Fourier transform

In physics, engineering and mathematics, the Fourier transform (FT) 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 this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform 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.

An example application of the Fourier transform is determining the constituent pitches in a musical waveform. This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed.

The Fourier transform relates the time domain, in red, with a function in the domain of the frequency, in blue. The component frequencies, extended for the whole frequency spectrum, are shown as peaks in the domain of the frequency.

The red sinusoid can be described by peak amplitude (1), peak-to-peak (2), RMS (3), and wavelength (4). The red and blue sinusoids have a phase difference of θ.

${\displaystyle \scriptstyle f(t)}$

${\displaystyle \scriptstyle {\widehat {f}}(\omega )}$

${\displaystyle \scriptstyle g(t)}$

${\displaystyle \scriptstyle {\widehat {g}}(\omega )}$

${\displaystyle \scriptstyle t}$

${\displaystyle \scriptstyle \omega }$

${\displaystyle \scriptstyle t}$

${\displaystyle \scriptstyle \omega }$

The top row shows a unit pulse as a function of time (f(t)) and its Fourier transform as a function of frequency (f̂(ω)). The bottom row shows a delayed unit pulse as a function of time (g(t)) and its Fourier transform as a function of frequency (ĝ(ω)). Translation (i.e. delay) in the time domain is interpreted as complex phase shifts in the frequency domain. The Fourier transform decomposes a function into eigenfunctions for the group of translations. The imaginary part of ĝ(ω) is negated because a negative sign exponent has been used in the Fourier transform, which is the default as derived from the Fourier series, but the sign does not matter for a transform that is not going to be reversed.

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 (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform 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.^{[note 1]} 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.^{[note 2]}

The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of 4-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.^{[note 3]} Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or Rⁿ, notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S¹, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.

Definition

The Fourier transform is an analysis process, decomposing a complex-valued function ${\displaystyle \textstyle f(x)}$  into its constituent frequencies and their amplitudes. The inverse process is synthesis, which recreates ${\displaystyle \textstyle f(x)}$  from its transform.

We can start with an analogy, the Fourier series, which analyzes ${\displaystyle \textstyle f(x)}$  on a bounded interval ${\displaystyle \textstyle x\in [-P/2,P/2],}$  for some positive real number ${\displaystyle P.}$  The constituent frequencies are a discrete set of harmonics at frequencies ${\displaystyle {\tfrac {n}{P}},n\in \mathbb {Z} ,}$  whose amplitude and phase are given by the analysis formula:

$${\displaystyle c_{n}={\tfrac {1}{P}}\int _{-P/2}^{P/2}f(x)\,e^{-i2\pi {\frac {n}{P}}x}\,dx.}$$

 

The actual Fourier series is the synthesis formula:

$${\displaystyle f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,e^{i2\pi {\tfrac {n}{P}}x},\quad \textstyle x\in [-P/2,P/2].}$$

 

The analogy for a function ${\displaystyle \textstyle f(x)}$  can be obtained formally from the analysis formula by taking the limit as ${\displaystyle P\to \infty }$ , while at the same time taking ${\displaystyle n}$  so that ${\displaystyle {\tfrac {n}{P}}\to \xi \in \mathbb {R} .}$ ^[1][2][3] Formally carrying this out, we obtain, for rapidly decreasing ${\displaystyle f}$ :^{[note 4][4]}

Fourier transform

${\displaystyle {\widehat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx.}$

(Eq.1)

It is easy to see, assuming the hypothesis of rapid decreasing, that the integral Eq.1 converges for all real ${\displaystyle \xi }$ , and (using the Riemann–Lebesgue lemma) that the transformed function ${\displaystyle {\widehat {f}}}$  is also rapidly decreasing. The validity of this definition for classes of functions ${\displaystyle f}$  that are not necessarily rapidly decreasing is discussed later in this section.

Evaluating Eq.1 for all values of ${\displaystyle \xi }$  produces the frequency-domain function. The complex number ${\displaystyle {\widehat {f}}(\xi )}$ , in polar coordinates, conveys both amplitude and phase of frequency ${\displaystyle \xi .}$  The intuitive interpretation of Eq.1 is that the effect of multiplying ${\displaystyle f(x)}$  by ${\displaystyle e^{-i2\pi \xi x}}$  is to subtract ${\displaystyle \xi }$  from every frequency component of function ${\displaystyle f(x).}$ ^{[note 5]} Only the component that was at frequency ${\displaystyle \xi }$  can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero. (see § Example)

The corresponding synthesis formula for such a function is:

Inverse transform

${\displaystyle f(x)=\int _{-\infty }^{\infty }{\widehat {f}}(\xi )\ e^{i2\pi \xi x}\,d\xi ,\quad \forall \ x\in \mathbb {R} .}$

(Eq.2)

Eq.2 is a representation of ${\displaystyle f(x)}$  as a weighted summation of complex exponential functions.

This is also known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat.^[5][6][7][8]

The functions ${\displaystyle f}$  and ${\displaystyle {\widehat {f}}}$  are referred to as a Fourier transform pair.^[9]  A common notation for designating transform pairs is:^[10]

$${\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ {\widehat {f}}(\xi ),}$$

 

  for example   ${\displaystyle \operatorname {rect} (x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ \operatorname {sinc} (\xi ).}$ 

Definition for Lebesgue integrable functions

Until now, we have been dealing with Schwartz functions, which decay rapidly at infinity, with all derivatives. This excludes many functions of practical importance from the definition, such as the rect function. A measurable function ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$  is called (Lebesgue) integrable if the Lebesgue integral of its absolute value is finite:

$${\displaystyle \|f\|_{1}=\int _{\mathbb {R} }|f(x)|\,dx<\infty .}$$

 

Two measurable functions are equivalent if they are equal except on a set of measure zero. The set of all equivalence classes of integrable functions is denoted ${\displaystyle L^{1}(\mathbb {R} )}$ . Then:^[11]

Definition — The Fourier transform of a Lebesgue integrable function ${\displaystyle f\in L^{1}(\mathbb {R} )}$  is defined by the formula Eq.1.

The integral Eq.1 is well-defined for all ${\displaystyle \xi \in \mathbb {R} ,}$  because of the assumption ${\displaystyle \|f\|_{1}<\infty }$ . (It can be shown that the function ${\displaystyle {\widehat {f}}\in L^{\infty }\cap C(\mathbb {R} )}$  is bounded and uniformly continuous in the frequency domain, and moreover, by the Riemann–Lebesgue lemma, it is zero at infinity.)

However, the class of Lebesgue integrable functions is not ideal from the point of view of the Fourier transform because there is no easy characterization of the image, and thus no easy characterization of the inverse transform.

Unitarity and definition for square integrable functions

While Eq.1 defines the Fourier transform for (complex-valued) functions in ${\displaystyle L^{1}(\mathbb {R} )}$ , it is easy to see that it is not well-defined for other integrability classes, most importantly ${\displaystyle L^{2}(\mathbb {R} )}$ . For functions in ${\displaystyle L^{1}\cap L^{2}(\mathbb {R} )}$ , and with the conventions of Eq.1, the Fourier transform is a unitary operator with respect to the Hilbert inner product on ${\displaystyle L^{2}(\mathbb {R} )}$ , restricted to the dense subspace of integrable functions. Therefore, it admits a unique continuous extension to a unitary operator on ${\displaystyle L^{2}(\mathbb {R} )}$ , also called the Fourier transform. This extension is important in part because the Fourier transform preserves the space ${\displaystyle L^{2}(\mathbb {R} )}$  so that, unlike the case of ${\displaystyle L^{1}}$ , the Fourier transform and inverse transform are on the same footing, being transformations of the same space of functions to itself.

Importantly, for functions in ${\displaystyle L^{2}}$ , the Fourier transform is no longer given by Eq.1 (interpreted as a Lebesgue integral). For example, the function ${\displaystyle f(x)=(1+x^{2})^{-1/2}}$  is in ${\displaystyle L^{2}}$  but not ${\displaystyle L^{1}}$ , so the integral Eq.1 diverges. 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 (pointwise) limits implicit in an improper integral. Titchmarsh (1986) and Dym & McKean (1985) each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure.

The conventions chosen in this article are those of harmonic analysis, and are characterized as the unique conventions such that the Fourier transform is both unitary on L² and an algebra homomorphism from L¹ to L^∞, without renormalizing the Lebesgue measure.^[12]

Angular frequency (ω)

When the independent variable (${\displaystyle x}$ ) represents time (often denoted by ${\displaystyle t}$ ), the transform variable (${\displaystyle \xi }$ ) represents frequency (often denoted by ${\displaystyle f}$ ). For example, if time is measured in seconds, then frequency is in hertz. The Fourier transform can also be written in terms of angular frequency, ${\displaystyle \omega =2\pi \xi ,}$  whose units are radians per second.

The substitution ${\displaystyle \xi ={\tfrac {\omega }{2\pi }}}$  into Eq.1 produces this convention, where function ${\displaystyle {\widehat {f}}}$  is relabeled ${\displaystyle {\widehat {f_{1}}}:}$ 

$${\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )&\triangleq \int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega \,x}\,dx={\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\cdot e^{i\omega \,x}\,d\omega .\end{aligned}}}$$

 

Unlike the Eq.1 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 ${\displaystyle 2\pi }$  factor evenly between the transform and its inverse, which leads to another convention:

$${\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )&\triangleq {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\cdot {\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\cdot e^{i\omega x}\,d\omega .\end{aligned}}}$$

 

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.

Summary of popular forms of the Fourier transform, one-dimensional

ordinary frequency ξ (Hz)	unitary	${\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i2\pi \xi x}\,dx={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{-\infty }^{\infty }{\widehat {f_{1}}}(\xi )\,e^{i2\pi x\xi }\,d\xi \end{aligned}}}$
angular frequency ω (rad/s)	unitary	${\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}$
	non-unitary	${\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}$

Generalization for n-dimensional functions

ordinary frequency ξ (Hz)	unitary	${\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{\mathbb {R} ^{n}}{\widehat {f_{1}}}(\xi )e^{i2\pi \xi \cdot x}\,d\xi \end{aligned}}}$
angular frequency ω (rad/s)	unitary	${\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{2}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}$
	non-unitary	${\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{3}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}$

Extension of the definition

For ${\displaystyle 1<p<2}$ , the Fourier transform can be defined on ${\displaystyle L^{p}(\mathbb {R} )}$  by Marcinkiewicz interpolation.

The Fourier transform can be defined on domains other than the real line. The Fourier transform on Euclidean space and the Fourier transform on locally abelian groups are discussed later in the article.

