Continuous Fourier transform

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Fourier transforms
Continuous Fourier transform
Fourier series
Discrete Fourier transform
Discrete-time Fourier transform
List of Fourier-related transforms>Related transforms [http://encycl.opentopia.com/ edit]

In mathematics, the continuous 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. In mathematical physics, the Fourier transform of a signal [x(t)\,] 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.)

Definition

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

[ X(\omega) = \frac} \int_^\infty x(t) e^\,dt ],   for every real number [\omega \,].

Other notations for this same function are:  [\hat(\omega)\,]  and  [\mathcal\(\omega)\,].  The function is complex-valued in general.   ([i\,], of course, represents the imaginary unit.)

If [X(\omega)\,] is defined as above, and [x(t)\,] is sufficiently smooth, then it can be reconstructed by the inverse transform:

[ x(t) = \frac} \int_^ X(\omega) e^\,d\omega ],   for every real number [t \,].

The interpretation of [X(\omega)\,] is aided by expressing it in polar coordinate form, [X(\omega) = A(\omega )e^ \,], where:

[A(\omega ) = |X(\omega)| \, ] the amplitude

[\phi (\omega ) = \angle X(\omega) \, ] the phase

Then the inverse transform can be written:

[ x(t) = \int_^ \frac}\cdot e^\,d\omega ]

which is a recombination of all the frequency components of [x(t)\,]. Each component is a complex sinusoid of the form [e^\,] whose amplitude is proportional to [A(\omega)\,] and whose initial phase (at t = 0) is [\phi (\omega )\,].

Normalization factors and alternative forms

The factors [1\over\sqrt] before each integral ensure that there is no net change in amplitude when one transforms from one domain to the other and back. A necessary and sufficient condition is that the product of the factors be [1 \over 2 \pi]. When they are chosen to be equal, the transform is referred to as unitary. A common non-unitary convention is shown here:

[ X(\omega) = \int_^\infty x(t) e^\,dt ]

[ x(t) = \frac \int_^ X(\omega) e^\,d\omega ]

As a rule of thumb, mathematicians generally prefer the unitary transform (for symmetry reasons), engineers commonly use the non-unitary form (as a special case of the bilateral Laplace transform), and physicists use either convention depending on the application.

Yet another popular form is a transform to or from the domain of ordinary frequency, [f = \frac\,].  In that case, the necessary product of the factors is just 1, so a unitary transform is the obvious choice:

[ X(f) = \int_^\infty x(t) e^\,dt ]

[ x(t) = \int_^\infty X(f) e^\,df ]

This form is commonly used in applications related to signal processing and communications systems.

And 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.

In many (but not all) applications of the CFT, the independent variable t represents time (with SI unit of seconds), while in mathematics this is rarely the case. In these situations, the transform variable ω represents the angular frequency (in radians per second), whereas f represents ordinary frequency (in hertz).

Summary of popular forms of the Fourier transform

angular frequency [ \omega \, ] (rad/s)	unitary	[ X_1(\omega) \equiv \frac} \int_^ x(t) \ e^\, dt \ = \frac} X_2(\omega) = \frac} X_3(\frac)\,] [ x(t) = \frac} \int_^ X_1(\omega) \ e^\, d \omega \ ]
	non-unitary	[ X_2(\omega) \equiv \int_^ x(t) \ e^ \ dt \ = \sqrt\ X_1(\omega) = X_3(\frac)\,] [ x(t) = \frac \int_^ X_2(\omega) \ e^ \ d \omega \ ]
ordinary frequency [ f \, ] (hertz)	unitary	[ X_3(f) \equiv \int_^ x(t) \ e^ \ dt \ = \sqrt\ X_1(2 \pi f) = X_2(2 \pi f)\,] [ x(t) = \int_^ X_3(f) \ e^\, df \ ]

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 [a] and [b], the most general definition of the forward 1-dimensional Fourier transform is given by:

[X(\omega) = \sqrt}} \int_^ x(t) e^ \, dt ]

and the inverse is given by:

[x(t) = \sqrt}} \int_^ X(\omega) e^ \, d\omega ]

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 [(a,b) = (0,1)]. 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 [(a,b)=(1,1)]. Another very common definition is [(a,b)=(0,2\pi)] which is often used in signal processing applications. In this case, the angular frequency [\omega] 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

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

Completeness

If we define the Fourier transform [\mathcal] 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

[ \mathcal:L^2(\mathbb)\rightarrow L^2(\mathbb).]

