# Fourier transform

The Fourier transformation (more precisely the continuous Fourier transformation; pronunciation: [fuʁie] ) is a mathematical method from the field of Fourier analysis with which continuous, aperiodic signals are broken down into a continuous spectrum . The function that describes this spectrum is also called a Fourier transform or spectral function. It is an integral transformation that is named after the mathematician Jean Baptiste Joseph Fourier . Fourier introduced the Fourier series in 1822 , which, however, is only defined for periodic signals and leads to a discrete frequency spectrum.

## definition

Be an integrable function. The (continuous) Fourier transform of is defined by ${\ displaystyle f \ in L ^ {1} (\ mathbb {R} ^ {n})}$${\ displaystyle {\ mathcal {F}} f}$${\ displaystyle f}$

${\ displaystyle ({\ mathcal {F}} f) (y) = {\ frac {1} {\ left (2 \ pi \ right) ^ {n / 2}}} \ int _ {\ mathbb {R} ^ {n}} f (x) \, e ^ {- \ mathrm {i} y \ cdot x} \, \ mathrm {d} x}$

and the corresponding inverse transformation is:

${\ displaystyle f (x) = {\ frac {1} {\ left (2 \ pi \ right) ^ {n / 2}}} \ int _ {\ mathbb {R} ^ {n}} ({\ mathcal {F}} f) (y) \, e ^ {\ mathrm {i} y \ cdot x} \, \ mathrm {d} y \ ;.}$

The following applies: and are -dimensional volume elements , the imaginary unit and the standard scalar product of the vectors and . ${\ displaystyle \ mathrm {d} x}$${\ displaystyle \ mathrm {d} y}$${\ displaystyle n}$${\ displaystyle \ mathrm {i}}$${\ displaystyle y \ cdot x}$${\ displaystyle y}$${\ displaystyle x}$

The normalization constant is not uniform in the literature. In the theory of pseudo differential operators and in signal processing, it is common to omit the factor in the transformation so that the inverse transformation receives the prefactor instead . The transformation is then: ${\ displaystyle 1 / (2 \ pi) ^ {n / 2}}$${\ displaystyle 1 / (2 \ pi) ^ {n}}$

${\ displaystyle ({\ mathcal {F}} f) (y) = \ int _ {\ mathbb {R} ^ {n}} f (x) \, e ^ {- \ mathrm {i} y \ cdot x } \, \ mathrm {d} x \ ;,}$
${\ displaystyle f (x) = {\ frac {1} {(2 \ pi) ^ {n}}} \ int _ {\ mathbb {R} ^ {n}} ({\ mathcal {F}} f) (y) \, e ^ {\ mathrm {i} y \ cdot x} \, \ mathrm {d} y \ ;.}$

This has the disadvantage that a prefactor appears in Parseval's theorem , which means that the Fourier transformation is then no longer a unitary mapping to . In other words: the signal power then changes as a result of the Fourier transformation. In the literature on signal processing and system theory, the following convention can also be found, which does not require any pre-factors: ${\ displaystyle L ^ {1} (\ mathbb {R} ^ {n}) \ cap L ^ {2} (\ mathbb {R} ^ {n})}$

${\ displaystyle ({\ mathcal {F}} f) (y) = \ int _ {\ mathbb {R} ^ {n}} f (x) \, e ^ {- 2 \ pi \ mathrm {i} y \ cdot x} \, \ mathrm {d} x \ ;,}$
${\ displaystyle f (x) = \ int _ {\ mathbb {R} ^ {n}} ({\ mathcal {F}} f) (y) \, e ^ {2 \ pi \ mathrm {i} y \ cdot x} \, \ mathrm {d} y \ ;.}$

The real form of the Fourier transform is called the Hartley transform . For real functions , the Fourier transformation can be substituted by the sine and cosine transformation . ${\ displaystyle f}$

## Use case

There is a special application in acoustics : The pure concert pitch is a sine wave with a frequency of 440 Hz, i.e. 440 oscillations per second. An ideal tuning fork emits exactly this sinusoidal signal. The same note played with another musical instrument (non-ideal tuning fork) is a composition / superposition of waves of different wavelengths. The composition of these waves is unique to the timbre of every musical instrument. Only the wave with the greatest wavelength, the fundamental tone of the signal, has a frequency of 440 Hz. The other waves, the overtones , have higher frequencies. Waves of higher frequency can only be perceived aurally up to a limit frequency of up to 20 kHz , which corresponds to the age of the person . ${\ displaystyle {\ bar {a}}}$

The different frequencies / wavelengths of the wave composition can be read directly from the Fourier transform of the audio signal. This property can be used, for example, for the automatic recognition of pitches and musical instruments in a sound signal. The continuous Fourier transformation is used for analog audio signals; the discrete Fourier transform is used for digital audio signals such as mp3 audio files . There are also runtime-optimized variants of the latter.