The Fourier transform can also be defined for tempered distributions, dual to the space of rapidly decreasing functions (Schwartz functions). A Schwartz function is a smooth function that decays at infinity, along with all of its derivatives. The space of Schwartz functions is denoted by ${\displaystyle {\mathcal {S}}(\mathbb {R} )}$ , and its dual ${\displaystyle {\mathcal {S}}'(\mathbb {R} )}$  is the space of tempered distributions. It is easy to see, by differentiating under the integral and applying the Riemann-Lebesgue lemma, that the Fourier transform of a Schwartz function (defined by the formula Eq.1) is again a Schwartz function. The Fourier transform of a tempered distribution ${\displaystyle T\in {\mathcal {S}}'(\mathbb {R} )}$  is defined by duality:

$${\displaystyle \langle {\widehat {T}},\phi \rangle =\langle T,{\widehat {\phi }}\rangle ;\quad \forall \phi \in {\mathcal {S}}(\mathbb {R} ).}$$

 

Many other characterizations of the Fourier transform exist. For example, one uses the Stone–von Neumann theorem: the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group.

Background

History

Main articles: Fourier analysis § History, and Fourier series § History

In 1822, Fourier claimed (see Joseph Fourier § The Analytic Theory of Heat) that any function, whether continuous or discontinuous, can be expanded into a series of sines.^[13] That important work was corrected and expanded upon by others to provide the foundation for the various forms of the Fourier transform used since.

 

Fig.1 When function ${\displaystyle A\cdot e^{i2\pi \xi t}}$  is depicted in the complex plane, the vector formed by its imaginary and real parts rotates around the origin. Its real part ${\displaystyle y(t)}$  is a cosine wave.

Complex sinusoids

In general, the coefficients ${\displaystyle {\widehat {f}}(\xi )}$  are complex numbers, which have two equivalent forms (see Euler's formula):

$${\displaystyle {\widehat {f}}(\xi )=\underbrace {Ae^{i\theta }} _{\text{polar coordinate form}}=\underbrace {A\cos(\theta )+iA\sin(\theta )} _{\text{rectangular coordinate form}}.}$$

 

The product with ${\displaystyle e^{i2\pi \xi x}}$  (Eq.2) has these forms:

$${\displaystyle {\widehat {f}}(\xi )\cdot e^{i2\pi \xi x}=Ae^{i\theta }\cdot e^{i2\pi \xi x}=\underbrace {Ae^{i(2\pi \xi x+\theta )}} _{\text{polar coordinate form}}=\underbrace {A\cos(2\pi \xi x+\theta )+iA\sin(2\pi \xi x+\theta )} _{\text{rectangular coordinate form}}.}$$

 

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

See also: Negative frequency § Simplifying the Fourier transform

Euler's formula introduces the possibility of negative ${\displaystyle \xi .}$   And Eq.1 is defined ${\displaystyle \forall \xi \in \mathbb {R} .}$  Only certain complex-valued ${\displaystyle f(x)}$  have transforms ${\displaystyle {\widehat {f}}=0,\ \forall \ \xi <0}$  (See Analytic signal. A simple example is ${\displaystyle e^{i2\pi \xi _{0}x}\ (\xi _{0}>0).}$ )  But negative frequency is necessary to characterize all other complex-valued ${\displaystyle f(x),}$  found in signal processing, partial differential equations, radar, nonlinear optics, quantum mechanics, and others.

For a real-valued ${\displaystyle f(x),}$  Eq.1 has the symmetry property ${\displaystyle {\widehat {f}}(-\xi )={\widehat {f}}^{*}(\xi )}$  (see § Conjugation below). This redundancy enables Eq.2 to distinguish ${\displaystyle f(x)=\cos(2\pi \xi _{0}x)}$  from ${\displaystyle e^{i2\pi \xi _{0}x}.}$   But of course it cannot tell us the actual sign of ${\displaystyle \xi _{0},}$  because ${\displaystyle \cos(2\pi \xi _{0}x)}$  and ${\displaystyle \cos(2\pi (-\xi _{0})x)}$  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 Eq.1 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 ${\displaystyle f(x)}$  is a periodic function, with period ${\displaystyle P}$ , that has a convergent Fourier series, then:

$${\displaystyle {\widehat {f}}(\xi )=\sum _{n=-\infty }^{\infty }c_{n}\cdot \delta \left(\xi -{\tfrac {n}{P}}\right),}$$

 

where ${\displaystyle c_{n}}$  are the Fourier series coefficients of ${\displaystyle f}$ , and ${\displaystyle \delta }$  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.

Sampling the Fourier transform

The Fourier transform of an integrable function ${\displaystyle f}$  can be sampled at regular intervals of arbitrary length ${\displaystyle {\tfrac {1}{P}}.}$  These samples can be deduced from one cycle of a periodic function ${\displaystyle f_{P}}$  which has Fourier series coefficients proportional to those samples by the Poisson summation formula:

$${\displaystyle f_{P}(x)\triangleq \sum _{n=-\infty }^{\infty }f(x+nP)={\frac {1}{P}}\sum _{k=-\infty }^{\infty }{\widehat {f}}\left({\tfrac {k}{P}}\right)e^{i2\pi {\frac {k}{P}}x},\quad \forall k\in \mathbb {Z} }$$

 

The integrability of ${\displaystyle f}$  ensures the periodic summation converges. Therefore, the samples ${\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)}$  can be determined by Fourier series analysis:

$${\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)=\int _{P}f_{P}(x)\cdot e^{-i2\pi {\frac {k}{P}}x}\,dx.}$$

 

When ${\displaystyle f(x)}$  has compact support, ${\displaystyle f_{P}(x)}$  has a finite number of terms within the interval of integration. When ${\displaystyle f(x)}$  does not have compact support, numerical evaluation of ${\displaystyle f_{P}(x)}$  requires an approximation, such as tapering ${\displaystyle f(x)}$  or truncating the number of terms.

Example

The following figures provide a visual illustration of how the Fourier transform measures whether a frequency is present in a particular function. The depicted function ${\displaystyle f(t)=\cos(2\pi \ 3t)\ e^{-\pi t^{2}}}$ oscillates at 3 Hz (if ${\displaystyle t}$  measures seconds) and tends quickly to 0. (The second factor in this equation is an envelope function that shapes the continuous sinusoid into a short pulse.). ${\displaystyle f(t)}$  was specially chosen to have a real Fourier transform that can be easily plotted. The first image is its graph. In order to calculate ${\displaystyle {\widehat {f}}(3)}$  we must integrate the product ${\displaystyle f(t)e^{-i2\pi 3t}.}$  The next 2 images are the real and imaginary parts of that product. The real part of the integrand has a non-negative average value, because the alternating signs of ${\displaystyle f(t)}$  and ${\displaystyle \operatorname {Re} (e^{-i2\pi 3t})}$  oscillate at the same rate and same phase, whereas ${\displaystyle f(t)}$  and ${\displaystyle \operatorname {Im} (e^{-i2\pi 3t})}$  are same rate but orthogonal phase. The result is that when you integrate the real part of the integrand you get a relatively large number (in this case ${\displaystyle {\tfrac {1}{2}}}$ ). Also, when you try to measure a frequency that is not present, as in the case when we look at ${\displaystyle {\widehat {f}}(5),}$  both real and imaginary component of the product vary rapidly between positive and negative values. Therefore, the integral is very small and the value for the Fourier transform for that frequency is nearly zero. The general situation is usually more complicated than this, but heuristically this is how the Fourier transform measures how much of an individual frequency is present in a function ${\displaystyle f(t).}$ 

 

Original function showing oscillation 3 Hz. Real and imaginary parts of integrand for Fourier transform at 3 Hz

•  
  Real and imaginary parts of integrand for Fourier transform at 5 Hz
•  
  Magnitude of Fourier transform, with 3 and 5 Hz labeled.

To re-enforce an earlier point, the reason for the response at  ${\displaystyle \xi =-3}$  Hz  is because  ${\displaystyle \cos(2\pi 3t)}$   and  ${\displaystyle \cos(2\pi (-3)t)}$   are indistinguishable. The transform of  ${\displaystyle e^{i2\pi 3t}\cdot e^{-\pi t^{2}}}$   would have just one response, whose amplitude is the integral of the smooth envelope: ${\displaystyle e^{-\pi t^{2}},}$   whereas  ${\displaystyle \operatorname {Re} (f(t)\cdot e^{-i2\pi 3t})}$  (second graph above) is  ${\displaystyle e^{-\pi t^{2}}(1+\cos(2\pi 6t))/2.}$ 

Properties of the Fourier transform

Let ${\displaystyle f(x)}$  and ${\displaystyle h(x)}$  represent integrable functions Lebesgue-measurable on the real line satisfying:

$${\displaystyle \int _{-\infty }^{\infty }|f(x)|\,dx<\infty .}$$

 

We denote the Fourier transforms of these functions as ${\displaystyle {\hat {f}}(\xi )}$  and ${\displaystyle {\hat {h}}(\xi )}$  respectively.

Basic properties

The Fourier transform has the following basic properties:^[14]

Linearity

$${\displaystyle a\ f(x)+b\ h(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ a\ {\widehat {f}}(\xi )+b\ {\widehat {h}}(\xi );\quad \ a,b\in \mathbb {C} }$$

 

Time shifting

$${\displaystyle f(x-x_{0})\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ e^{-i2\pi x_{0}\xi }\ {\widehat {f}}(\xi );\quad \ x_{0}\in \mathbb {R} }$$

 

Frequency shifting

$${\displaystyle e^{i2\pi \xi _{0}x}f(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(\xi -\xi _{0});\quad \ \xi _{0}\in \mathbb {R} }$$

 

Time scaling

$${\displaystyle f(ax)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\xi }{a}}\right);\quad \ a\neq 0}$$

 

The case ${\displaystyle a=-1}$  leads to the time-reversal property:

$${\displaystyle f(-x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(-\xi )}$$

 

Symmetry

When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:

$${\displaystyle {\begin{aligned}{\mathsf {Time\ domain}}\quad &\ f\quad &=\quad &f_{_{RE}}\quad &+\quad &f_{_{RO}}\quad &+\quad i\ &f_{_{IE}}\quad &+\quad &\underbrace {i\ f_{_{IO}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\mathsf {Frequency\ domain}}\quad &{\widehat {f}}\quad &=\quad &{\widehat {f}}_{RE}\quad &+\quad &\overbrace {i\ {\widehat {f}}_{IO}} \quad &+\quad i\ &{\widehat {f}}_{IE}\quad &+\quad &{\widehat {f}}_{RO}\end{aligned}}}$$

 

From this, various relationships are apparent, for example:

• The transform of a real-valued function ${\displaystyle (f_{_{RE}}+f_{_{RO}})}$  is the even symmetric function ${\displaystyle {\hat {f}}_{RE}+i\ {\hat {f}}_{IO}.}$  Conversely, an even-symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function ${\displaystyle (i\ f_{_{IE}}+i\ f_{_{IO}})}$  is the odd symmetric function ${\displaystyle {\hat {f}}_{RO}+i\ {\hat {f}}_{IE},}$  and the converse is true.
• The transform of an even-symmetric function ${\displaystyle (f_{_{RE}}+i\ f_{_{IO}})}$  is the real-valued function ${\displaystyle {\hat {f}}_{RE}+{\hat {f}}_{RO},}$  and the converse is true.
• The transform of an odd-symmetric function ${\displaystyle (f_{_{RO}}+i\ f_{_{IE}})}$  is the imaginary-valued function ${\displaystyle i\ {\hat {f}}_{IE}+i{\hat {f}}_{IO},}$  and the converse is true.