Moreover,

[ \mathcal^2 f(x)=f(-x),\quad\mbox\quad\mathcal^*=\mathcal^=\mathcal^3.]

Note that in this relation, [\mathcal^*] refers to adjoint of the Fourier Transform operator.

Extensions

The Fourier transform can also be extended to integrable functions

[f: \, \mathbb^n \to \mathbb.]

In this case the definition usually appears as

[ \mathcal\(w) \equiv \int_ f(x)e^\,dx.]

Where [\omega\in \mathbb^n] and [ \omega \cdot x] is the inner product of the two vectors [\omega] and [x]. All the above properties and formulas remain valid.

Again one may also use this to define the continuous Fourier transform for compactly supported smooth funcitions, which are dense in [L^2(\mathbb^n).] Parseval's theorem then allows us to extend the definition of the Fourier transform to functions on [L^2(\mathbb^n)] by continuity.

Unfortunately, further extensions become more technical. One may use the Hausdorff-Young inequality inequality to define the Fourier transform for [f\in L^p(\mathbb^n)] for [ 1\leq p\leq 2]. The Fourier transform of functions in [ L^p ] for the range [ 2distribution.

The Plancherel theorem and Parseval's theorem

If [ f(t) ] and [ g(t) ] are square-integrable and [ F(\omega) ] and [ G(\omega)] are their Fourier transforms, then we have the Parseval's theorem:

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

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

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

[\int_^\infty \left| x(t) \right|^2 dt = \int_^\infty \left| X(\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 is [F(\omega)]. 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 function

[f(t) = \exp \left( \frac \right).]

This function is related to Gaussian distributions, and in fact, is an eigenfunction of the Fourier transform operators. The Fourier transform also translates between smoothness and decay: if [f(t)] is several times differentiable, then [F(\omega)] decays rapidly towards zero for [s \to \plusmn \infin].

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

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

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

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

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

[\langle B\rangle \equiv \int_^\infty B(\omega)F(\omega)F^*(\omega)\,d\omega]

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

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

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

[\Delta t \Delta \omega \ge \frac.]

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 [h \over 2 \pi] 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

Main article: Convolution theorem

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

In the current normalization convention, this means that if

[y(t) = h(t)*x(t) = \int_^\infty h(\tau)x(t - \tau)\,d\tau]

then

[Y(\omega) = \sqrt\cdot H(\omega)X(\omega).\,]

In LTI system theory, it is common to interpret [h(t)] as the impulse response of a linear, time-invariant system with input [x(t)] and output [y(t)], since substituting the unit impulse for [x(t)] yields [y(t)=h(t)]. In this case, [H(\omega)] represents the frequency response of the system.

Conversely, if [x(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(\omega)] and [Q(\omega)], again with a constant scaling factor.

In the current normalization convention, this means that if

[x(t) = p(t) q(t)\,]

then

[X(\omega) = \frac} \bigg( P(\omega) * Q(\omega) \bigg) = \frac} \int_^\infty P(\alpha)Q(\omega - \alpha)\,d\alpha.]

Cross-correlation theorem

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

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

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

[H(\omega) = \sqrt\,F^*(\omega)\,G(\omega)]

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 [1/\sqrt]. Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.

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

[F(\mathbf)=\left(\frac}\right)^N\int_^N} f (\mathbf)\,e^ \cdot \mathbf}\,d\mathbf]

where dx is an N-dimensional infinitesimal volume element in the space and the product in the exponential is the dot product. Thinking of the function [ f(x)=1 ] as a tempered distribution one can derive the relationship:

[\delta(\mathbf)=\left(\frac\right)^N\int_^N} e^ \cdot \mathbf}\,d\mathbf]

as tempered distributions .This yields the inverse transform:

[f(\mathbf)=\left(\frac}\right)^N\int_^N} F (\mathbf)\,e^ \cdot \mathbf}\,d\mathbf]

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.

