Fourier transform


 * This article specifically discusses Fourier transformation of functions on the real line; for other kinds of Fourier transformation, see Fourier analysis and list of Fourier-related transforms. For generalizations, see fractional Fourier transform and linear canonical transform.

In mathematics, the continuous Fourier transform is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation of the original function (where the original function is often a function in the time-domain). In this specific case, both domains are continuous and unbounded. The term Fourier transform can refer to either the frequency domain representation of a function or to the process/formula that "transforms" one function into the other.

Definitions
There are several common conventions for defining the Fourier transform of a complex-valued Lebesgue integrable function f:R→C. One common definition is:


 * $$F(\nu) = \int_{-\infty}^{\infty} f(t)\ e^{-2\pi i \nu t}\,dt, $$  for every real number ν.

When the independent variable t represents time (with SI unit of seconds), the transform variable ν represents ordinary frequency (in hertz). If f is Hölder continuous, then it can be reconstructed from F by the inverse transform:


 * $$f(t) = \int_{-\infty}^{\infty} F(\nu)\ e^{2 \pi i \nu t}\,d\nu,$$  for every real number t.

Other notations for $$F(\nu)\,$$ are: $$\hat{f}(\nu),$$  $$\mathcal{F} \big\{f(t)\big\},$$  $$\mathcal{F}\{f\}(\nu),$$ and $$(\mathcal{F}f)(\nu).$$

The interpretation of $$F(\nu)\,$$ is aided by expressing it in polar coordinate form: $$F(\nu) = A(\nu)\ e^{i \phi (\nu)},\,$$  where:


 * $$A(\nu) = |F(\nu)|, \, $$  the amplitude
 * $$\phi (\nu) = \arg \big( F(\nu) \big), \, $$  the phase.  (see arg function)

Then the inverse transform can be written:


 * $$f(t) = \int _{-\infty}^{\infty} A(\nu)\ e^{ i(2\pi \nu t +\phi (\nu))}\,d\nu,$$

which is a recombination of all the frequency components of f(t). Each component is a complex sinusoid of the form e2πiνt whose amplitude is A(ν) and whose initial phase angle (at t=0) is φ(ν).

The Fourier transform is often written in terms of angular frequency:  ω = 2πν whose units are radians per second.

The substitution ν = ω/(2π) into the formulas above produces this convention:


 * $$F(\omega) = \int _{-\infty}^\infty f(t)\ e^{- i\omega t}\,dt $$


 * $$f(t) = \frac{1}{2\pi} \int _{-\infty}^{\infty} F(\omega)\ e^{ i\omega t}\,d\omega, $$

which is also a bilateral Laplace transform evaluated at s=iω.

The 2π factor can be split evenly between the Fourier transform and the inverse, which leads to another popular convention:


 * $$ F(\omega) = \frac{1}{\sqrt{2\pi}} \int _{-\infty}^\infty f(t)\ e^{- i\omega t}\,dt $$


 * $$f(t) = \frac{1}{\sqrt{2\pi}} \int _{-\infty}^{\infty} F(\omega)\ e^{ i\omega t}\,d\omega. $$

This makes the transform a unitary one.

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. Other than that, the choice is (again) a matter of convention.

Some Fourier transform properties
Notation: $$f(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad F(\omega)$$ denotes that f(t) and F(ω) are a Fourier transform pair.


 * Linearity
 * $$a\cdot f(t) + b\cdot g(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad a\cdot F(\omega) + b\cdot G(\omega) $$


 * Multiplication

\frac{1}{\sqrt{2\pi}}\cdot (F*G)(\omega) \,$$ \frac{1}{2\pi}\cdot (F*G)(\omega) \,$$ (F*G)(\nu) \,$$
 * $$f(t)\cdot g(t) \,$$
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * (unitary normalization convention)
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * (non-unitary convention)
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * (ordinary frequency)
 * }
 * Modulation

\begin{align} f(t)\cdot \cos \omega_{0}t &\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{1}{2}[F(\omega+\omega_{0})+F(\omega-\omega_{0})],\qquad \omega_{0} \in \mathbb{R} \\ f(t)\cdot \sin \omega_{0}t &\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{i}{2}[F(\omega+\omega_{0})-F(\omega-\omega_{0})] \\ f(t)\cdot e^{i\omega_{0}t} &\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad F(\omega-\omega_{0}) \end{align} \,$$