Conjugation

$${\displaystyle {\bigl (}f(x){\bigr )}^{*}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ \left({\widehat {f}}(-\xi )\right)^{*}}$$

 

(Note: the ∗ denotes complex conjugation.)

In particular, if ${\displaystyle f}$  is real, then ${\displaystyle {\widehat {f}}}$  is even symmetric (aka Hermitian function):

$${\displaystyle {\widehat {f}}(-\xi )={\bigl (}{\widehat {f}}(\xi ){\bigr )}^{*}.}$$

 

And if ${\displaystyle f}$  is purely imaginary, then ${\displaystyle {\widehat {f}}}$  is odd symmetric:

$${\displaystyle {\widehat {f}}(-\xi )=-({\widehat {f}}(\xi ))^{*}.}$$

 

Real and imaginary part in time

$${\displaystyle \operatorname {Re} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2}}\left({\widehat {f}}(\xi )+{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}$$

 

$${\displaystyle \operatorname {Im} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2i}}\left({\widehat {f}}(\xi )-{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}$$

 

Zero frequency component

Substituting ${\displaystyle \xi =0}$  in the definition, we obtain:

$${\displaystyle {\widehat {f}}(0)=\int _{-\infty }^{\infty }f(x)\,dx.}$$

 

The integral of ${\displaystyle f}$  over its domain is known as the average value or DC bias of the function.

Invertibility and periodicity

Further information: Fourier inversion theorem and Fractional Fourier transform

Under suitable conditions on the function ${\displaystyle f}$ , it can be recovered from its Fourier transform ${\displaystyle {\hat {f}}}$ . Indeed, denoting the Fourier transform operator by ${\displaystyle {\mathcal {F}}}$ , so ${\displaystyle {\mathcal {F}}f:={\hat {f}}}$ , then for suitable functions, applying the Fourier transform twice simply flips the function: ${\displaystyle \left({\mathcal {F}}^{2}f\right)(x)=f(-x)}$ , which can be interpreted as "reversing time". Since reversing time is two-periodic, applying this twice yields ${\displaystyle {\mathcal {F}}^{4}(f)=f}$ , so the Fourier transform operator is four-periodic, and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times: ${\displaystyle {\mathcal {F}}^{3}\left({\hat {f}}\right)=f}$ . In particular the Fourier transform is invertible (under suitable conditions).

More precisely, defining the parity operator ${\displaystyle {\mathcal {P}}}$  such that ${\displaystyle ({\mathcal {P}}f)(x)=f(-x)}$ , we have:

$${\displaystyle {\begin{aligned}{\mathcal {F}}^{0}&=\mathrm {id} ,\\{\mathcal {F}}^{1}&={\mathcal {F}},\\{\mathcal {F}}^{2}&={\mathcal {P}},\\{\mathcal {F}}^{3}&={\mathcal {F}}^{-1}={\mathcal {P}}\circ {\mathcal {F}}={\mathcal {F}}\circ {\mathcal {P}},\\{\mathcal {F}}^{4}&=\mathrm {id} \end{aligned}}}$$

 

These equalities of operators require careful definition of the space of functions in question, defining equality of functions (equality at every point? equality almost everywhere?) and defining equality of operators – that is, defining the topology on the function space and operator space in question. These are not true for all functions, but are true under various conditions, which are the content of the various forms of the Fourier inversion theorem.

This fourfold periodicity of the Fourier transform is similar to a rotation of the plane by 90°, particularly as the two-fold iteration yields a reversal, and in fact this analogy can be made precise. While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically it can be interpreted as a rotation by 90° in the time–frequency domain (considering time as the x-axis and frequency as the y-axis), and the Fourier transform can be generalized to the fractional Fourier transform, which involves rotations by other angles. This can be further generalized to linear canonical transformations, which can be visualized as the action of the special linear group SL₂(R) on the time–frequency plane, with the preserved symplectic form corresponding to the uncertainty principle, below. This approach is particularly studied in signal processing, under time–frequency analysis.

Units

See also: Spectral density § Units

The frequency variable must have inverse units to the units of the original function's domain (typically named t or x). For example, if t is measured in seconds, ξ should be in cycles per second or hertz. If the scale of time is in units of 2π seconds, then another Greek letter ω typically is used instead to represent angular frequency (where ω = 2πξ) in units of radians per second. If using x for units of length, then ξ must be in inverse length, e.g., wavenumbers. That is to say, there are two versions of the real line: one which is the range of t and measured in units of t, and the other which is the range of ξ and measured in inverse units to the units of t. These two distinct versions of the real line cannot be equated with each other. Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition.

In general, ξ must always be taken to be a linear form on the space of its domain, which is to say that the second real line is the dual space of the first real line. See the article on linear algebra for a more formal explanation and for more details. This point of view becomes essential in generalizations of the Fourier transform to general symmetry groups, including the case of Fourier series.

That there is no one preferred way (often, one says "no canonical way") to compare the two versions of the real line which are involved in the Fourier transform—fixing the units on one line does not force the scale of the units on the other line—is the reason for the plethora of rival conventions on the definition of the Fourier transform. The various definitions resulting from different choices of units differ by various constants.

In other conventions, the Fourier transform has i in the exponent instead of −i, and vice versa for the inversion formula. This convention is common in modern physics^[15] and is the default for Wolfram Alpha, and does not mean that the frequency has become negative, since there is no canonical definition of positivity for frequency of a complex wave. It simply means that ${\displaystyle {\hat {f}}(\xi )}$  is the amplitude of the wave  ${\displaystyle e^{-i2\pi \xi x}}$   instead of the wave  ${\displaystyle e^{i2\pi \xi x}}$  (the former, with its minus sign, is often seen in the time dependence for Sinusoidal plane-wave solutions of the electromagnetic wave equation, or in the time dependence for quantum wave functions). Many of the identities involving the Fourier transform remain valid in those conventions, provided all terms that explicitly involve i have it replaced by −i. In Electrical engineering the letter j is typically used for the imaginary unit instead of i because i is used for current.

When using dimensionless units, the constant factors might not even be written in the transform definition. For instance, in probability theory, the characteristic function Φ of the probability density function f of a random variable X of continuous type is defined without a negative sign in the exponential, and since the units of x are ignored, there is no 2π either:

$${\displaystyle \phi (\lambda )=\int _{-\infty }^{\infty }f(x)e^{i\lambda x}\,dx.}$$

 

(In probability theory, and in mathematical statistics, the use of the Fourier—Stieltjes transform is preferred, because so many random variables are not of continuous type, and do not possess a density function, and one must treat not functions but distributions, i.e., measures which possess "atoms".)

From the higher point of view of group characters, which is much more abstract, all these arbitrary choices disappear, as will be explained in the later section of this article, which treats the notion of the Fourier transform of a function on a locally compact Abelian group.

Uniform continuity and the Riemann–Lebesgue lemma

 

The rectangular function is Lebesgue integrable.

 

The sinc function, which is the Fourier transform of the rectangular function, is bounded and continuous, but not Lebesgue integrable.

The Fourier transform may be defined in some cases for non-integrable functions, but the Fourier transforms of integrable functions have several strong properties.

The Fourier transform f̂ of any integrable function f is uniformly continuous and^[16]

$${\displaystyle \left\|{\hat {f}}\right\|_{\infty }\leq \left\|f\right\|_{1}}$$

 

By the Riemann–Lebesgue lemma,^[11]

$${\displaystyle {\hat {f}}(\xi )\to 0{\text{ as }}|\xi |\to \infty .}$$

 

However, ${\displaystyle {\hat {f}}}$  need not be integrable. For example, the Fourier transform of the rectangular function, which is integrable, is the sinc function, which is not Lebesgue integrable, because its improper integrals behave analogously to the alternating harmonic series, in converging to a sum without being absolutely convergent.

It is not generally possible to write the inverse transform as a Lebesgue integral. However, when both f and ${\displaystyle {\hat {f}}}$  are integrable, the inverse equality

$${\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{i2\pi x\xi }\,d\xi }$$

 

holds holds for almost every x. As a result, the Fourier transform is injective on L¹(R).

Plancherel theorem and Parseval's theorem

Let f(x) and g(x) be integrable, and let f̂(ξ) and ĝ(ξ) be their Fourier transforms. If f(x) and g(x) are also square-integrable, then the Parseval formula follows:^[17]

$${\displaystyle \langle f,g\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }{\hat {f}}(\xi ){\overline {{\hat {g}}(\xi )}}\,d\xi ,}$$

 

where the bar denotes complex conjugation.

The Plancherel theorem, which follows from the above, states that^[18]

$${\displaystyle \|f\|_{L^{2}}^{2}=\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx=\int _{-\infty }^{\infty }\left|{\hat {f}}(\xi )\right|^{2}\,d\xi .}$$

 

Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary operator on L²(R). On L¹(R) ∩ L²(R), this extension agrees with original Fourier transform defined on L¹(R), thus enlarging the domain of the Fourier transform to L¹(R) + L²(R) (and consequently to L^p(R) for 1 ≤ p ≤ 2). Plancherel's theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity. The terminology of these formulas is not quite standardised. Parseval's theorem was proved only for Fourier series, and was first proved by Lyapunov. But Parseval's formula makes sense for the Fourier transform as well, and so even though in the context of the Fourier transform it was proved by Plancherel, it is still often referred to as Parseval's formula, or Parseval's relation, or even Parseval's theorem.

See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.

