# Continuous Fourier transform

In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. Loosely, the Fourier transform decomposes a function into a continuous spectrum of the frequencies that comprise that function. In mathematical physics, the Fourier transform of a signal [itex]f(t)[itex] can be thought of as that signal in the "frequency domain." This is similar to the basic idea of the various other Fourier transforms including the Fourier series of a periodic function. (See also fractional Fourier transform for a generalization.)

Suppose [itex]f[itex] is a complex-valued Lebesgue integrable function. We then define its continuous Fourier transform [itex]F[itex] to be also a complex-valued function:

[itex] F(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(t) e^{- i\omega t}\,dt [itex]

for every real number [itex]\omega[itex]. (Here, [itex]i[itex] is the imaginary unit). We think of [itex]\omega[itex] as an angular frequency and [itex]F(\omega)[itex] as the complex number which is the amplitude and phase of the component of the signal [itex]f(t)[itex] at that frequency.

The Fourier transform is close to a self-inverse mapping: if [itex]F(\omega)[itex] is defined as above, and [itex]f[itex] is sufficiently smooth, then

[itex] f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(\omega) e^{ i\omega t}\,d\omega [itex]

for every real number [itex]t[itex].

The factors [itex]1\over\sqrt{2\pi}[itex] before each integral are normalization factors. These factors are arbitrary so long as their product is equal to [itex]1 \over 2 \pi[itex]. The particular values chosen above are referred to as unitary normalization constants; another common choice is 1 and [itex]1/2\pi[itex] for the forward and inverse transforms, respectively. As a rule of thumb, mathematicians generally use the former (for symmetry reasons), while physicists and engineers use the latter.

In addition, the Fourier coordinate [itex]\omega[itex] is sometimes replaced by [itex]2 \pi \nu[itex], integrating over frequency [itex]\nu[itex], in which case the unitary normalization constants are both equal to unity. Another arbitrary convention choice is whether the exponent is [itex]+i\omega t[itex] or [itex]-i\omega t[itex] in the forward transform—the only real requirement is that the forward- and inverse-transform exponents have opposite signs.

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## Properties

### Completeness

If we define the Fourier transform [itex]\mathcal{F}[itex] in this way on the set of complex-valued functions on the line with compact support and extend by continuity to the Hilbert space of square-integrable functions, then it is a unitary operator

[itex] \mathcal{F}:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R}).[itex]

Moreover,

Note that in this relation, conjugation refers to the operator only, not to the entire Fourier transform of the function.

### Orthogonality

The Fourier transform can also be defined for functions (and distributions)

[itex]f: \, \mathbb{R}^n \to \mathbb{C}.[itex]

In the definition, the product [itex]\omega t[itex] is then to be interpreted as the inner product of the two vectors [itex]\omega[itex] and [itex]t[itex]. All the above properties and formulas remain valid. In this context, the functions [itex]\frac{\exp \left(i \omega t \right)}{\sqrt{2 \pi}}[itex] form an orthonormal basis in the space of tempered distributions

[itex]

\int_{-\infty}^\infty

\left(\frac{e^{i\alpha t}}{\sqrt{2\pi}}\right)
\,
\left(\frac{e^{-i\beta t}}{\sqrt{2\pi}}\right)
\,
dt


= \delta(\alpha - \beta). [itex]

The Fourier transform can be thought of as a transformation of coordinate basis in this space.

### The Plancherel theorem and Parseval's theorem

If f(t) and g(t) are square-integrable and F(ω) and G(ω) are their Fourier transforms, then we have the Plancherel theorem:

[itex]\int_{-\infty}^\infty f(t) g(t)^* \, dt = \int_{-\infty}^\infty F(\omega) G(\omega)^* \, d\omega[itex]

(where the star denotes complex conjugation). Therefore, the Fourier transformation yields an isometric automorphism of the Hilbert space L2(R).

Parseval's theorem, a special case of the Plancherel theorem, states that

[itex]\int_{-\infty}^\infty \left| f(t) \right|^2 dt = \int_{-\infty}^\infty \left| F(\omega) \right|^2 d\omega. [itex]

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 [itex]f(t)[itex] is, the more spread out is [itex]F(\omega)[itex]. The only functions which coincide with their own Fourier transforms are the constant multiples of the function [itex]f(t) = \exp \left( \frac{-t^2}{2} \right)[itex]. In a certain sense, this function therefore strikes the precise balance between being concentrated and being spread out. The Fourier transform also translates between smoothness and decay: if [itex]f(t)[itex] is several times differentiable, then [itex]F(\omega)[itex] decays rapidly towards zero for [itex]s \to \plusmn \infin[itex].