Functional relationships

G \left( \frac \right)\,] G \left( \frac \right)\,] G \left( \frac \right)\,] spreads out and flattens

	Signal	Fourier transform unitary, angular frequency	Fourier transform unitary, ordinary frequency	Remarks
	[ g(t)\!\equiv\!] [ \frac} \int_^\!\!G(\omega) e^ d \omega \,]	[ G(\omega)\!\equiv\!] [\frac} \int_^\!\!g(t) e^ dt \,]	[ G(f)\!\equiv ] [\int_^\!\!g(t) e^ dt \,]
1	[a\cdot g(t) + b\cdot h(t)\,]	[a\cdot G(\omega) + b\cdot H(\omega)\,]	[a\cdot G(f) + b\cdot H(f)\,]	Linearity
2	[g(t - a)\,]	[e^ G(\omega)\,]	[e^ G(f)\,]	Shift in time domain
3	[e^ g(t)\,]	[G(\omega - a)\,]	[G \left(f - \frac\right)\,]	Shift in frequency domain, dual of 2
4	[g(a t)\,]	[\frac>
[\frac>
a|\,] is large, then [g(a t)\,] is concentrated around 0 and [\frac>
5	[G(t)\,]	[ g(-\omega)\,]	[ g(-f)\,]	Duality property of the Fourier transform. Results from swapping "dummy" variables of [ t \,] and [ \omega \,].
6	[\frac\,]	[ (i\omega)^n G(\omega)\,]	[ (i 2\pi f)^n G(f)\,]	Generalized derivative property of the Fourier transform
7	[t^n g(t)\,]	[i^n \frac\,]	[\left (\frac\right)^n \frac\,]	This is the dual to 6
8	[(g * h)(t)\,]	[\sqrt G(\omega) H(\omega)\,]	[G(f) H(f)\,]	[g * h\,] denotes the convolution of [g\,] and [h\,] — this rule is the convolution theorem
9	[g(t) h(t)\,]	[(G * H)(\omega) \over \sqrt\,]	[(G * H)(f)\,]	This is the dual of 8

Square-integrable functions

} \int_^\!\!G(\omega) e^ d \omega \,] |align="center"|[ G(\omega)\!\equiv\!]

[\frac} \int_^\!\!g(t) e^ dt \,] |align="center"|[ G(f)\!\equiv ]

[\int_^\!\!g(t) e^ dt \,] | |- | 10 |[\mathrm(a t) \,] |[\frac}\cdot \mathrm\left(\frac\right)] |[\frac\cdot \mathrm\left(\frac\right)] |The rectangular pulse and the normalized sinc function |- | 11 |[ \mathrm(a t)\,] |[\frac}\cdot \mathrm\left(\frac\right)] |[\frac\cdot \mathrm\left(\frac \right)\,] |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 |[ \mathrm^2 (a t) \,] |[ \frac}\cdot \mathrm \left( \frac \right) ] |[ \frac\cdot \mathrm \left( \frac \right) ] | tri is the triangular function |- | 13 |[ \mathrm (a t) \,] |[\frac} \cdot \mathrm^2 \left( \frac \right) ] |[\frac\cdot \mathrm^2 \left( \frac \right) \,] | Dual of rule 12. |- | 14 |[e^\,] |[\frac}\cdot e^}] |[\sqrt}\cdot e^}] |Shows that the Gaussian function [\exp(-\alpha t^2)] is its own Fourier transform. For this to be integrable we must have [\mathrm(\alpha)>0]. |- | |[ e^ = \left. e^\right|_ \,] |[ \frac} \cdot e^ -\frac\right)}] |[ \sqrt} \cdot e^ -\frac\right)} ] | common in optics |- | |[\cos ( a t^2 ) \,] |[ \frac} \cos \left( \frac - \frac \right) ] |[ \sqrt} \cos \left( \frac - \frac \right) ] | |- | |[\sin ( a t^2 ) \,] |[ \frac} \sin \left( \frac - \frac \right) ] |[ - \sqrt} \sin \left( \frac - \frac \right) ] | |- | |[e^ \,] |[ \sqrt} \cdot \frac ] |[ \frac ] | a>0 |- | |[ \frac} \,] |[ \frac}] |[ \frac} ] | the transform is the function itself |- | |[ J_0 (t)\,] |[ \sqrt} \cdot \frac \left( \frac \right)}} ] |[ \frac (\pi f)}} ] | J₀(t) is the Bessel function of first kind of order 0, rect is the rectangular function |- | |[ J_n (t) \,] |[ \sqrt} \frac \left( \frac \right)}} ] |[ \frac (\pi f)}} ] | it's the generalization of the previous transform; T_n (t) is the Chebyshev polynomial of the first kind. |- | |[ \frac \,] |[ \sqrt} \frac (-i)^n \cdot U_ (\omega)\,]
  [\cdot \ \sqrt \mathrm \left( \frac \right) ] |[ \frac (-i)^n \cdot U_ (2 \pi f)\,]
  [\cdot \ \sqrt \mathrm ( \pi f ) ] | U_n (t) is the Chebyshev polynomial of the second kind |}