Poisson summation formula

Main article: Poisson summation formula

The Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. The Poisson summation formula says that for sufficiently regular functions f,

$${\displaystyle \sum _{n}{\hat {f}}(n)=\sum _{n}f(n).}$$

 

It has a variety of useful forms that are derived from the basic one by application of the Fourier transform's scaling and time-shifting properties. The formula has applications in engineering, physics, and number theory. The frequency-domain dual of the standard Poisson summation formula is also called the discrete-time Fourier transform.

Poisson summation is generally associated with the physics of periodic media, such as heat conduction on a circle. The fundamental solution of the heat equation on a circle is called a theta function. It is used in number theory to prove the transformation properties of theta functions, which turn out to be a type of modular form, and it is connected more generally to the theory of automorphic forms where it appears on one side of the Selberg trace formula.

Differentiation

Suppose f(x) is an absolutely continuous differentiable function, and both f and its derivative f′ are integrable. Then the Fourier transform of the derivative is given by

$${\displaystyle {\widehat {f'\,}}(\xi )={\mathcal {F}}\left\{{\frac {d}{dx}}f(x)\right\}=i2\pi \xi {\hat {f}}(\xi ).}$$

 

More generally, the Fourier transformation of the nth derivative f⁽ⁿ⁾ is given by

$${\displaystyle {\widehat {f^{(n)}}}(\xi )={\mathcal {F}}\left\{{\frac {d^{n}}{dx^{n}}}f(x)\right\}=(i2\pi \xi )^{n}{\hat {f}}(\xi ).}$$

 

Analogically, ${\displaystyle {\mathcal {F}}\left\{{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi )\right\}=(i2\pi x)^{n}f(x)}$ , so ${\displaystyle {\mathcal {F}}\left\{x^{n}f(x)\right\}=\left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi ).}$ 

By applying the Fourier transform and using these formulas, some ordinary differential equations can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb "f(x) is smooth if and only if f̂(ξ) quickly falls to 0 for |ξ| → ∞." By using the analogous rules for the inverse Fourier transform, one can also say "f(x) quickly falls to 0 for |x| → ∞ if and only if f̂(ξ) is smooth."

Convolution theorem

Main article: Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. If f(x) and g(x) are integrable functions with Fourier transforms f̂(ξ) and ĝ(ξ) respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms f̂(ξ) and ĝ(ξ) (under other conventions for the definition of the Fourier transform a constant factor may appear).

This means that if:

$${\displaystyle h(x)=(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy,}$$

 

where ∗ denotes the convolution operation, then:

$${\displaystyle {\hat {h}}(\xi )={\hat {f}}(\xi )\,{\hat {g}}(\xi ).}$$

 

In linear time invariant (LTI) system theory, it is common to interpret g(x) as the impulse response of an LTI system with input f(x) and output h(x), since substituting the unit impulse for f(x) yields h(x) = g(x). In this case, ĝ(ξ) represents the frequency response of the system.

Conversely, if f(x) can be decomposed as the product of two square integrable functions p(x) and q(x), then the Fourier transform of f(x) is given by the convolution of the respective Fourier transforms p̂(ξ) and q̂(ξ).

Cross-correlation theorem

Main articles: Cross-correlation and Wiener–Khinchin_theorem

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

$${\displaystyle h(x)=(f\star g)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}g(x+y)\,dy}$$

 

then the Fourier transform of h(x) is:

$${\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}\,{\hat {g}}(\xi ).}$$

 

As a special case, the autocorrelation of function f(x) is:

$${\displaystyle h(x)=(f\star f)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}f(x+y)\,dy}$$

 

for which

$${\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}{\hat {f}}(\xi )=\left|{\hat {f}}(\xi )\right|^{2}.}$$

 

Eigenfunctions

See also: Mehler kernel and Hermite polynomials § Hermite functions as eigenfunctions of the Fourier transform

The Fourier transform is a linear transform which has eigenfunctions obeying ${\displaystyle {\mathcal {F}}[\psi ]=\lambda \psi ,}$  with ${\displaystyle \lambda \in \mathbb {C} .}$ 

A set of eigenfunctions is found by noting that the homogeneous differential equation

$${\displaystyle \left[U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)+U(x)\right]\psi (x)=0}$$

 

leads to eigenfunctions ${\displaystyle \psi (x)}$  of the Fourier transform ${\displaystyle {\mathcal {F}}}$  as long as the form of the equation remains invariant under Fourier transform.^{[note 6]} In other words, every solution ${\displaystyle \psi (x)}$  and its Fourier transform ${\displaystyle {\hat {\psi }}(\xi )}$  obey the same equation. Assuming uniqueness of the solutions, every solution ${\displaystyle \psi (x)}$  must therefore be an eigenfunction of the Fourier transform. The form of the equation remains unchanged under Fourier transform if ${\displaystyle U(x)}$  can be expanded in a power series in which for all terms the same factor of either one of ${\displaystyle \pm 1,\pm i}$  arises from the factors ${\displaystyle i^{n}}$  introduced by the differentiation rules upon Fourier transforming the homogeneous differential equation because this factor may then be cancelled. The simplest allowable ${\displaystyle U(x)=x}$  leads to the standard normal distribution.^[19]

More generally, a set of eigenfunctions is also found by noting that the differentiation rules imply that the ordinary differential equation

$${\displaystyle \left[W\left({\frac {i}{2\pi }}{\frac {d}{dx}}\right)+W(x)\right]\psi (x)=C\psi (x)}$$

 

with ${\displaystyle C}$  constant and ${\displaystyle W(x)}$  being a non-constant even function remains invariant in form when applying the Fourier transform ${\displaystyle {\mathcal {F}}}$  to both sides of the equation. The simplest example is provided by ${\displaystyle W(x)=x^{2}}$  which is equivalent to considering the Schrödinger equation for the quantum harmonic oscillator.^[20] The corresponding solutions provide an important choice of an orthonormal basis for L²(R) and are given by the "physicist's" Hermite functions. Equivalently one may use

$${\displaystyle \psi _{n}(x)={\frac {\sqrt[{4}]{2}}{\sqrt {n!}}}e^{-\pi x^{2}}\mathrm {He} _{n}\left(2x{\sqrt {\pi }}\right),}$$

 

where He_n(x) are the "probabilist's" Hermite polynomials, defined as

$${\displaystyle \mathrm {He} _{n}(x)=(-1)^{n}e^{{\frac {1}{2}}x^{2}}\left({\frac {d}{dx}}\right)^{n}e^{-{\frac {1}{2}}x^{2}}.}$$

 

Under this convention for the Fourier transform, we have that

$${\displaystyle {\hat {\psi }}_{n}(\xi )=(-i)^{n}\psi _{n}(\xi ).}$$

 

In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier transform on L²(R).^[14][21] However, this choice of eigenfunctions is not unique. Because of ${\displaystyle {\mathcal {F}}^{4}=\mathrm {id} }$  there are only four different eigenvalues of the Fourier transform (the fourth roots of unity ±1 and ±i) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction.^[22] As a consequence of this, it is possible to decompose L²(R) as a direct sum of four spaces H₀, H₁, H₂, and H₃ where the Fourier transform acts on He_k simply by multiplication by i^k.

Since the complete set of Hermite functions ψ_n provides a resolution of the identity they diagonalize the Fourier operator, i.e. the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed:

$${\displaystyle {\mathcal {F}}[f](\xi )=\int dxf(x)\sum _{n\geq 0}(-i)^{n}\psi _{n}(x)\psi _{n}(\xi )~.}$$

 

This approach to define the Fourier transform was first proposed by Norbert Wiener.^[23] Among other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are thus used to define a generalization of the Fourier transform, namely the fractional Fourier transform used in time–frequency analysis.^[24] In physics, this transform was introduced by Edward Condon.^[25] This change of basis functions becomes possible because the Fourier transform is a unitary transform when using the right conventions. Consequently, under the proper conditions it may be expected to result from a self-adjoint generator ${\displaystyle N}$  via^[26]

$${\displaystyle {\mathcal {F}}[\psi ]=e^{-itN}\psi .}$$

 

The operator ${\displaystyle N}$  is the number operator of the quantum harmonic oscillator written as^[27][28]