The vibrations of the sound signal of an instrument can be visualized with a microphone and oscilloscope. There is also a clear introduction to the Fourier transform of the sound signal of a horn, a clarinet and a tuning fork.

## example

As an example, the frequency spectrum of a damped oscillation with sufficiently weak damping is to be examined. This can be described by the following function:

${\ displaystyle f (t) = x_ {0} \ cdot e ^ {- {\ frac {t} {\ tau}}} \ cdot \ cos (\ omega _ {\ rm {s}} t) \ Theta ( t)}$

or in complex notation:

${\ displaystyle f (t) = x_ {0} \ cdot e ^ {- {\ frac {t} {\ tau}}} \ cdot {\ tfrac {1} {2}} (e ^ {\ mathrm {i } \ omega _ {\ rm {s}} t} + e ^ {- \ mathrm {i} \ omega _ {\ rm {s}} t}) \ Theta (t) \ ;.}$

Here is the amplitude and the angular frequency of the oscillation, the time in which the amplitude by a factor drops, and the Heaviside function . That is, the function is only non-zero for positive times. ${\ displaystyle x_ {0}}$${\ displaystyle \ omega _ {\ rm {s}}}$${\ displaystyle \ tau}$${\ displaystyle 1 / e}$${\ displaystyle \ Theta (t)}$

You get

{\ displaystyle {\ begin {aligned} F (\ omega) = ({\ mathcal {F}} f) (\ omega) & = {\ frac {1} {\ sqrt {2 \ pi}}} \ int _ {- \ infty} ^ {\ infty} f (t) e ^ {- \ mathrm {i} \ omega t} \, \ mathrm {d} t \\ & = {\ frac {1} {\ sqrt {2 \ pi}}} \ int _ {- \ infty} ^ {\ infty} x_ {0} \ cdot e ^ {- t / \ tau} \ cdot {\ tfrac {1} {2}} \ left (e ^ {\ mathrm {i} \ omega _ {\ rm {s}} t} + e ^ {- \ mathrm {i} \ omega _ {\ rm {s}} t} \ right) \ Theta (t) \ cdot e ^ {- \ mathrm {i} \ omega t} \, \ mathrm {d} t \\ & = {\ frac {x_ {0}} {\ sqrt {2 \ pi}}} \ int _ {0} ^ {\ infty} e ^ {- t / \ tau} \ cdot {\ tfrac {1} {2}} \ left (e ^ {\ mathrm {i} \ omega _ {\ rm {s}} t} + e ^ {- \ mathrm {i} \ omega _ {\ rm {s}} t} \ right) \ cdot e ^ {- \ mathrm {i} \ omega t} \, \ mathrm {d} t \\ & = {\ frac {x_ {0}} {2 {\ sqrt {2 \ pi}}}} \ int _ {0} ^ {\ infty} \ left (e ^ {- t \ left (1 / \ tau - \ mathrm {i} (\ omega _ {\ rm {s}} - \ omega) \ right)} + e ^ {- t \ left (1 / \ tau + \ mathrm {i} (\ omega _ {\ rm {s}} + \ omega) \ right)} \ right) \, \ mathrm {d} t \\ & = {\ frac {x_ {0}} {2 {\ sqrt {2 \ pi}}}} \ left [- {\ frac {1} {1 / \ tau - \ mathrm {i} (\ omega _ {\ rm {s}} - \ omega)}} e ^ {- t \ left (1 / \ tau - \ mathrm {i} (\ omega _ {\ rm {s}} - \ omega) \ right)} - ​​{\ frac {1} {1 / \ tau + \ mathrm {i} (\ omega _ {\ rm {s}} + \ omega)}} e ^ {- t \ left (1 / \ tau + \ mathrm {i} (\ omega _ {\ rm {s}} + \ omega) \ right)} \ right] _ {0} ^ {\ infty} \\ & = {\ frac {x_ {0}} {2 {\ sqrt {2 \ pi}}}} \ left ({\ frac {1} {1 / \ tau - \ mathrm {i} (\ omega _ {\ rm {s}} - \ omega)}} + {\ frac {1} {1 / \ tau + \ mathrm {i} (\ omega _ {\ rm {s}} + \ omega)}} \ right) \\ & = {\ frac {x_ {0} } {\ sqrt {2 \ pi}}} {\ frac {1 / \ tau + \ mathrm {i} \ omega} {(1 / \ tau + \ mathrm {i} \ omega) ^ {2} + \ omega _ {\ rm {s}} ^ {2}}} \,. \ end {aligned}}}

