Fourier transform
In mathematics, the Fourier transform 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. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.
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 . The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced sine and cosine transforms 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. 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.
The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of "position space" to a function of 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. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on Real number#Arithmetic| or, notably includes the discrete-time Fourier transform, the discrete Fourier transform and the Fourier series or circular Fourier transform. The latter is routinely employed to handle periodic functions. The fast Fourier transform is an algorithm for computing the DFT.
Definition
The Fourier transform of a complex-valued function on the real line, is the complex valued function, defined by the integralIn this case is integrable over the whole real line, i.e., the above integral converges to a continuous function at all .
However, the Fourier transform can also be defined for functions for which the Lebesgue integral does not make sense. Interpreting the integral suitably extends the Fourier transform to functions that are not necessarily integrable over the whole real line. More generally, the Fourier transform also applies to generalized functions like the Dirac delta, in which case it is defined by duality rather than an integral.
First introduced in Fourier's Analytical Theory of Heat., the corresponding inversion formula for "sufficiently nice" functions is given by the Fourier inversion theorem, i.e.,
The functions and are referred to as a Fourier transform pair. A common notation for designating transform pairs is:
For example, the Fourier transform of the delta function is the constant function :
Angular frequency (''ω'')
When the independent variable represents time, the transform variable represents frequency. For example, if time has the unit second, then frequency has the unit hertz. The transform variable can also be written in terms of angular frequency, with the unit radian per second.The substitution into produces this convention, where function is relabeled
Unlike the definition, the Fourier transform is no longer a unitary transformation, and there is less symmetry between the formulas for the transform and its inverse. Those properties are restored by splitting the factor evenly between the transform and its inverse, which leads to another convention:
Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites.
| ordinary frequency | unitary | |
| angular frequency | unitary | |
| angular frequency | non-unitary |
| ordinary frequency | unitary | |
| angular frequency | unitary | |
| angular frequency | non-unitary |
Lebesgue integrable functions
A measurable function is called integrable if the Lebesgue integral of its absolute value is finite:If is Lebesgue integrable then the Fourier transform, given by, is well-defined for all. Furthermore, is bounded, uniformly continuous and vanishing at infinity. Here denotes the space of continuous functions on that approach 0 as x approaches positive or negative infinity.
The space is the space of measurable functions for which the norm is finite, modulo the equivalence relation of equality almost everywhere. The Fourier transform on is one-to-one. However, there is no easy characterization of the image, and thus no easy characterization of the inverse transform. In particular, is no longer valid, as it was stated only under the hypothesis that was "sufficiently nice".
While defines the Fourier transform for functions in, it is not well-defined for other integrability classes, most importantly the space of square-integrable functions. For example, the function is in but not and therefore the Lebesgue integral does not exist. However, the Fourier transform on the dense subspace admits a unique continuous extension to a unitary operator on. This extension is important in part because, unlike the case of, the Fourier transform is an automorphism of the space.
In such cases, the Fourier transform can be obtained explicitly by regularizing the integral, and then passing to a limit. In practice, the integral is often regarded as an improper integral instead of a proper Lebesgue integral, but sometimes for convergence one needs to use weak limit or principal value instead of the limits implicit in an improper integral. and each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure. A general principle in working with the Fourier transform is that Gaussians are dense in, and the various features of the Fourier transform, such as its unitarity, are easily inferred for Gaussians. Many of the properties of the Fourier transform can then be proven from two facts about Gaussians:
- that is its own Fourier transform; and
- that the Gaussian integral
Background
History
In 1822, Fourier claimed that any function, whether continuous or discontinuous, can be expanded into a series of sines. That important work was corrected and expanded upon by others to provide the foundation for the various forms of the Fourier transform used since.Complex sinusoids
In general, the coefficients are complex numbers, which have two equivalent forms :
The product with has these forms:
which conveys both amplitude and phase of frequency Likewise, the intuitive interpretation of is that multiplying by has the effect of subtracting from every frequency component of function Only the component that was at frequency can produce a non-zero value of the infinite integral, because all the other shifted components are oscillatory and integrate to zero.
It is noteworthy how easily the product was simplified using the polar form, and how easily the rectangular form was deduced by an application of Euler's formula.
Negative frequency
Euler's formula introduces the possibility of negative And is defined Only certain complex-valued have transforms But negative frequency is necessary to characterize all other complex-valued found in signal processing, partial differential equations, radar, nonlinear optics, quantum mechanics, and others.For a real-valued has the symmetry property . This redundancy enables to distinguish from But it cannot determine the actual sign of because and are indistinguishable on just the real numbers line.
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 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. If is a periodic function, with period, that has a convergent Fourier series, then:
where are the Fourier series coefficients of, and is the Dirac delta function. In other words, the Fourier transform is a Dirac comb function whose teeth are multiplied by the Fourier series coefficients.