum Fouriertransform (QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fouriertransform. The quantum...
The decomposition process itself is called a Fourier transformation. Its output, the Fouriertransform, is often given a more specific name, which depends...
Laplace Transform and the z-transform are closely related to the FourierTransform. Laplace Transform is somewhat more general in scope than the Fourier Transform...
mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fouriertransform. Intuitively...
classical Fouriertransform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis. The Graph Fouriertransform is...
fractional Fouriertransform (FRFT) is a family of linear transformations generalizing the Fouriertransform. It can be thought of as the Fouriertransform to...
is also known as the Fourier–Bessel transform. Just as the Fouriertransform for an infinite interval is related to the Fourier series over a finite interval...
Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fouriertransform) is an...
In mathematics, the Fourier sine and cosine transforms are forms of the Fouriertransform that do not use complex numbers or require negative frequency...
the Fouriertransform on finite groups is a generalization of the discrete Fouriertransform from cyclic to arbitrary finite groups. The Fourier transform...
formula for the inverse Laplace transform, called the Mellin's inverse formula, the Bromwich integral, or the Fourier–Mellin integral, is given by the...
In mathematics, the discrete Fouriertransform over a ring generalizes the discrete Fouriertransform (DFT), of a function whose values are commonly complex...
heat transfer and vibrations. The Fouriertransform and Fourier's law of conduction are also named in his honour. Fourier is also generally credited with...
by using the Fouriertransform for functions on the real line or by Fourier series for periodic functions. Generalizing these transforms to other domains...
applied mathematics, a DFT matrix is an expression of a discrete Fouriertransform (DFT) as a transformation matrix, which can be applied to a signal...
The Radon transform is closely related to the Fouriertransform. We define the univariate Fouriertransform here as: f ^ ( ω ) = ∫ − ∞ ∞ f ( x ) e − 2 π...
in time resolution at ascending frequencies for the Fouriertransform and the wavelet transform is shown below. Note however, that the frequency resolution...
the sign of the frequency (see § Relationship with the Fouriertransform). The Hilbert transform is important in signal processing, where it is a component...
mathematics, the Hartley transform (HT) is an integral transform closely related to the Fouriertransform (FT), but which transforms real-valued functions...
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