 * Convolution

\sqrt{2\pi}\cdot F(\omega)\cdot G(\omega) \,$$ F(\omega)\cdot G(\omega) \,$$ F(\nu)\cdot G(\nu) \,$$
 * $$ (f*g)(t) \,$$
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * (unitary convention)
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * (non-unitary convention)
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * $$\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
 * (ordinary frequency)
 * }
 * Integration (example of convolution)

\int_{-\infty}^{t} f(\tau)\, d\tau = (f*u)(t)$$  $$ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \frac{1}{i\omega}F(\omega)+\pi F(0)\cdot \delta(\omega), \,$$


 * Conjugation
 * $$\overline{f(t)} \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \overline{F(-\omega)}$$


 * Scaling
 * $$ f(at) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{1}{|a|}F\biggl(\frac{\omega}{a}\biggr), \qquad a \in \mathbb{R}, a \ne 0$$


 * Time reversal
 * $$f(-t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad F(-\omega)$$


 * Time shift
 * $$f(t-t_0) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad e^{-i\omega t_0}\cdot F(\omega)$$


 * Parseval's theorem


 * $$\int_{-\infty}^{\infty} f(t)\cdot \overline{g(t)}\, dt \,$$
 * $$= \int_{-\infty}^{\infty} F(\omega)\cdot \overline{G(\omega)}\, d\omega \,$$
 * (unitary convention)
 * $$= \frac{1}{2\pi}\cdot \int_{-\infty}^{\infty} F(\omega)\cdot \overline{G(\omega)}\, d\omega \,$$
 * (non-unitary convention)
 * $$= \int_{-\infty}^{\infty} F(\nu)\cdot \overline{G(\nu)}\, df \,$$
 * (ordinary frequency)
 * }
 * $$= \int_{-\infty}^{\infty} F(\nu)\cdot \overline{G(\nu)}\, df \,$$
 * (ordinary frequency)
 * }
 * }

The section "Table of important Fourier transforms" (below) documents more properties of the continuous Fourier transform.

Generalization
Using two arbitrary real constants a and b, the most general definition of the forward 1-dimensional Fourier transform is given by:


 * $$F(\omega) = \sqrt{\frac{|b|}{(2 \pi)^{1-a}}} \int_{-\infty}^{+\infty} f(t) \cdot e^{-i b \omega t} \, dt, $$

and the inverse is given by:


 * $$f(t) = \sqrt{\frac{|b|}{(2 \pi)^{1+a}}} \int_{-\infty}^{+\infty} F(\omega) \cdot e^{i b \omega t} \, d\omega. $$

Note that the transform definitions are symmetric; they can be reversed by simply changing the signs of a and b.

The ordinary frequency convention corresponds to (a,b) = (0,2π), and in that case the variable ω is changed to ν. If ν and t carry units, their product must be dimensionless. For example, t may be in units of time, specifically seconds, and ν would be in hertz.

The unitary, angular frequency convention is (a,b) = (0,1), and the non-unitary convention (above) is (a,b) = (1,1).

The "forward" and "inverse" transforms are always defined so that the operation of both transforms in either order on a function will return the original function. In other words, the composition of the transform pair is defined to be the identity transformation.

Completeness
We define the Fourier transform on the set of integrable complex-valued functions of R and then extend it by continuity to the Hilbert space of square-integrable functions with the usual inner-product. Then $$ \mathcal{F}$$: L2(R) → L2(R) is a unitary operator. That is $$\mathcal{F}^*=\mathcal{F}^{-1}$$ and the transform preserves inner-products (see Parseval's theorem, also described below). Note that, $$\mathcal{F}^*$$ refers to adjoint of the Fourier Transform operator. Moreover we can check that:


 * $$ \mathcal{F}^2 = \mathcal{J},\quad \mathcal{F}^3 = \mathcal{F}^* = \mathcal{F}^{-1}, \quad \mbox{and} \quad \mathcal{F}^4 = \mathcal{I}, $$

where $$\mathcal{J}$$ is the Time-Reversal operator defined as:


 * $$ \mathcal{J}(f)(t) = f(-t), $$

and $$\mathcal{I}$$ is the Identity operator defined as:


 * $$ \mathcal{I}(f)(t) = f(t).$$

Multi-dimensional version
The Fourier transform, can be expanded to arbitrary dimension $$n$$. In the unitary, angular frequency convention, the definition is:


 * $$F(\boldsymbol{\omega}) = \mathcal{F}\{f\}(\boldsymbol{\omega}) \ \stackrel{\mathrm{def}}{=}\

\left(\frac{1}{\sqrt{2\pi}}\right)^{n}\int_{\R^n} f(\mathbf{x})\cdot e^{-i(\boldsymbol{\omega}\cdot \mathbf{x})}\,d\mathbf{x},$$

where $$\mathbf{x}$$ and $$\boldsymbol{\omega}$$ are $$n$$-dimensional vectors, and $$\boldsymbol{\omega}\cdot \mathbf{x}$$ is the inner product, also written $$\left\langle \boldsymbol{\omega},\mathbf{x} \right\rangle,$$ of the 2 vectors. The integration is performed over all $$n$$ dimensions.

The function $$f(\mathbf{x})$$ is assumed to belong to the "space" of integrable functions defined on Rn:


 * $$ \mathcal{F}:L^1(\mathbb{R}^n)\rightarrow C(\mathbb{R}^n),$$

where:


 * $$ L^1(\mathbb{R}^n) = \{f: \, \mathbb{R}^n \to \mathbb{C} \;\big|\; \int_{\mathbb{R}^n} |f(\mathbf{x})|\, d\mathbf{x} < \infty\},$$

and C(Rn) is the space of continuous functions on Rn.

One may now use this to define the Fourier transform for compactly supported smooth functions, which are dense in L2(Rn). The Plancherel theorem then allows us to extend the definition of the Fourier transform to functions on L2(Rn) (even those not compactly supported) by continuity arguments. All the properties and formulas listed on this page apply to the Fourier transform so defined.

Unfortunately, further extensions become more technical. One may use the Hausdorff-Young inequality to define the Fourier transform for f ∈ Lp(Rn) for 1 ≤ p ≤ 2. The Fourier transform of functions in Lp for the range 2 < p < ∞ requires the study of distributions, since the Fourier transform of some functions in these spaces is no longer a function, but rather a distribution.

The Plancherel theorem and Parseval's theorem
It should be noted that depending on the author either of these theorems might be referred to as the Plancherel theorem or as Parseval's theorem.

If f(t) and g(t) are square-integrable and F(ω) and G(ω) are their unitary Fourier transforms, then we have Parseval's theorem:


 * $$\int_{\mathbb{R}^n} f(t) \bar{g}(t) \, dt = \int_{\mathbb{R}^n} F(\omega) \bar{G}(\omega) \, d\omega,$$

where the bar denotes complex conjugation. Therefore, the Fourier transform yields an isometric automorphism of the Hilbert space L2(Rn).

The Plancherel theorem, which is equivalent to Parseval's theorem, states:


 * $$\int_{\mathbb{R}^n} \left| f(t) \right|^2\, dt = \int_{\mathbb{R}^n} \left| F(\omega) \right|^2\, d\omega. $$

This theorem is usually interpreted as asserting the unitary property of the Fourier transform. See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.

Localization property
As a rule of thumb: the more concentrated f(t) is, the more spread out F(ω) is. In particular, if we "squeeze" a function in t, it spreads out in ω and vice-versa; and we cannot arbitrarily concentrate both the function and its Fourier transform.

Therefore a function which equals its Fourier transform strikes a precise balance between being concentrated and being spread out. It is easy in theory to construct examples of such functions (called self-dual functions) because the Fourier transform has order 4 (that is, iterating it four times on a function returns the original function). The sum of the four iterated Fourier transforms of any function will be self-dual. There are also some explicit examples of self-dual functions, the most important being constant multiples of the Gaussian function


 * $$f(t) = \exp \left( \frac{-t^2}{2} \right).$$

This function is related to Gaussian distributions, and in fact, is an eigenfunction of the Fourier transform operators. Again, it is worth stressing that the mere fact that the Gaussian is self-dual does not make it in any way special: many self-dual functions exist.