$$ N ≡ 1 2 January 01, 1970 Latest articles January 03, 2023I. M. Ibrahim January 07, 2023I. Y. Yunioshi January 05, 2023I've Got the Music in Me January 06, 2023I've Never Been To Me January 06, 2023I'll Always Love You (Taylor Dayne song) January 04, 2023I'll CKBC January 15, 2023I'm Not the Man I Used to Be January 02, 2023I'm Alive and on Fire January 07, 2023I'm Easy (Keith Carradine song) January 04, 2023I'm From Rolling Stone Most ReadDecember 30, 2022Sunset in NaplesJanuary 11, 2023SunseraJanuary 02, 2023Sunshine (wrestling)January 06, 2023Sunshine CruzJanuary 06, 2023Sunshine Biscuits fourier, transform, physics, engineering, mathematics, integral, transform, that, takes, function, input, outputs, another, function, that, describes, extent, which, various, frequencies, present, original, function, output, transform, complex, valued, functio. In physics engineering and mathematics the Fourier transform FT 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 this complex valued function and the mathematical operation When a distinction needs to be made the Fourier transform 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 An example application of the Fourier transform is determining the constituent pitches in a musical waveform This image is the result of applying a constant Q transform a Fourier related transform to the waveform of a C major piano chord The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord C E G The remaining smaller peaks are higher frequency overtones of the fundamental pitches A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed The Fourier transform relates the time domain in red with a function in the domain of the frequency in blue The component frequencies extended for the whole frequency spectrum are shown as peaks in the domain of the frequency The red sinusoid can be described by peak amplitude 1 peak to peak 2 RMS 3 and wavelength 4 The red and blue sinusoids have a phase difference of 8 f t displaystyle scriptstyle f t f w displaystyle scriptstyle widehat f omega g t displaystyle scriptstyle g t g w displaystyle scriptstyle widehat g omega t displaystyle scriptstyle t w displaystyle scriptstyle omega t displaystyle scriptstyle t w displaystyle scriptstyle omega The top row shows a unit pulse as a function of time f t and its Fourier transform as a function of frequency f w The bottom row shows a delayed unit pulse as a function of time g t and its Fourier transform as a function of frequency ĝ w Translation i e delay in the time domain is interpreted as complex phase shifts in the frequency domain The Fourier transform decomposes a function into eigenfunctions for the group of translations The imaginary part of ĝ w is negated because a negative sign exponent has been used in the Fourier transform which is the default as derived from the Fourier series but the sign does not matter for a transform that is not going to be reversed 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 e g diffusion The Fourier transform of a Gaussian function is another Gaussian function Joseph Fourier introduced the transform 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 note 1 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 note 2 The Fourier transform can also be generalized to functions of several variables on Euclidean space sending a function of 3 dimensional position space to a function of 3 dimensional momentum or a function of space and time to a function of 4 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 note 3 Still further generalization is possible to functions on groups which besides the original Fourier transform on R or Rn notably includes the discrete time Fourier transform DTFT group Z the discrete Fourier transform DFT group Z mod N and the Fourier series or circular Fourier transform group S1 the unit circle closed finite interval with endpoints identified The latter is routinely employed to handle periodic functions The fast Fourier transform FFT is an algorithm for computing the DFT Contents 1 Definition 1 1 Definition for Lebesgue integrable functions 1 2 Unitarity and definition for square integrable functions 1 3 Angular frequency w 1 4 Extension of the definition 2 Background 2 1 History 2 2 Complex sinusoids 2 3 Negative frequency 2 4 Fourier transform for periodic functions 2 5 Sampling the Fourier transform 3 Example 4 Properties of the Fourier transform 4 1 Basic properties 4 1 1 Linearity 4 1 2 Time shifting 4 1 3 Frequency shifting 4 1 4 Time scaling 4 1 5 Symmetry 4 1 6 Conjugation 4 1 7 Real and imaginary part in time 4 1 8 Zero frequency component 4 2 Invertibility and periodicity 4 3 Units 4 4 Uniform continuity and the Riemann Lebesgue lemma 4 5 Plancherel theorem and Parseval s theorem 4 6 Poisson summation formula 4 7 Differentiation 4 8 Convolution theorem 4 9 Cross correlation theorem 4 10 Eigenfunctions 4 11 Connection with the Heisenberg group 5 Complex domain 5 1 Laplace transform 5 2 Inversion 6 Fourier transform on Euclidean space 6 1 Uncertainty principle 6 2 Sine and cosine transforms 6 3 Spherical harmonics 6 4 Restriction problems 7 Fourier transform on function spaces 7 1 On Lp spaces 7 1 1 On L1 7 1 2 On L2 7 1 3 On other Lp 7 2 Tempered distributions 8 Generalizations 8 1 Fourier Stieltjes transform 8 2 Locally compact abelian groups 8 3 Gelfand transform 8 4 Compact non abelian groups 9 Alternatives 10 Applications 10 1 Analysis of differential equations 10 2 Fourier transform spectroscopy 10 3 Quantum mechanics 10 4 Signal processing 11 Other notations 12 Computation methods 12 1 Discrete Fourier transforms and fast Fourier transforms 12 2 Analytic integration of closed form functions 12 3 Numerical integration of closed form continuous functions 12 4 Numerical integration of a series of ordered pairs 13 Tables of important Fourier transforms 13 1 Functional relationships one dimensional 13 2 Square integrable functions one dimensional 13 3 Distributions one dimensional 13 4 Two dimensional functions 13 5 Formulas for general n dimensional functions 14 See also 15 Notes 16 Citations 17 References 18 External linksDefinition editThe Fourier transform is an analysis process decomposing a complex valued function f x displaystyle textstyle f x nbsp into its constituent frequencies and their amplitudes The inverse process is synthesis which recreates f x displaystyle textstyle f x nbsp from its transform We can start with an analogy the Fourier series which analyzes f x displaystyle textstyle f x nbsp on a bounded interval x P 2 P 2 displaystyle textstyle x in P 2 P 2 nbsp for some positive real number P displaystyle P nbsp The constituent frequencies are a discrete set of harmonics at frequencies n P n Z displaystyle tfrac n P n in mathbb Z nbsp whose amplitude and phase are given by the analysis formula c n 1 P P 2 P 2 f x e i 2 p n P x d x displaystyle c n tfrac 1 P int P 2 P 2 f x e i2 pi frac n P x dx nbsp The actual Fourier series is the synthesis formula f x n c n e i 2 p n P x x P 2 P 2 displaystyle f x sum n infty infty c n e i2 pi tfrac n P x quad textstyle x in P 2 P 2 nbsp The analogy for a function f x displaystyle textstyle f x nbsp can be obtained formally from the analysis formula by taking the limit as P displaystyle P to infty nbsp while at the same time taking n displaystyle n nbsp so that n P 3 R displaystyle tfrac n P to xi in mathbb R nbsp 1 2 3 Formally carrying this out we obtain for rapidly decreasing f displaystyle f nbsp note 4 4 Fourier transform f 3 f x e i 2 p 3 x d x displaystyle widehat f xi int infty infty f x e i2 pi xi x dx nbsp Eq 1 It is easy to see assuming the hypothesis of rapid decreasing that the integral Eq 1 converges for all real 3 displaystyle xi nbsp and using the Riemann Lebesgue lemma that the transformed function f displaystyle widehat f nbsp is also rapidly decreasing The validity of this definition for classes of functions f displaystyle f nbsp that are not necessarily rapidly decreasing is discussed later in this section Evaluating Eq 1 for all values of 3 displaystyle xi nbsp produces the frequency domain function The complex number f 3 displaystyle widehat f xi nbsp in polar coordinates conveys both amplitude and phase of frequency 3 displaystyle xi nbsp The intuitive interpretation of Eq 1 is that the effect of multiplying f x displaystyle f x nbsp by e i 2 p 3 x displaystyle e i2 pi xi x nbsp is to subtract 3 displaystyle xi nbsp from every frequency component of function f x displaystyle f x nbsp note 5 Only the component that was at frequency 3 displaystyle xi nbsp can produce a non zero value of the infinite integral because at least formally all the other shifted components are oscillatory and integrate to zero see Example The corresponding synthesis formula for such a function is Inverse transform f x f 3 e i 2 p 3 x d 3 x R displaystyle f x int infty infty widehat f xi e i2 pi xi x d xi quad forall x in mathbb R nbsp Eq 2 Eq 2 is a representation of f x displaystyle f x nbsp as a weighted summation of complex exponential functions This is also known as the Fourier inversion theorem and was first introduced in Fourier s Analytical Theory of Heat 5 6 7 8 The functions f displaystyle f nbsp and f displaystyle widehat f nbsp are referred to as a Fourier transform pair 9 A common notation for designating transform pairs is 10 f x F f 3 displaystyle f x stackrel mathcal F longleftrightarrow widehat f xi nbsp for example rect x F sinc 3 displaystyle operatorname rect x stackrel mathcal F longleftrightarrow operatorname sinc xi nbsp Definition for Lebesgue integrable functions edit Until now we have been dealing with Schwartz functions which decay rapidly at infinity with all derivatives This excludes many functions of practical importance from the definition such as the rect function A measurable function f R C displaystyle f mathbb R to mathbb C nbsp is called Lebesgue integrable if the Lebesgue integral of its absolute value is finite f 1 R f x d x lt displaystyle f 1 int mathbb R f x dx lt infty nbsp Two measurable functions are equivalent if they are equal except on a set of measure zero The set of all equivalence classes of integrable functions is denoted L 1 R displaystyle L 1 mathbb R nbsp Then 11 Definition The Fourier transform of a Lebesgue integrable function f L 1 R displaystyle f in L 1 mathbb R nbsp is defined by the formula Eq 1 The integral Eq 1 is well defined for all 3 R displaystyle xi in mathbb R nbsp because of the assumption f 1 lt displaystyle f 1 lt infty nbsp It can be shown that the function f L C R displaystyle widehat f in L infty cap C mathbb R nbsp is bounded and uniformly continuous in the frequency domain and moreover by the Riemann Lebesgue lemma it is zero at infinity However the class of Lebesgue integrable functions is not ideal from the point of view of the Fourier transform because there is no easy characterization of the image and thus no easy characterization of the inverse transform Unitarity and definition for square integrable functions edit While Eq 1 defines the Fourier transform for complex valued functions in L 1 R displaystyle L 1 mathbb R nbsp it is easy to see that it is not well defined for other integrability classes most importantly L 2 R displaystyle L 2 mathbb R nbsp For functions in L 1 L 2 R displaystyle L 1 cap L 2 mathbb R nbsp and with the conventions of Eq 1 the Fourier transform is a unitary operator with respect to the Hilbert inner product on L 2 R displaystyle L 2 mathbb R nbsp restricted to the dense subspace of integrable functions Therefore it admits a unique continuous extension to a unitary operator on L 2 R displaystyle L 2 mathbb R nbsp also called the Fourier transform This extension is important in part because the Fourier transform preserves the space L 2 R displaystyle L 2 mathbb R nbsp so that unlike the case of L 1 displaystyle L 1 nbsp the Fourier