Distributions

} \int_^\!\!G(\omega) e^ d \omega \,] |align="center"|[ G(\omega)\!\equiv\!]

[\frac} \int_^\!\!g(t) e^ dt \,] |align="center"|[ G(f)\!\equiv ]

[\int_^\!\!g(t) e^ dt \,] | |- | 15 |[1\,] |[\sqrt\cdot \delta(\omega)\,] |[\delta(f)\,] |[\delta(\omega)] 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. |- | 16 |[\delta(t)\,] |[\frac}\,] |[1\,] |Dual of rule 15. |- | 17 |[e^\,] |[\sqrt\cdot \delta(\omega - a)\,] |[\delta(f - \frac)\,] |This follows from and 3 and 15. |- | 18 |[\cos (a t)\,] |[\sqrt \frac\,] |[\frac\frac\end)\!+\!\delta(f\!+\!\begin\frac\end)}\,] |Follows from rules 1 and 17 using Euler's formula: [\cos(a t) = (e^ + e^)/2.] |- | 19 |[\sin( at)\,] |[\sqrt\frac\,] |[\frac\frac\end)\!-\!\delta(f\!+\!\begin\frac\end)}\,] |Also from 1 and 17. |- | 20 |[t^n\,] |[i^n \sqrt \delta^ (\omega)\,] |[\left(\frac\right)^n \delta^ (f)\,] |Here, [n] is a natural number. [\delta^n(\omega)] is the [n]-th distribution derivative of the Dirac delta. This rule follows from rules 7 and 15. Combining this rule with 1, we can transform all polynomials. |- | 21 |[\frac\,] |[-i\sqrt}\sgn(\omega)\,] |[-i\pi\cdot \sgn(f)\,] |Here [\sgn(\omega)] is the sign function; note that this is consistent with rules 7 and 15. |- | 22 |[\frac\,] |[-i \begin \sqrt}\cdot \frac}\end \sgn(\omega)\,] |[-i\pi \begin \frac}\end \sgn(f)\,] |Generalization of rule 21. |- | 23 |[\sgn(t)\,] |[\sqrt}\cdot \frac\,] |[\frac\,] |The dual of rule 21. |- | 24 |[ u(t) \,] |[\sqrt} \left( \frac + \delta(\omega)\right)\,] |[\frac\left(\frac + \delta(f)\right)\,] |Here [u(t)] is the Heaviside unit step function; this follows from rules 1 and 21. |- | |[ e^ u(t) \,] |[\frac (a + i \omega)}] |[\frac] |[u(t)] is the Heaviside unit step function and [a > 0]. |- | 25 |[\sum_^ \delta (t - n T) \,] |[\begin \frac}\end \sum_^ \delta \left( \omega -k \begin \frac\end \right)\,] |[\frac \sum_^ \delta \left( f -\frac\right) \,] |The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time. |}

See also

• Fourier transform
• Fourier series
• Laplace transform
• Discrete Fourier 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 3540586547
• 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

External links

• [Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.

 

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