## properties

### Linearity

The Fourier transform is a linear operator . That is, it applies . ${\ displaystyle {\ mathcal {F}}}$${\ displaystyle {\ mathcal {F}} (a \ cdot f + b \ cdot g) = a \ cdot {\ mathcal {F}} (f) + b \ cdot {\ mathcal {F}} (g)}$

### continuity

The Fourier transformation is a continuous operator from the space of integrable functions into the space of functions that vanish at infinity . The set of continuous functions which vanish for is denoted by . The fact that the Fourier transforms vanish at infinity is also known as the Riemann-Lebesgue lemma . The inequality also applies ${\ displaystyle L ^ {1} (\ mathbb {R} ^ {n})}$${\ displaystyle C_ {0} (\ mathbb {R} ^ {n})}$${\ displaystyle C_ {0} (\ mathbb {R} ^ {n})}$${\ displaystyle | x | \ to \ infty}$

${\ displaystyle \ | {\ mathcal {F}} f \ | _ {L ^ {\ infty} (\ mathbb {R} ^ {n})} \ leq {\ frac {1} {({\ sqrt {2 \ pi}}) ^ {n}}} \ | f \ | _ {L ^ {1} (\ mathbb {R} ^ {n})}}$.

### Differentiation rules

Be a Schwartz function and a multi-index . Then applies ${\ displaystyle f \ in S (\ mathbb {R} ^ {n}) \ subset L ^ {1} (\ mathbb {R} ^ {n})}$${\ displaystyle \ alpha \ in \ mathbb {N} _ {0} ^ {n}}$

• ${\ displaystyle {\ mathcal {F}} f \ in S (\ mathbb {R} ^ {n})}$and .${\ displaystyle D ^ {\ alpha} ({\ mathcal {F}} f) = (- {\ rm {i}}) ^ {| \ alpha |} {\ mathcal {F}} (x ^ {\ alpha } f)}$
• ${\ displaystyle ({\ mathcal {F}} (D ^ {\ alpha} f)) (\ xi) = {\ rm {i}} ^ {| \ alpha |} \ xi ^ {\ alpha} ({\ mathcal {F}} f) (\ xi)}$.

### fixed point

The density function

${\ displaystyle \ varphi (x) = {\ frac {1} {(2 \ pi) ^ {n / 2}}} \ cdot e ^ {- {\ frac {1} {2}} \ | x \ | ^ {2}}}$

with the ( -dimensional) Gaussian normal distribution is a fixed point of the Fourier transformation. That is, the equation applies to everyone${\ displaystyle x \ in \ mathbb {R} ^ {n}}$${\ displaystyle n}$${\ displaystyle x \ in \ mathbb {R} ^ {n}}$

${\ displaystyle ({\ mathcal {F}} \ varphi) (x) = \ varphi (x)}$.

In particular, is an eigenfunction of the Fourier transform to the eigenvalue . With the help of the residual theorem or with the help of partial integration and solving an ordinary differential equation, the Fourier integral can be determined in this case . ${\ displaystyle \ varphi}$ ${\ displaystyle 1}$${\ displaystyle \ textstyle {\ tfrac {1} {(2 \ pi) ^ {n}}} \ int _ {\ mathbb {R}} e ^ {{\ rm {i}} x \ cdot \ xi} e ^ {- {\ frac {1} {2}} x ^ {2}} \ mathrm {d} x}$

### Mirror symmetry

The equation applies to all${\ displaystyle f \ in S (\ mathbb {R} ^ {n})}$${\ displaystyle x \ in \ mathbb {R} ^ {n}}$

${\ displaystyle ({\ mathcal {F}} ^ {2} f) (x) = ({\ mathcal {F}} ({\ mathcal {F}} f)) (x) = f (-x)}$.