transform and inverse transform are on the same footing being transformations of the same space of functions to itself Importantly for functions in L 2 displaystyle L 2 nbsp the Fourier transform is no longer given by Eq 1 interpreted as a Lebesgue integral For example the function f x 1 x 2 1 2 displaystyle f x 1 x 2 1 2 nbsp is in L 2 displaystyle L 2 nbsp but not L 1 displaystyle L 1 nbsp so the integral Eq 1 diverges 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 pointwise limits implicit in an improper integral Titchmarsh 1986 and Dym amp McKean 1985 each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure The conventions chosen in this article are those of harmonic analysis and are characterized as the unique conventions such that the Fourier transform is both unitary on L2 and an algebra homomorphism from L1 to L without renormalizing the Lebesgue measure 12 Angular frequency w edit When the independent variable x displaystyle x nbsp represents time often denoted by t displaystyle t nbsp the transform variable 3 displaystyle xi nbsp represents frequency often denoted by f displaystyle f nbsp For example if time is measured in seconds then frequency is in hertz The Fourier transform can also be written in terms of angular frequency w 2 p 3 displaystyle omega 2 pi xi nbsp whose units are radians per second The substitution 3 w 2 p displaystyle xi tfrac omega 2 pi nbsp into Eq 1 produces this convention where function f displaystyle widehat f nbsp is relabeled f 1 displaystyle widehat f 1 nbsp f 3 w f x e i w x d x f 1 w 2 p f x 1 2 p f 3 w e i w x d w displaystyle begin aligned widehat f 3 omega amp triangleq int infty infty f x cdot e i omega x dx widehat f 1 left tfrac omega 2 pi right f x amp frac 1 2 pi int infty infty widehat f 3 omega cdot e i omega x d omega end aligned nbsp Unlike the Eq 1 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 2 p displaystyle 2 pi nbsp factor evenly between the transform and its inverse which leads to another convention f 2 w 1 2 p f x e i w x d x 1 2 p f 1 w 2 p f x 1 2 p f 2 w e i w x d w displaystyle begin aligned widehat f 2 omega amp triangleq frac 1 sqrt 2 pi int infty infty f x cdot e i omega x dx frac 1 sqrt 2 pi cdot widehat f 1 left tfrac omega 2 pi right f x amp frac 1 sqrt 2 pi int infty infty widehat f 2 omega cdot e i omega x d omega end aligned nbsp 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 Summary of popular forms of the Fourier transform one dimensional ordinary frequency 3 Hz unitary f 1 3 f x e i 2 p 3 x d x 2 p f 2 2 p 3 f 3 2 p 3 f x f 1 3 e i 2 p x 3 d 3 displaystyle begin aligned widehat f 1 xi amp triangleq int infty infty f x e i2 pi xi x dx sqrt 2 pi widehat f 2 2 pi xi widehat f 3 2 pi xi f x amp int infty infty widehat f 1 xi e i2 pi x xi d xi end aligned nbsp angular frequency w rad s unitary f 2 w 1 2 p f x e i w x d x 1 2 p f 1 w 2 p 1 2 p f 3 w f x 1 2 p f 2 w e i w x d w displaystyle begin aligned widehat f 2 omega amp triangleq frac 1 sqrt 2 pi int infty infty f x e i omega x dx frac 1 sqrt 2 pi widehat f 1 left frac omega 2 pi right frac 1 sqrt 2 pi widehat f 3 omega f x amp frac 1 sqrt 2 pi int infty infty widehat f 2 omega e i omega x d omega end aligned nbsp non unitary f 3 w f x e i w x d x f 1 w 2 p 2 p f 2 w f x 1 2 p f 3 w e i w x d w displaystyle begin aligned widehat f 3 omega amp triangleq int infty infty f x e i omega x dx widehat f 1 left frac omega 2 pi right sqrt 2 pi widehat f 2 omega f x amp frac 1 2 pi int infty infty widehat f 3 omega e i omega x d omega end aligned nbsp Generalization for n dimensional functions ordinary frequency 3 Hz unitary f 1 3 R n f x e i 2 p 3 x d x 2 p n 2 f 2 2 p 3 f 3 2 p 3 f x R n f 1 3 e i 2 p 3 x d 3 displaystyle begin aligned widehat f 1 xi amp triangleq int mathbb R n f x e i2 pi xi cdot x dx 2 pi frac n 2 widehat f 2 2 pi xi widehat f 3 2 pi xi f x amp int mathbb R n widehat f 1 xi e i2 pi xi cdot x d xi end aligned nbsp angular frequency w rad s unitary f 2 w 1 2 p n 2 R n f x e i w x d x 1 2 p n 2 f 1 w 2 p 1 2 p n 2 f 3 w f x 1 2 p n 2 R n f 2 w e i w x d w displaystyle begin aligned widehat f 2 omega amp triangleq frac 1 2 pi frac n 2 int mathbb R n f x e i omega cdot x dx frac 1 2 pi frac n 2 widehat f 1 left frac omega 2 pi right frac 1 2 pi frac n 2 widehat f 3 omega f x amp frac 1 2 pi frac n 2 int mathbb R n widehat f 2 omega e i omega cdot x d omega end aligned nbsp non unitary f 3 w R n f x e i w x d x f 1 w 2 p 2 p n 2 f 2 w f x 1 2 p n R n f 3 w e i w x d w displaystyle begin aligned widehat f 3 omega amp triangleq int mathbb R n f x e i omega cdot x dx widehat f 1 left frac omega 2 pi right 2 pi frac n 2 widehat f 2 omega f x amp frac 1 2 pi n int mathbb R n widehat f 3 omega e i omega cdot x d omega end aligned nbsp Extension of the definition edit For 1 lt p lt 2 displaystyle 1 lt p lt 2 nbsp the Fourier transform can be defined on L p R displaystyle L p mathbb R nbsp by Marcinkiewicz interpolation The Fourier transform can be defined on domains other than the real line The Fourier transform on Euclidean space and the Fourier transform on locally abelian groups are discussed later in the article The Fourier transform can also be defined for tempered distributions dual to the space of rapidly decreasing functions Schwartz functions A Schwartz function is a smooth function that decays at infinity along with all of its derivatives The space of Schwartz functions is denoted by S R displaystyle mathcal S mathbb R nbsp and its dual S R displaystyle mathcal S mathbb R nbsp is the space of tempered distributions It is easy to see by differentiating under the integral and applying the Riemann Lebesgue lemma that the Fourier transform of a Schwartz function defined by the formula Eq 1 is again a Schwartz function The Fourier transform of a tempered distribution T S R displaystyle T in mathcal S mathbb R nbsp is defined by duality T ϕ T ϕ ϕ S R displaystyle langle widehat T phi rangle langle T widehat phi rangle quad forall phi in mathcal S mathbb R nbsp Many other characterizations of the Fourier transform exist For example one uses the Stone von Neumann theorem the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrodinger representations of the Heisenberg group Background editHistory edit Main articles Fourier analysis History and Fourier series History In 1822 Fourier claimed see Joseph Fourier The Analytic Theory of Heat that any function whether continuous or discontinuous can be expanded into a series of sines 13 That important work was corrected and expanded upon by others to provide the foundation for the various forms of the Fourier transform used since nbsp Fig 1 When function A e i 2 p 3 t displaystyle A cdot e i2 pi xi t nbsp is depicted in the complex plane the vector formed by its imaginary and real parts rotates around the origin Its real part y t displaystyle y t nbsp is a cosine wave Complex sinusoids edit In general the coefficients f 3 displaystyle widehat f xi nbsp are complex numbers which have two equivalent forms see Euler s formula f 3 A e i 8 polar coordinate form A cos 8 i A sin 8 rectangular coordinate form displaystyle widehat f xi underbrace Ae i theta text polar coordinate form underbrace A cos theta iA sin theta text rectangular coordinate form nbsp The product with e i 2 p 3 x displaystyle e i2 pi xi x nbsp Eq 2 has these forms f 3 e i 2 p 3 x A e i 8 e i 2 p 3 x A e i 2 p 3 x 8 polar coordinate form A cos 2 p 3 x 8 i A sin 2 p 3 x 8 rectangular coordinate form displaystyle widehat f xi cdot e i2 pi xi x Ae i theta cdot e i2 pi xi x underbrace Ae i 2 pi xi x theta text polar coordinate form underbrace A cos 2 pi xi x theta iA sin 2 pi xi x theta text rectangular coordinate form nbsp 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 edit See also Negative frequency Simplifying the Fourier transform Euler s formula introduces the possibility of negative 3 displaystyle xi nbsp And Eq 1 is defined 3 R displaystyle forall xi in mathbb R nbsp Only certain complex valued f x displaystyle f x nbsp have transforms f 0 3 lt 0 displaystyle widehat f 0 forall xi lt 0 nbsp See Analytic signal A simple example is e i 2 p 3 0 x 3 0 gt 0 displaystyle e i2 pi xi 0 x xi 0 gt 0 nbsp But negative frequency is necessary to characterize all other complex valued f x displaystyle f x nbsp found in signal processing partial differential equations radar nonlinear optics quantum mechanics and others For a real valued f x displaystyle f x nbsp Eq 1 has the symmetry property f 3 f 3 displaystyle widehat f xi widehat f xi nbsp see Conjugation below This redundancy enables Eq 2 to distinguish f x cos 2 p 3 0 x displaystyle f x cos 2 pi xi 0 x nbsp from e i 2 p 3 0 x displaystyle e i2 pi xi 0 x nbsp But of course it cannot tell us the actual sign of 3 0 displaystyle xi 0 nbsp because cos 2 p 3 0 x displaystyle cos 2 pi xi 0 x nbsp and cos 2 p 3 0 x displaystyle cos 2 pi xi 0 x nbsp are indistinguishable on just the real numbers line Fourier transform for periodic functions edit The Fourier transform of a periodic function cannot be defined using the integral formula directly In order for integral in Eq 1 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 f x displaystyle f x nbsp is a periodic function with period P displaystyle P nbsp that has a convergent Fourier series then f 3 n c n d 3 n P displaystyle widehat f xi sum n infty infty c n cdot delta left xi tfrac n P right nbsp where c n displaystyle c n nbsp are the Fourier series coefficients of f displaystyle f nbsp and d displaystyle delta nbsp 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 Sampling the Fourier transform edit The Fourier transform of an integrable function f displaystyle f nbsp can be sampled at regular intervals of arbitrary length 1 P displaystyle tfrac 1 P nbsp These samples can be deduced from one cycle of a periodic function f P displaystyle f P nbsp which has Fourier series coefficients proportional to those samples by the Poisson summation formula f P x n f x n P 1 P k f k P e i 2 p k P x k Z displaystyle f P x triangleq sum n infty infty f x nP frac 1 P sum k infty infty widehat f left tfrac k P right e i2 pi frac k P x quad forall k in mathbb Z nbsp The integrability of f displaystyle f nbsp ensures the periodic summation converges Therefore the samples f k P displaystyle widehat f left tfrac k P right nbsp can be determined by Fourier series analysis f k P P f P x e i 2 p k P x d x displaystyle widehat f left tfrac k P right int P f P x cdot e i2 pi frac k P x dx nbsp When f x displaystyle f x nbsp has compact support f P x displaystyle f P x nbsp has a finite number of terms within the interval of integration When f x displaystyle f x nbsp does not have compact support numerical evaluation of f P x displaystyle f P x nbsp requires an approximation such as tapering f x displaystyle f x nbsp or truncating the number of terms Example editThe following figures provide a visual illustration of how the Fourier transform measures whether a frequency is present in a particular function The depicted function f t cos 2 p 3 t e p t 2 displaystyle f t cos 2 pi 3t e pi t 2 nbsp oscillates at 3 Hz if t displaystyle t nbsp measures seconds and tends quickly to 0 The second factor in this equation is an envelope function that shapes the continuous sinusoid into a short pulse f t displaystyle f t nbsp was specially chosen to have a real Fourier transform that can be easily plotted The first image is its graph In order to calculate f 3 displaystyle widehat f 3 nbsp we must integrate the product f t e i 2 p 3 t displaystyle f t e i2 pi 3t nbsp The next 2 images are the real and imaginary parts of that product The real part of the integrand has a non negative average value because the alternating signs of f t displaystyle f t nbsp and Re e i 2 p 3 t displaystyle operatorname Re e i2 pi 3t nbsp oscillate at the same rate and same phase whereas f t