The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of a Fourier Uncertainty Principle. Suppose f(t) and F(ω) are a Fourier transform pair for a finite-energy (i.e. square-integrable) function. Without loss of generality, we assume that f(t) is normalized:


 * $$\int_{-\infty}^\infty |f(t)|^2 \,dt=1.$$

It follows from Parseval's theorem that F(ω) is also normalized.

Define the expected location of a particle (with probability density |f(t)|2) as


 * $$u_f \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty t|f(t)|^2\,dt.$$

and the expectation value of the momentum of the particle (with probability density |F(ω)|2) as


 * $$\xi_F \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty \omega |F(\omega)|^2\,d\omega.$$

Also define the variances around the above-defined average values as


 * $$\sigma^2_{f} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty (t-u_f)^2|f(t)|^2\,dt $$

and


 * $$\sigma^2_{F} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty (\omega-\xi_F)^2 |F(\omega)|^2\,d\omega. $$

Then it can be shown that


 * $$\sigma^2_{f}\, \sigma^2_{F} \ge \frac{1}{4}.$$

The equality is achieved for the Gaussian function listed above, which shows that the Gaussian function is maximally concentrated in "time-frequency". The most famous practical application of this property is found in quantum mechanics. Following from the axioms of quantum mechanics, the momentum and position wave functions are Fourier transform pairs to within a factor of h/2π and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle.

The Fourier transform also translates between smoothness and decay. If f(t) is several times differentiable, then F(ω) decays rapidly towards zero for ω → ± ∞.

Eigenfunctions
One important choice of orthonormal (under $$ L^2 $$ norm) eigenfunctions of the Fourier transform are the Hermite functions


 * $${\psi}_n(t) = \frac{1}{\sqrt{n!\,2^n\sqrt{\pi}}}\,e^{-t^2/2}H_n(t).\,\!$$

where $$H_n(t)$$ is the physicist-defined Hermite polynomial. When using the unitary definition of the Fourier Transform, the relation is


 * $$ \mathcal{F} \left\{ {\psi}_n(t) \right\} = (-i)^n {\psi}_n(\omega) \,\!$$

However, this choice of eigenfunctions is not unique. Because there are only four different eigenvalues of the Fourier transform (±1 and ±i), each highly degenerate, any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction. However, the choice of Hermite functions is convenient because they are exponentially localized in both frequency and time domains, and thus give rise to the fractional Fourier transform used in time-frequency analyses.

Analysis of differential equations
Fourier transforms, and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with differentiation in the following sense: if f(t) is a differentiable function with Fourier transform F(ω), then the Fourier transform of its derivative is given by iω F(ω). This can be used to transform differential equations into algebraic equations. Note that this technique only applies to problems whose domain is the whole set of real numbers. By extending the Fourier transform to functions of several variables (as outlined below), partial differential equations with domain Rn can also be translated into algebraic equations.

Convolution theorem

 * Main article: Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. If f(t) and h(t) are integrable functions with Fourier transforms F(ω) and H(ω) respectively, and if the convolution of f and h exists and is absolutely integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms F(ω) H(ω) (possibly multiplied by a constant factor depending on the Fourier normalization convention).

In the unitary normalization convention, this means that if:


 * $$g(t) = \{f*h\}(t) = \int_{-\infty}^\infty f(\tau)h(t - \tau)\,d\tau,$$

where * denotes the convolution operation, then:


 * $$G(\omega) = \sqrt{2\pi}\cdot F(\omega)H(\omega).\,$$

The above formulas hold true for functions defined on both one- and multi-dimension real space. In linear time invariant (LTI) system theory, it is common to interpret h(t) as the impulse response of an LTI system with input f(t) and output g(t), since substituting the unit impulse for f(t) yields g(t)=h(t). In this case, H(ω) represents the frequency response of the system.

Conversely, if f(t) can be decomposed as the product of two other functions p(t) and q(t) such that their product p(t)q(t) is integrable, then the Fourier transform of this product is given by the convolution of the respective Fourier transforms P(ω) and Q(ω), again with a constant scaling factor.

