Fourier Transform

In Mathematics , the Fourier transform, named in honor of French mathematician Joseph Fourier , is a certain Linear Operator that maps Function s to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its ''frequency components'', and the inverse transform synthesizes a function from its spectrum of frequency components. A useful analogy is the relationship between a series of pure Notes (the frequency components) and a musical chord (the function itself). In mathematical physics, the Fourier transform of a signal ''x''(''t'') can be thought of as that signal in the " Frequency Domain ." This is similar to the basic idea of the various other Fourier transforms including the Fourier Series of a periodic function.
(''See also Fractional Fourier Transform and Linear Canonical Transform for generalizations.'')

DEFINITIONS

There are several common conventions for defining the Fourier transform of a Complex-valued Lebesgue Integrable function,

x.\,

In communications and Signal Processing , for instance, it is often the function:

:

X(f) = \int_{-\infty}^{\infty} x(t)\ e^{-i 2\pi f t}\,dt,

for every Real Number

f.\,

When the independent variable

t\,

represents ''time'' (with SI unit of Second s), the transform variable

f\,

represents Ordinary Frequency (in Hertz ). The complex-valued function,

X,\,

is said to represent

x\,

in the frequency domain. I.e., if

x\,

is a continuous function, then it can be reconstructed from

X\,

by the '''inverse transform:'''

:

x(t) = \int_{-\infty}^{\infty} X(f)\ e^{ i 2 \pi f t}\,df,

for every real number

t.\,

Other notations for

X(f)\,

are:

\hat{x}(f)\,

and

\mathcal{F}\{x\}(f).\,

The interpretation of

X\,

is aided by expressing it in Polar Coordinate form:

X(f) = A(f)\ e^{i \phi (f)},\,

where:

x(t) = rac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} X(\omega)\ e^{ i\omega t}\,d\omega.

This convention and the

X(f)

convention are unitary transforms.

Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.

x(t) = rac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} X_1(\omega) \ e^{i \omega t}\, d \omega \

Align

"center" style="color: darkblue" '''non-unitary'''

Rowspan "2" align="center" style="color: darkred" '''ordinary frequency <math> f \, </math> (hertz)
Align "center" style="color: darkblue" '''unitary'''

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

Completeness

=\mathcal{F}^{-1} and the transform preserves inner-products (see Parseval's Theorem , also described below). Note that, $\mathcal{F}^---$ refers to Adjoint of the Fourier Transform operator.

Moreover we can check that,

= \mathcal{F}^{-1}, \quad \mbox{and} \quad \mathcal{F}^4 = \mathcal{I}\quad

where

\mathcal{J}

is the Time-Reversal operator defined as,

:<math> \mathcal{I}\{f\}(t) - F(t) 2 0</math>
:<math> L^1(\mathbb{R}^n) \{f: \, \mathbb{R}^n o \mathbb{C} \\big\ \int_{\mathbb{R}^n} f(x)\, dx < \infty\}</math>

where ω ∈ Rⁿ and ω · ''x'' is the Inner Product of the two vectors ω and ''x''.

One may now use this to define the continuous Fourier transform for compactly supported smooth functions, which are dense in ''L''²(Rⁿ). The Plancherel Theorem then allows us to extend the definition of the Fourier transform to functions on ''L''²(R^''n'') (even those not compactly supported) by continuity arguments. All the properties and formulas listed on this page apply to the Fourier transform so defined.

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

The Plancherel theorem and Parseval's theorem

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

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

:

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

where the bar denotes Complex Conjugation . Therefore, the Fourier transformation yields an Isometric Automorphism of the Hilbert Space ''L''²(Rⁿ).

The Plancherel Theorem , which is equivalent to Parseval's Theorem , states that

:<math>\int {-\infty}^\infty f(t)^2 \,dt 1</math>
Define The "http://wwwinformationdelightinfo/information/entry/expected_value" class="copylinks">Expected ''location''Location, momentum and particle do not have any physical meaning here they are simply convenient monikers chosen with analogy to the interpretation used in the Heisenberg Uncertainty Principle of a particle (with probability density ''f''(''t)''2) as

Cross-correlation theorem

In an analogous manner, it can be shown that if

g(t)

is the Cross-correlation of

f(t)

and

h(t)

:

:

g(t)=(f\star h)(t) = \int_{-\infty}^\infty \bar{f}(s)\,h(t+s)\,ds

then the Fourier transform of

g(t)

is:

:

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

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

Tempered distributions

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

TABLE OF IMPORTANT FOURIER TRANSFORMS

The following table records some important Fourier transforms. ''G'' and ''H'' denote Fourier transforms of ''g''(''t'') and ''h''(''t''), respectively. ''g'' and ''h'' may be integrable functions or tempered distributions. Note that the two most common unitary conventions are included.