This can be more quantitatively expressed as follows. Suppose [itex]f(t)[itex] and [itex]F(\omega)[itex] are a Fourier transform pair. Without loss of generality, we can assume that [itex]f(t)[itex] is normalized:

[itex]\int_{-\infty}^\infty f(t)f^*(t)\,dt=1.[itex]

It follows from Parseval's theorem that F(ω) is also normalized. If we define the expectation value of a function A(t) as:

[itex]\langle A\rangle \equiv \int_{-\infty}^\infty A(t)f(t)f^*(t)\,dt[itex]

and the expectation value of a function [itex]B(\omega)[itex] as:

[itex]\langle B\rangle \equiv \int_{-\infty}^\infty B(\omega)F(t)F^*(\omega)\,d\omega[itex]

and then define the variance of [itex]A(t)[itex] as:

[itex]\Delta^2 A\equiv\langle A^2-\langle A\rangle ^2\rangle [itex]

and similarly for the variance of [itex]B(\omega)[itex], then it can be shown that

[itex]\Delta t \Delta \omega \ge \frac{1}{2}.[itex]

The most famous practical example of this property is found in quantum mechanics. The momentum and position wave functions are Fourier transform pairs to within a factor of [itex]h \over 2 \pi[itex] and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle.

### 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 and cross-correlation theorem

Main article: Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. If [itex]f(t)[itex] and [itex]g(t)[itex] are integrable functions with Fourier transforms [itex]F(\omega)[itex] and [itex]G(\omega)[itex], respectively, and if the convolution of [itex]f[itex] and [itex]g[itex] exists and is integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms [itex]F(\omega) G(\omega)[itex] (possibly multiplied by a constant factor depending on the Fourier normalization convention).

In the current normalization convention, this means that if

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

then

[itex]H(\omega) = \sqrt{2\pi} F(\omega)G(\omega).\,[itex]

Also, if the product [itex]f(t) g(t)[itex] is integrable, then the Fourier transform of this product is given by the convolution of [itex]F(\omega)[itex] and [itex]G(\omega)[itex], again with a constant factor.

In the current normalization convention, this means that if

[itex]h(t) = f(t) g(t)\,[itex]

then

[itex]H(\omega) = \sqrt{2\pi} \int_{-\infty}^\infty F(\tau)G(\omega - \tau)\,d\tau.[itex]

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

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

then the Fourier transform of [itex]h(t)[itex] is:

[itex]H(\omega) = \sqrt{2\pi}\,F^*(\omega)\,G(\omega)[itex]

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

### Tempered distributions

The most general and natural context for studying the continuous 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 the constant function [itex]1/\sqrt{2\pi}[itex]. Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.

If a function [itex]f: \, \mathbb{R} \to \mathbb{C}[itex] is square-integrable, that is

[itex]\int_{-\infty}^\infty |f(t)|^2 \, dt < \infty,[itex]

then it can be viewed as a tempered distribution and hence has a Fourier transform. This transform is again square integrable.

## Extension to higher dimensions

The Fourier transform can be extended to an N-dimensional space in a straightforward manner. If f(x) is a function of an N-dimensional vector x in the space, and k is the corresponding vector in the transform space (sometimes called the wavevector), then

[itex]F(\mathbf{k})=

\left(\frac{1}{\sqrt{2\pi}}\right)^N \int_{\mathbb{R}^N} f (\mathbf{x})\,e^{-i\,\mathbf{k} \cdot \mathbf{x}}\,d\mathbf{x} [itex]

where dx is an N-dimensional infinitesimal volume element in the space and the product in the exponential is the dot product. Using the N-dimensional orthogonality relationship:

[itex]\delta(\mathbf{k})=\left(\frac{1}{2\pi}\right)^N

\int_{\mathbb{R}^N} e^{\pm i\,\mathbf{k} \cdot \mathbf{x}}\,d\mathbf{x} [itex]

yields the inverse transform:

[itex]f(\mathbf{x})=

\left(\frac{1}{\sqrt{2\pi}}\right)^N \int_{\mathbb{R}^N} F (\mathbf{k})\,e^{+i\,\mathbf{k} \cdot \mathbf{x}}\,d\mathbf{k} [itex]

## Table of important Fourier transforms

The following table records some important Fourier transforms. [itex]F(\omega)[itex] and [itex]G(\omega)[itex] denote the Fourier transforms of [itex]f(t)[itex] and [itex]g(t)[itex], respectively. [itex]f[itex] and [itex]g[itex] may be integrable functions or tempered distributions. Note that this table's relations, and in particular constant factors such as [itex]\sqrt{2\pi}[itex], depend upon the convention used for the Fourier transform definition above (although the general form of the relations is always the same).