displaystyle f t nbsp and Im e i 2 p 3 t displaystyle operatorname Im e i2 pi 3t nbsp are same rate but orthogonal phase The result is that when you integrate the real part of the integrand you get a relatively large number in this case 1 2 displaystyle tfrac 1 2 nbsp Also when you try to measure a frequency that is not present as in the case when we look at f 5 displaystyle widehat f 5 nbsp both real and imaginary component of the product vary rapidly between positive and negative values Therefore the integral is very small and the value for the Fourier transform for that frequency is nearly zero The general situation is usually more complicated than this but heuristically this is how the Fourier transform measures how much of an individual frequency is present in a function f t displaystyle f t nbsp nbsp Original function showing oscillation 3 Hz Real and imaginary parts of integrand for Fourier transform at 3 Hz nbsp Real and imaginary parts of integrand for Fourier transform at 5 Hz nbsp Magnitude of Fourier transform with 3 and 5 Hz labeled To re enforce an earlier point the reason for the response at 3 3 displaystyle xi 3 nbsp Hz is because cos 2 p 3 t displaystyle cos 2 pi 3t nbsp and cos 2 p 3 t displaystyle cos 2 pi 3 t nbsp are indistinguishable The transform of e i 2 p 3 t e p t 2 displaystyle e i2 pi 3t cdot e pi t 2 nbsp would have just one response whose amplitude is the integral of the smooth envelope e p t 2 displaystyle e pi t 2 nbsp whereas Re f t e i 2 p 3 t displaystyle operatorname Re f t cdot e i2 pi 3t nbsp second graph above is e p t 2 1 cos 2 p 6 t 2 displaystyle e pi t 2 1 cos 2 pi 6t 2 nbsp Properties of the Fourier transform editLet f x displaystyle f x nbsp and h x displaystyle h x nbsp represent integrable functions Lebesgue measurable on the real line satisfying f x d x lt displaystyle int infty infty f x dx lt infty nbsp We denote the Fourier transforms of these functions as f 3 displaystyle hat f xi nbsp and h 3 displaystyle hat h xi nbsp respectively Basic properties edit The Fourier transform has the following basic properties 14 Linearity edit a f x b h x F a f 3 b h 3 a b C displaystyle a f x b h x stackrel mathcal F Longleftrightarrow a widehat f xi b widehat h xi quad a b in mathbb C nbsp Time shifting edit f x x 0 F e i 2 p x 0 3 f 3 x 0 R displaystyle f x x 0 stackrel mathcal F Longleftrightarrow e i2 pi x 0 xi widehat f xi quad x 0 in mathbb R nbsp Frequency shifting edit e i 2 p 3 0 x f x F f 3 3 0 3 0 R displaystyle e i2 pi xi 0 x f x stackrel mathcal F Longleftrightarrow widehat f xi xi 0 quad xi 0 in mathbb R nbsp Time scaling edit f a x F 1 a f 3 a a 0 displaystyle f ax stackrel mathcal F Longleftrightarrow frac 1 a widehat f left frac xi a right quad a neq 0 nbsp The case a 1 displaystyle a 1 nbsp leads to the time reversal property f x F f 3 displaystyle f x stackrel mathcal F Longleftrightarrow widehat f xi nbsp Symmetry edit When the real and imaginary parts of a complex function are decomposed into their even and odd parts there are four components denoted below by the subscripts RE RO IE and IO And there is a one to one mapping between the four components of a complex time function and the four components of its complex frequency transform T i m e d o m a i n f f R E f R O i f I E i f I O F F F F F F r e q u e n c y d o m a i n f f R E i f I O i f I E f R O displaystyle begin aligned mathsf Time domain quad amp f quad amp quad amp f RE quad amp quad amp f RO quad amp quad i amp f IE quad amp quad amp underbrace i f IO amp Bigg Updownarrow mathcal F amp amp Bigg Updownarrow mathcal F amp amp Bigg Updownarrow mathcal F amp amp Bigg Updownarrow mathcal F amp amp Bigg Updownarrow mathcal F mathsf Frequency domain quad amp widehat f quad amp quad amp widehat f RE quad amp quad amp overbrace i widehat f IO quad amp quad i amp widehat f IE quad amp quad amp widehat f RO end aligned nbsp From this various relationships are apparent for example The transform of a real valued function f R E f R O displaystyle f RE f RO nbsp is the even symmetric function f R E i f I O displaystyle hat f RE i hat f IO nbsp Conversely an even symmetric transform implies a real valued time domain The transform of an imaginary valued function i f I E i f I O displaystyle i f IE i f IO nbsp is the odd symmetric function f R O i f I E displaystyle hat f RO i hat f IE nbsp and the converse is true The transform of an even symmetric function f R E i f I O displaystyle f RE i f IO nbsp is the real valued function f R E f R O displaystyle hat f RE hat f RO nbsp and the converse is true The transform of an odd symmetric function f R O i f I E displaystyle f RO i f IE nbsp is the imaginary valued function i f I E i f I O displaystyle i hat f IE i hat f IO nbsp and the converse is true Conjugation edit f x F f 3 displaystyle bigl f x bigr stackrel mathcal F Longleftrightarrow left widehat f xi right nbsp Note the denotes complex conjugation In particular if f displaystyle f nbsp is real then f displaystyle widehat f nbsp is even symmetric aka Hermitian function f 3 f 3 displaystyle widehat f xi bigl widehat f xi bigr nbsp And if f displaystyle f nbsp is purely imaginary then f displaystyle widehat f nbsp is odd symmetric f 3 f 3 displaystyle widehat f xi widehat f xi nbsp Real and imaginary part in time edit Re f x F 1 2 f 3 f 3 displaystyle operatorname Re f x stackrel mathcal F Longleftrightarrow tfrac 1 2 left widehat f xi bigl widehat f xi bigr right nbsp Im f x F 1 2 i f 3 f 3 displaystyle operatorname Im f x stackrel mathcal F Longleftrightarrow tfrac 1 2i left widehat f xi bigl widehat f xi bigr right nbsp Zero frequency component edit Substituting 3 0 displaystyle xi 0 nbsp in the definition we obtain f 0 f x d x displaystyle widehat f 0 int infty infty f x dx nbsp The integral of f displaystyle f nbsp over its domain is known as the average value or DC bias of the function Invertibility and periodicity edit Further information Fourier inversion theorem and Fractional Fourier transform Under suitable conditions on the function f displaystyle f nbsp it can be recovered from its Fourier transform f displaystyle hat f nbsp Indeed denoting the Fourier transform operator by F displaystyle mathcal F nbsp so F f f displaystyle mathcal F f hat f nbsp then for suitable functions applying the Fourier transform twice simply flips the function F 2 f x f x displaystyle left mathcal F 2 f right x f x nbsp which can be interpreted as reversing time Since reversing time is two periodic applying this twice yields F 4 f f displaystyle mathcal F 4 f f nbsp so the Fourier transform operator is four periodic and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times F 3 f f displaystyle mathcal F 3 left hat f right f nbsp In particular the Fourier transform is invertible under suitable conditions More precisely defining the parity operator P displaystyle mathcal P nbsp such that P f x f x displaystyle mathcal P f x f x nbsp we have F 0 i d F 1 F F 2 P F 3 F 1 P F F P F 4 i d displaystyle begin aligned mathcal F 0 amp mathrm id mathcal F 1 amp mathcal F mathcal F 2 amp mathcal P mathcal F 3 amp mathcal F 1 mathcal P circ mathcal F mathcal F circ mathcal P mathcal F 4 amp mathrm id end aligned nbsp These equalities of operators require careful definition of the space of functions in question defining equality of functions equality at every point equality almost everywhere and defining equality of operators that is defining the topology on the function space and operator space in question These are not true for all functions but are true under various conditions which are the content of the various forms of the Fourier inversion theorem This fourfold periodicity of the Fourier transform is similar to a rotation of the plane by 90 particularly as the two fold iteration yields a reversal and in fact this analogy can be made precise While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain with the inverse Fourier transform switching them back more geometrically it can be interpreted as a rotation by 90 in the time frequency domain considering time as the x axis and frequency as the y axis and the Fourier transform can be generalized to the fractional Fourier transform which involves rotations by other angles This can be further generalized to linear canonical transformations which can be visualized as the action of the special linear group SL2 R on the time frequency plane with the preserved symplectic form corresponding to the uncertainty principle below This approach is particularly studied in signal processing under time frequency analysis Units edit See also Spectral density Units The frequency variable must have inverse units to the units of the original function s domain typically named t or x For example if t is measured in seconds 3 should be in cycles per second or hertz If the scale of time is in units of 2p seconds then another Greek letter w typically is used instead to represent angular frequency where w 2p3 in units of radians per second If using x for units of length then 3 must be in inverse length e g wavenumbers That is to say there are two versions of the real line one which is the range of t and measured in units of t and the other which is the range of 3 and measured in inverse units to the units of t These two distinct versions of the real line cannot be equated with each other Therefore the Fourier transform goes from one space of functions to a different space of functions functions which have a different domain of definition In general 3 must always be taken to be a linear form on the space of its domain which is to say that the second real line is the dual space of the first real line See the article on linear algebra for a more formal explanation and for more details This point of view becomes essential in generalizations of the Fourier transform to general symmetry groups including the case of Fourier series That there is no one preferred way often one says no canonical way to compare the two versions of the real line which are involved in the Fourier transform fixing the units on one line does not force the scale of the units on the other line is the reason for the plethora of rival conventions on the definition of the Fourier transform The various definitions resulting from different choices of units differ by various constants In other conventions the Fourier transform has i in the exponent instead of i and vice versa for the inversion formula This convention is common in modern physics 15 and is the default for Wolfram Alpha and does not mean that the frequency has become negative since there is no canonical definition of positivity for frequency of a complex wave It simply means that f 3 displaystyle hat f xi nbsp is the amplitude of the wave e i 2 p 3 x displaystyle e i2 pi xi x nbsp instead of the wave e i 2 p 3 x displaystyle e i2 pi xi x nbsp the former with its minus sign is often seen in the time dependence for Sinusoidal plane wave solutions of the electromagnetic wave equation or in the time dependence for quantum wave functions Many of the identities involving the Fourier transform remain valid in those conventions provided all terms that explicitly involve i have it replaced by i In Electrical engineering the letter j is typically used for the imaginary unit instead of i because i is used for current When using dimensionless units the constant factors might not even be written in the transform definition For instance in probability theory the characteristic function F of the probability density function f of a random variable X of continuous type is defined without a negative sign in the exponential and since the units of x are ignored there is no 2p either ϕ l f x e i l x d x displaystyle phi lambda int infty infty f x e i lambda x dx nbsp In probability theory and in mathematical statistics the use of the Fourier Stieltjes transform is preferred