In the unitary normalization convention, this means that if f(t) = p(t) q(t) then:


 * $$F(\omega) = \frac{1}{\sqrt{2\pi}}  \bigg( P(\omega) * Q(\omega)  \bigg) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty P(\alpha)Q(\omega - \alpha)\,d\alpha.$$

Cross-correlation theorem
In an analogous manner, it can be shown that if $$g(t)$$ is the cross-correlation of $$f(t)$$ and $$h(t)$$:


 * $$g(t)=(f\star h)(t) = \int_{-\infty}^\infty \bar{f}(\tau)\,h(t+\tau)\,d\tau$$

then the Fourier transform of $$g(t)$$ is:


 * $$G(\omega) = \sqrt{2\pi}\,\overline{F}(\omega)\,H(\omega)$$

where capital letters are again used to denote the Fourier transform.

Tempered distributions
The most general and useful context for studying the Fourier transform is given by the tempered distributions; these include all the integrable functions mentioned above and have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution and the rule for the inverse of the Fourier transform is universally valid. Furthermore, the useful Dirac delta is a tempered distribution but not a function; its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used). Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.

Domain and range of the Fourier transform
The domain and range of the Fourier transform cannot be described as two well-defined sets of functions. Instead, they can be chosen in several different ways depending on exactly what is meant by a function and an integral. Furthermore, for some pair of domain and range which can be described for the Fourier transform, it may sometimes be of interest to consider the restriction of the transform to a proper subset of the domain. In general, however, it is of interest to describe as "large" sets as possible for the domain. Such extensions can be done in different ways and may lead to domains where either one is not a subset of the other. Some examples of domains and ranges which are described in the literature are


 * The set of Schwartz functions is closed under the Fourier transform. Schwartz functions are rapidly decaying functions and does not include all functions which are relevant for the Fourier transform.


 * $$L^2$$ is closed under the Fourier transform. This is a much richer set than Schwartz functions but is, technically speaking, not a set of functions but rather equivalence classes of functions.


 * The set of tempered distributions is closed under the Fourier transform. Tempered distributions are also a form a generalization of functions.  It is in this generality that one can define the Fourier Transform of objects like the Dirac delta function.

Table of important Fourier transforms
The following table records some important Fourier transforms. G and H denote Fourier transforms of g(t) and h(t), respectively. g and h may be integrable functions or tempered distributions. Note that the two most common unitary conventions are included.

About notation
The Fourier transform is a mapping on a function space. This mapping is here denoted $$\mathcal{F}$$ and $$\mathcal{F}\{s\}$$ is used to denote the Fourier transform of the function s. This mapping is linear, which means that $$\mathcal{F}$$ can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the signal s) can be used to write $$\mathcal{F} s$$ instead of $$\mathcal{F}\{s\}$$. Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value $$\omega$$ for its variable, and this is denoted either as $$\mathcal{F}\{s\}(\omega)$$ or as $$(\mathcal{F} s)(\omega)$$. Notice that in the former case, it is implicitly understood that $$\mathcal{F}$$ is applied first to s and then the resulting function is evaluated at $$\omega$$, not the other way around.

In mathematics and various applied sciences it is often necessary to distinguish between a function s and the value of s when its variable equals t, denoted s(t). This means that a notation like $$\mathcal{F}\{s(t)\}$$ formally can be interpreted as the Fourier transform of the values of s at t, which must be considered as an ill-formed expression since it describes the Fourier transform of a function value rather than of a function. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed. For example, $$\mathcal{F}\{ \mathrm{rect}(t) \} = \mathrm{sinc}(\omega)$$ is sometimes used to express that the Fourier transform of a rectangular function is a sinc function, or $$\mathcal{F}\{s(t+t_{0})\} = \mathcal{F}\{s(t)\} e^{i \omega t_{0}}$$ is used to express the shift property of the Fourier transform. Notice, that the last example is only correct under the assumption that the transformed function is a function of t, not of $$t_{0}$$. If possible, this informal usage of the $$\mathcal{F}$$ operator should be avoided, in particular when it is not perfectly clear which variable the function to be transformed depends on.