Signal Fourier transform Remarks
1 [itex]a f(t) + b g(t)\,[itex] [itex]a F(\omega) + b G(\omega)\,[itex] Linearity
2 [itex]f(t - a)\,[itex] [itex]e^{- i\omega a} F(\omega)\,[itex] Shift in time domain
3 [itex]e^{ iat} f(t)\,[itex] [itex]F(\omega - a)\,[itex] Shift in frequency domain
4 [itex]f(a t)\,[itex] [itex]|a|^{-1} F \left( \frac{\omega}{a} \right)\,[itex] If [itex]a\,[itex] is large, then [itex]f(a t)\,[itex] is concentrated around 0 and [itex]|a|^{-1}F(\frac{\omega}{a})\,[itex] spreads out and flattens
5 [itex]\frac{d^n f(t)}{dt^n}\,[itex] [itex] (i\omega)^n F(\omega)\,[itex] Generalized derivative property of the Fourier transform
6 [itex]t^n f(t)\,[itex] [itex]i^n \frac{d^n F(\omega)}{d\omega^n}\,[itex] This is the inverse rule to 5
7 [itex](f * g)(t)\,[itex] [itex]\sqrt{2\pi} F(\omega) G(\omega)\,[itex] [itex]f * g\,[itex] denotes the convolution of [itex]f\,[itex] and [itex]g\,[itex] — this rule is the convolution theorem
8 [itex]f(t) g(t)\,[itex] [itex](F * G)(\omega) \over \sqrt{2\pi}\,[itex] This is the inverse of 7
9 [itex]\delta(t)\,[itex] [itex]\frac{1}{\sqrt{2\pi}}\,[itex] [itex]\delta(t)\,[itex] denotes the Dirac delta distribution
10 [itex]1\,[itex] [itex]\sqrt{2\pi}\delta(\omega)\,[itex] Inverse of 9. This rule shows why the Dirac delta is important: it shows up as the Fourier transform of everyday functions
11 [itex]t^n\,[itex] [itex]i^n \sqrt{2\pi} \delta^{(n)} (\omega)\,[itex] Here, [itex]n\,[itex] is a natural number. [itex]\delta^n(\omega)\,[itex] is the n-th distribution derivative of the Dirac delta. This rule follows from rules 6 and 10. Combining this rule with 1, we can transform all polynomials
12 [itex]e^{i a t}\,[itex] [itex]\sqrt{2 \pi} \delta(\omega - a)\,[itex] This follows from and 3 and 10
13 [itex]\cos (a t)\,[itex] [itex]\sqrt{2 \pi} \frac{\delta(\omega - a) + \delta(\omega + a)}{2}\,[itex] Follows from rules 1 and 12 using [itex]\cos(a t) = \frac{1}{2} \left( e^{i a t} + e^{-i a t}\right)\,[itex] (Euler's formula)
14 [itex]\sin( at)\,[itex] [itex]\sqrt{2 \pi}\frac{\delta(\omega - a) - \delta(\omega + a)}{2i}\,[itex] Also from 1 and 12
15 [itex]\exp(-a t^2)\,[itex] [itex]\frac{1}{\sqrt{2a}} \exp\left(\frac{-\omega^2}{4a}\right)\,[itex] Shows that the Gaussian function [itex]\exp(-t^2/2)\,[itex] is its own Fourier transform
16 [itex]W \sqrt{\frac{2}{\pi}} \mathrm{sinc}(W t)\,[itex] [itex]\mathrm{rect}\left(\frac{\omega}{2 W}\right)\,[itex] The rectangular function is a perfect low-pass filter and the sinc function is its time equivalent
17 [itex]\frac{1}{t}\,[itex] [itex]-i\sqrt{\frac{\pi}{2}}\sgn(\omega)\,[itex] Here [itex]\sgn(\omega)\,[itex] is the sign function; note that this is consistent with rules 6 and 10
18 [itex]\frac{1}{t^n}\,[itex] [itex]-i\sqrt{\frac{\pi}{2}}\frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega)\,[itex] Generalization of rule 17
19 [itex]\sgn(t)\,[itex] [itex]\sqrt{\frac{2}{\pi}}(i\omega)^{-1}\,[itex] The inverse of rule 17
20 [itex]\sqrt{2\pi}\mathrm{H}(t)\,[itex] [itex]\frac{1}{i\omega} + \pi\delta(\omega)\,[itex] Here [itex]\mathrm{H}(t)\,[itex] is the Heaviside step function; this follows from rules 1 and 19

## References

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