Functional relationships

	<math>g H\,</math> Denotes The	"http://wwwinformationdelightinfo/information/entry/convolution" class="copylinks">Convolution of <math>g\,</math> and <math>h\,</math> &mdash this rule is the Convolution Theorem
	<math>g(t)\,</math> Is Purely Real, And An	"http://wwwinformationdelightinfo/information/entry/even_function" class="copylinks">Even Function
	colspan	"2" align="center"<math>G(\omega)\,</math> and <math>G(f)\,</math> are purely real, and Even Function s
	<math>g(t)\,</math> Is Purely Real, And An	"http://wwwinformationdelightinfo/information/entry/odd_function" class="copylinks">Odd Function
	colspan	"2" align="center"<math>G(\omega)\,</math> and <math>G(f)\,</math> are purely Imaginary , and Odd Function s

{ Class

"wikitable"

The

"http://wwwinformationdelightinfo/information/entry/rectangular_function" class="copylinks">Rectangular Pulse and the normalized Sinc Function

Dual Of Rule 201 The

"http://wwwinformationdelightinfo/information/entry/rectangular_function" class="copylinks">Rectangular Function is an idealized Low-pass Filter , and the Sinc Function is the Non-causal impulse response of such a filter

''tri'' Is The

"http://wwwinformationdelightinfo/information/entry/triangular_function" class="copylinks">Triangular Function

Shows That The "http://wwwinformationdelightinfo/information/entry/Gaussian_function" class="copylinks">Gaussian Function <math>\exp(-\alpha t^2)</math> is its own Fourier transform For this to be integrable we must have <math>\operatorname{Re}(\alpha)>0</math>
<math> E^{iat^2} \left e^{-\alpha t^2} ight_{\alpha = -i a} \,</math>

Common In "http://wwwinformationdelightinfo/information/entry/optics" class="copylinks">Optics

''J0(t)'' Is The "http://wwwinformationdelightinfo/information/entry/Bessel_function" class="copylinks">Bessel Function of first kind of order 0

It's The Generalization Of The Previous Transform ''Tn (t)'' Is The "http://wwwinformationdelightinfo/information/entry/Chebyshev_polynomials" class="copylinks">Chebyshev Polynomial Of The First Kind

''Un (t)'' Is The "http://wwwinformationdelightinfo/information/entry/Chebyshev_polynomials" class="copylinks">Chebyshev Polynomial Of The Second Kind

"http://wwwinformationdelightinfo/information/entry/Hyperbolic_function" class="copylinks">Hyperbolic Secant is its own Fourier transform

{ Class

"wikitable"

<math>\displaystyle\delta(\omega)</math> Denotes The "http://wwwinformationdelightinfo/information/entry/Dirac_delta" class="copylinks">Dirac Delta distribution

Follows From Rules 101 And 303 Using "http://wwwinformationdelightinfo/information/entry/Eulers_formula_in_complex_analysis" class="copylinks">Euler's Formula : <math>\displaystyle\cos(a t) = (e^{i a t} + e^{-i a t})/2</math>
Also From 101 And 303 Using <math>\displaystyle\sin(a T) (e^{i a t} - e^{-i a t})/(2i)</math>
Here, <math>\displaystyle N</math> Is A "http://wwwinformationdelightinfo/information/entry/natural_number" class="copylinks">Natural Number <math>\displaystyle\delta^n(\omega)</math> is the <math>\displaystyle n</math>-th distribution derivative of the Dirac delta This rule follows from rules 107 and 302 Combining this rule with 1, we can transform all Polynomial s

Here <math>\displaystyle\sgn(\omega)</math> Is The "http://wwwinformationdelightinfo/information/entry/sign_function" class="copylinks">Sign Function note that this is consistent with rules 107 and 302

Here <math>u(t)</math> Is The Heaviside "http://wwwinformationdelightinfo/information/entry/Heaviside_step_function" class="copylinks">Unit Step Function this follows from rules 101 and 309
<math>u(t)</math> Is The Heaviside "http://wwwinformationdelightinfo/information/entry/Heaviside_step_function" class="copylinks">Unit Step Function and <math>a > 0</math>
<math>\sum {n -\infty}^{\infty} \delta (t - n T) \,</math>

<math>Rac{1}{T} \sum {k -\infty}^{\infty} \delta \left( f -rac{k }{T} ight) \,</math>
The "http://wwwinformationdelightinfo/information/entry/Dirac_comb" class="copylinks">Dirac Comb &mdash helpful for explaining or understanding the transition from continuous to discrete time

;Linearity
::::

af_1(t) + bf_2(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad aF_1(\omega) + bF_2(\omega)

;Convolution

f_2(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad F_1(\omega)F_2(\omega)

;Conjugation
::::

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

;Scaling

;Time shift
::::

f(t-t_0) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad e^{-i\omega t_0}F(\omega)

;Modulation (multiplication by complex exponential)
::::

f(t)\cdot e^{i\omega_{0}t} \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad F(\omega-\omega_{0})\qquad \omega_{0} \in \mathbb{R},

;Multiplication by sin

\omega

₀t
::::

f(t)\sin \omega_{0}t \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad rac{i}{2}

{Link without Title} \,

;Multiplication by cos

\omega

₀t
::::

f(t)\cos \omega_{0}t \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad rac{1}{2}

{Link without Title} \,

;Integration
::::

\int_{-\infty}^{t} f(u)\, du \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad rac{1}{i\omega}F(\omega)+\pi F(0)\delta(\omega)\,

;Parseval's theorem
::::

\int_{\mathbb{R}} f(t)\cdot \overline{g(t)}\, dt = \int_{\mathbb{R}} F(\omega)\cdot \overline{G(\omega)}\, d\omega \,

	CATEGORIES ABOUT FOURIER TRANSFORM

	fundamental physics concepts

	fourier analysis

	integral transforms

	unitary operators

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Fourier Transform