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.

In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components. A useful analogy is the relationship between a series of pure notes (the frequency components) and a musical chord (the function itself). In mathematical physics, the Fourier transform of a signal 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 and linear canonical transform for generalizations.)

## Definition

Suppose is a complex-valued Lebesgue integrable function. The Fourier transform to the frequency domain, , is given by the function:

,   for every real number

When the independent variable t represents time (with SI unit of seconds), the transform variable ω represents angular frequency (in radians per second).

Other notations for this same function are:    and  .  The function is complex-valued in general.   ( represents the imaginary unit.)

If is defined as above, and is sufficiently smooth, then it can be reconstructed by the inverse transform:

,   for every real number

The interpretation of is aided by expressing it in polar coordinate form, , where:

the amplitude
the phase

Then the inverse transform can be written:

which is a recombination of all the frequency components of .  Each component is a complex sinusoid of the form whose amplitude is proportional to and whose initial phase angle (at t = 0) is .

## Normalization factors and alternative forms

The factors before each integral ensure that there is no net change in amplitude when one transforms from one domain to the other and back. The actual requirement is that their product be  .  When they are chosen to be equal, the transform is referred to as unitary. A common non-unitary convention is shown here:

As a rule of thumb, mathematicians generally prefer the unitary transform (for symmetry reasons), and physicists use either convention depending on the application.

The non-unitary form is preferred by some engineers as a special case of the bilateral Laplace transform. The substitution , where is ordinary frequency (hertz), results in another unitary transform that is popular in the field of signal processing and communications systems:

We note that and represent different, but related, functions, as shown in the table below.

Variations of all three forms 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.

## Generalization

There are several ways to define the Fourier transform pair. 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. Using two arbitrary real constants and , the most general definition of the forward 1-dimensional Fourier transform is given by

and the inverse is given by

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

The convention adopted in this article is . The choice of a and b is usually chosen so that it is geared towards the context in which the transform pairs are being used. The non-unitary convention above is . Another very common definition is which is often used in signal processing applications. In this case, the angular frequency becomes ordinary frequency f. If f (or ω) and t carry units, then their product must be dimensionless. For example, t may be in units of time, specifically seconds, and f (or ω) would be in hertz (or radian/s).

## Properties

In this section, all the results are derived for the following definition (normalization) of the Fourier transform:

See also the "Table of important Fourier transforms" section below for other properties of the continuous Fourier transform.

### Completeness

We define the Fourier transform on the set of compactly-supported complex-valued functions of and then extend it by continuity to the Hilbert space of square-integrable functions with the usual inner-product. Then is a unitary operator. That is. and the transform preserves inner-products (see Parseval's theorem, also described below). Note that, refers to adjoint of the Fourier Transform operator.

Moreover we can check that,

where is the Time-Reversal operator defined as,

and is the Identity operator defined as,

### Extensions

The Fourier transform can also be extended to the space integrable functions defined on

where,

and is the space of continuous functions on .

In this case the definition usually appears as

where and is the inner product of the two vectors and .

One may now use this to define the continuous Fourier transform for compactly supported smooth functions, which are dense in The Plancherel theorem then allows us to extend the definition of the Fourier transform to functions on (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 for . The Fourier transform of functions in for the range 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 and are square-integrable and and are their Fourier transforms, then we have Parseval's theorem:

where the bar denotes complex conjugation. Therefore, the Fourier transformation yields an isometric automorphism of the Hilbert space .

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

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 is, the more spread out is . In particular, if we "squeeze" a function in , it spreads out in 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

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. Suppose and are a Fourier transform pair. Without loss of generality, we assume that is normalized:

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

and the expectation value of a function as:

Also define the variance of as:

and similarly define the variance of . Then it can be shown that

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. The momentum and position wave functions are Fourier transform pairs to within a factor of 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 is several times differentiable, then decays rapidly towards zero for .

### 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 can also be translated into algebraic equations.

### Convolution theorem

Main article: Convolution theorem

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

In the current normalization convention, this means that if

where * denotes the convolution operation; then

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 as the impulse response of an LTI system with input and output , since substituting the unit impulse for yields . In this case, represents the frequency response of the system.

Conversely, if can be decomposed as the product of two other functions and such that their product is integrable, then the Fourier transform of this product is given by the convolution of the respective Fourier transforms and , again with a constant scaling factor.

In the current normalization convention, this means that if

then

### Cross-correlation theorem

In an analogous manner, it can be shown that if is the cross-correlation of and :

then the Fourier transform of is:

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

### Tempered distributions

The most general and useful 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 . Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.

## Table of important Fourier transforms

The following table records some important Fourier transforms. and denote Fourier transforms of and , respectively. and may be integrable functions or tempered distributions. Note that the two most common unitary conventions are included.

### Functional relationships

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks

1 Linearity
2 Shift in time domain
3 Shift in frequency domain, dual of 2
4 If is large, then is concentrated around 0 and spreads out and flattens. It is interesting to consider the limit of this as tends to infinity - the delta function
5 Duality property of the Fourier transform. Results from swapping "dummy" variables of and .
6 Generalized derivative property of the Fourier transform
7 This is the dual of 6
8 denotes the convolution of and — this rule is the convolution theorem
9 This is the dual of 8

### Square-integrable functions

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks

10 The rectangular pulse and the normalized sinc function
11 Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter.
12 tri is the triangular function
13 Dual of rule 12.
14 Shows that the Gaussian function is its own Fourier transform. For this to be integrable we must have .
15 common in optics
16
17
18 a>0
19 the transform is the function itself
20 J0(t) is the Bessel function of first kind of order 0
21 it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.
22

Un (t) is the Chebyshev polynomial of the second kind
23 Hyperbolic secant is its own Fourier transform

### Distributions

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks

23 denotes the Dirac delta distribution. This rule shows why the Dirac delta is important: it shows up as the Fourier transform of a constant function.
24 Dual of rule 23.
25 This follows from and 3 and 24.
26 Follows from rules 1 and 25 using Euler's formula:
27 Also from 1 and 25.
28 Here, is a natural number. is the -th distribution derivative of the Dirac delta. This rule follows from rules 7 and 24. Combining this rule with 1, we can transform all polynomials.
29 Here is the sign function; note that this is consistent with rules 7 and 24.
30 Generalization of rule 29.
31 The dual of rule 29.
32 Here is the Heaviside unit step function; this follows from rules 1 and 31.
33 is the Heaviside unit step function and .
34 The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time.

## Fourier transform properties

Notation: denotes that and are a Fourier transform pair.

Conjugation

Scaling

Time reversal

Time shift

Modulation (multiplication by complex exponential)

Multiplication by sin 0t

Multiplication by cos 0t

Integration

Parseval's theorem

• Fourier series
• Laplace transform
• Discrete Fourier transform
• Fractional Fourier transform
• Linear canonical transform

## References

• Fourier Transforms from eFunda - includes tables
• Dym & McKean, Fourier Series and Integrals. (For readers with a background in mathematical analysis.)
• K. Yosida, Functional Analysis, Springer-Verlag, 1968. ISBN 3-540-58654-7
• L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, 1976. (Somewhat terse.)
• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
• R. G. Wilson, "Fourier Series and Optical Transform Techniques in Contemporary Optics", Wiley, 1995. ISBN-10: 0471303577