because so many random variables are not of continuous type and do not possess a density function and one must treat not functions but distributions i e measures which possess atoms From the higher point of view of group characters which is much more abstract all these arbitrary choices disappear as will be explained in the later section of this article which treats the notion of the Fourier transform of a function on a locally compact Abelian group Uniform continuity and the Riemann Lebesgue lemma edit nbsp The rectangular function is Lebesgue integrable nbsp The sinc function which is the Fourier transform of the rectangular function is bounded and continuous but not Lebesgue integrable The Fourier transform may be defined in some cases for non integrable functions but the Fourier transforms of integrable functions have several strong properties The Fourier transform f of any integrable function f is uniformly continuous and 16 f f 1 displaystyle left hat f right infty leq left f right 1 nbsp By the Riemann Lebesgue lemma 11 f 3 0 as 3 displaystyle hat f xi to 0 text as xi to infty nbsp However f displaystyle hat f nbsp need not be integrable For example the Fourier transform of the rectangular function which is integrable is the sinc function which is not Lebesgue integrable because its improper integrals behave analogously to the alternating harmonic series in converging to a sum without being absolutely convergent It is not generally possible to write the inverse transform as a Lebesgue integral However when both f and f displaystyle hat f nbsp are integrable the inverse equalityf x f 3 e i 2 p x 3 d 3 displaystyle f x int infty infty hat f xi e i2 pi x xi d xi nbsp holds holds for almost every x As a result the Fourier transform is injective on L1 R Plancherel theorem and Parseval s theorem edit Let f x and g x be integrable and let f 3 and ĝ 3 be their Fourier transforms If f x and g x are also square integrable then the Parseval formula follows 17 f g L 2 f x g x d x f 3 g 3 d 3 displaystyle langle f g rangle L 2 int infty infty f x overline g x dx int infty infty hat f xi overline hat g xi d xi nbsp where the bar denotes complex conjugation The Plancherel theorem which follows from the above states that 18 f L 2 2 f x 2 d x f 3 2 d 3 displaystyle f L 2 2 int infty infty left f x right 2 dx int infty infty left hat f xi right 2 d xi nbsp Plancherel s theorem makes it possible to extend the Fourier transform by a continuity argument to a unitary operator on L2 R On L1 R L2 R this extension agrees with original Fourier transform defined on L1 R thus enlarging the domain of the Fourier transform to L1 R L2 R and consequently to Lp R for 1 p 2 Plancherel s theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity The terminology of these formulas is not quite standardised Parseval s theorem was proved only for Fourier series and was first proved by Lyapunov But Parseval s formula makes sense for the Fourier transform as well and so even though in the context of the Fourier transform it was proved by Plancherel it is still often referred to as Parseval s formula or Parseval s relation or even Parseval s theorem See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups Poisson summation formula edit Main article Poisson summation formula The Poisson summation formula PSF is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function s continuous Fourier transform The Poisson summation formula says that for sufficiently regular functions f n f n n f n displaystyle sum n hat f n sum n f n nbsp It has a variety of useful forms that are derived from the basic one by application of the Fourier transform s scaling and time shifting properties The formula has applications in engineering physics and number theory The frequency domain dual of the standard Poisson summation formula is also called the discrete time Fourier transform Poisson summation is generally associated with the physics of periodic media such as heat conduction on a circle The fundamental solution of the heat equation on a circle is called a theta function It is used in number theory to prove the transformation properties of theta functions which turn out to be a type of modular form and it is connected more generally to the theory of automorphic forms where it appears on one side of the Selberg trace formula Differentiation edit Suppose f x is an absolutely continuous differentiable function and both f and its derivative f are integrable Then the Fourier transform of the derivative is given byf 3 F d d x f x i 2 p 3 f 3 displaystyle widehat f xi mathcal F left frac d dx f x right i2 pi xi hat f xi nbsp More generally the Fourier transformation of the n th derivative f n is given by f n 3 F d n d x n f x i 2 p 3 n f 3 displaystyle widehat f n xi mathcal F left frac d n dx n f x right i2 pi xi n hat f xi nbsp Analogically F d n d 3 n f 3 i 2 p x n f x displaystyle mathcal F left frac d n d xi n hat f xi right i2 pi x n f x nbsp so F x n f x i 2 p n d n d 3 n f 3 displaystyle mathcal F left x n f x right left frac i 2 pi right n frac d n d xi n hat f xi nbsp By applying the Fourier transform and using these formulas some ordinary differential equations can be transformed into algebraic equations which are much easier to solve These formulas also give rise to the rule of thumb f x is smooth if and only if f 3 quickly falls to 0 for 3 By using the analogous rules for the inverse Fourier transform one can also say f x quickly falls to 0 for x if and only if f 3 is smooth Convolution theorem edit Main article Convolution theorem The Fourier transform translates between convolution and multiplication of functions If f x and g x are integrable functions with Fourier transforms f 3 and ĝ 3 respectively then the Fourier transform of the convolution is given by the product of the Fourier transforms f 3 and ĝ 3 under other conventions for the definition of the Fourier transform a constant factor may appear This means that if h x f g x f y g x y d y displaystyle h x f g x int infty infty f y g x y dy nbsp where denotes the convolution operation then h 3 f 3 g 3 displaystyle hat h xi hat f xi hat g xi nbsp In linear time invariant LTI system theory it is common to interpret g x as the impulse response of an LTI system with input f x and output h x since substituting the unit impulse for f x yields h x g x In this case ĝ 3 represents the frequency response of the system Conversely if f x can be decomposed as the product of two square integrable functions p x and q x then the Fourier transform of f x is given by the convolution of the respective Fourier transforms p 3 and q 3 Cross correlation theorem edit Main articles Cross correlation and Wiener Khinchin theorem In an analogous manner it can be shown that if h x is the cross correlation of f x and g x h x f g x f y g x y d y displaystyle h x f star g x int infty infty overline f y g x y dy nbsp then the Fourier transform of h x is h 3 f 3 g 3 displaystyle hat h xi overline hat f xi hat g xi nbsp As a special case the autocorrelation of function f x is h x f f x f y f x y d y displaystyle h x f star f x int infty infty overline f y f x y dy nbsp for which h 3 f 3 f 3 f 3 2 displaystyle hat h xi overline hat f xi hat f xi left hat f xi right 2 nbsp Eigenfunctions edit See also Mehler kernel and Hermite polynomials Hermite functions as eigenfunctions of the Fourier transform The Fourier transform is a linear transform which has eigenfunctions obeying F ps l ps displaystyle mathcal F psi lambda psi nbsp with l C displaystyle lambda in mathbb C nbsp A set of eigenfunctions is found by noting that the homogeneous differential equation U 1 2 p d d x U x ps x 0 displaystyle left U left frac 1 2 pi frac d dx right U x right psi x 0 nbsp leads to eigenfunctions ps x displaystyle psi x nbsp of the Fourier transform F displaystyle mathcal F nbsp as long as the form of the equation remains invariant under Fourier transform note 6 In other words every solution ps x displaystyle psi x nbsp and its Fourier transform ps 3 displaystyle hat psi xi nbsp obey the same equation Assuming uniqueness of the solutions every solution ps x displaystyle psi x nbsp must therefore be an eigenfunction of the Fourier transform The form of the equation remains unchanged under Fourier transform if U x displaystyle U x nbsp can be expanded in a power series in which for all terms the same factor of either one of 1 i displaystyle pm 1 pm i nbsp arises from the factors i n displaystyle i n nbsp introduced by the differentiation rules upon Fourier transforming the homogeneous differential equation because this factor may then be cancelled The simplest allowable U x x displaystyle U x x nbsp leads to the standard normal distribution 19 More generally a set of eigenfunctions is also found by noting that the differentiation rules imply that the ordinary differential equation W i 2 p d d x W x ps x C ps x displaystyle left W left frac i 2 pi frac d dx right W x right psi x C psi x nbsp with C displaystyle C nbsp constant and W x displaystyle W x nbsp being a non constant even function remains invariant in form when applying the Fourier transform F displaystyle mathcal F nbsp to both sides of the equation The simplest example is provided by W x x 2 displaystyle W x x 2 nbsp which is equivalent to considering the Schrodinger equation for the quantum harmonic oscillator 20 The corresponding solutions provide an important choice of an orthonormal basis for L2 R and are given by the physicist s Hermite functions Equivalently one may use ps n x 2 4 n e p x 2 H e n 2 x p displaystyle psi n x frac sqrt 4 2 sqrt n e pi x 2 mathrm He n left 2x sqrt pi right nbsp where Hen x are the probabilist s Hermite polynomials defined as H e n x 1 n e 1 2 x 2 d d x n e 1 2 x 2 displaystyle mathrm He n x 1 n e frac 1 2 x 2 left frac d dx right n e frac 1 2 x 2 nbsp Under this convention for the Fourier transform we have thatps n 3 i n ps n 3 displaystyle hat psi n xi i n psi n xi nbsp In other words the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier transform on L2 R 14 21 However this choice of eigenfunctions is not unique Because of F 4 i d displaystyle mathcal F 4 mathrm id nbsp there are only four different eigenvalues of the Fourier transform the fourth roots of unity 1 and i and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction 22 As a consequence of this it is possible to decompose L2 R as a direct sum of four spaces H0 H1 H2 and H3 where the Fourier transform acts on Hek simply by multiplication by ik Since the complete set of Hermite functions psn provides a resolution of the identity they diagonalize the Fourier operator i e the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues and these sums can be explicitly summed F f 3 d x f x n 0 i n ps n x ps n 3 displaystyle mathcal F f xi int dxf x sum n geq 0 i n psi n x psi n xi nbsp This approach to define the Fourier transform was first proposed by Norbert Wiener 23 Among other properties Hermite functions decrease exponentially fast in both frequency and time domains and they are thus used to define a generalization of the Fourier transform namely the fractional Fourier transform used in time frequency analysis 24 In physics this transform was introduced by Edward Condon 25 This change of basis functions becomes possible because the Fourier transform is a unitary transform when using the right conventions Consequently under the proper conditions it may be expected to result from a self adjoint generator N displaystyle N nbsp via 26 F ps e i t N ps displaystyle mathcal F psi e itN psi nbsp The operator N displaystyle N nbsp is the number operator of the quantum harmonic oscillator written as 27 28 N 1 2, wikipedia, wiki, book, books, library, article, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games. © Copyright 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