This can be equivalent to the operator equation on the Schwartz space ${\ displaystyle S (\ mathbb {R} ^ {n})}$

${\ displaystyle {\ mathcal {F}} ^ {2} = {\ mathcal {P}}}$

write, where

${\ displaystyle {\ mathcal {P}}: f \ mapsto (x \ mapsto f (-x))}$

denotes the parity operator.

### Inverse transformation formula

Let be an integrable function such that also holds. Then the inverse transformation applies ${\ displaystyle f \ in L ^ {1} (\ mathbb {R} ^ {n})}$${\ displaystyle {\ mathcal {F}} (f) \ in L ^ {1} (\ mathbb {R} ^ {n})}$

${\ displaystyle {\ mathcal {F}} ^ {- 1} ({\ mathcal {F}} (f)) (x) = f (x) = {\ frac {1} {(2 \ pi) ^ { \ frac {n} {2}}}} \ int _ {\ mathbb {R} ^ {n}} e ^ {\ mathrm {i} tx} {\ mathcal {F}} (f) (t) \, \ mathrm {d} t.}$

This is also called Fourier synthesis . In the Schwartz space , the Fourier transform is an automorphism . ${\ displaystyle S (\ mathbb {R} ^ {n})}$

### Convolution theorem

The convolution theorem for the Fourier transformation says that the convolution of two functions is converted into a multiplication of real numbers by the Fourier transformation in their image space. For so true ${\ displaystyle f, g \ in L ^ {1} (\ mathbb {R} ^ {n})}$

${\ displaystyle {\ mathcal {F}} (f * g) = (2 \ pi) ^ {\ tfrac {n} {2}} \, {\ mathcal {F}} (f) \ cdot {\ mathcal { F}} (g)}$.

The reverse of the convolution theorem says

${\ displaystyle {\ mathcal {F}} (f) * {\ mathcal {F}} (g) = (2 \ pi) ^ {\ tfrac {n} {2}} {\ mathcal {F}} (f \ cdot g)}$.

## Fourier transform of L 2 functions

### definition

For a function , the Fourier transform is defined by means of a density argument ${\ displaystyle f \ in L ^ {2} (\ mathbb {R} ^ {n})}$

${\ displaystyle {\ mathcal {F}} (f) (\ xi) = \ lim _ {r \ to \ infty} {\ frac {1} {\ left (2 \ pi \ right) ^ {\ frac {n } {2}}}} \ int _ {B_ {r} (0)} f (x) e ^ {- \ mathrm {i} x \ xi} \ mathrm {d} x}$.

The convergence is to be understood in the sense of and is the sphere around the origin with a radius . For functions , this definition is the same as in the first section. Since the Fourier transform with respect to the is unitary -Skalarproduktes (see below) and in is dense, it follows that the Fourier transform is an isometric of the automorphism is. This is the statement of Plancherel's theorem . ${\ displaystyle L ^ {2}}$${\ displaystyle B_ {r} (0) = \ {x \ in \ mathbb {R} ^ {n}: | x | \ leq r \}}$${\ displaystyle r}$${\ displaystyle f \ in L ^ {2} (\ mathbb {R} ^ {n}) \ cap L ^ {1} (\ mathbb {R} ^ {n})}$${\ displaystyle L ^ {2}}$${\ displaystyle L ^ {2} (\ mathbb {R} ^ {n}) \ cap L ^ {1} (\ mathbb {R} ^ {n})}$${\ displaystyle L ^ {2} (\ mathbb {R} ^ {n})}$${\ displaystyle L ^ {2} (\ mathbb {R} ^ {n})}$

### Hausdorff-Young inequality

Be and . For is and it applies ${\ displaystyle 1 \ leq p \ leq 2}$${\ displaystyle {\ tfrac {1} {p}} + {\ tfrac {1} {q}} = 1}$${\ displaystyle f \ in L ^ {p} (\ mathbb {R} ^ {n})}$${\ displaystyle {\ mathcal {F}} (f) \ in L ^ {q} (\ mathbb {R} ^ {n})}$

${\ displaystyle \ | {\ mathcal {F}} (f) \ | _ {L ^ {q} (\ mathbb {R} ^ {n})} \ leq {\ frac {1} {(2 \ pi) ^ {n \ left ({\ frac {1} {p}} - {\ frac {1} {2}} \ right)}}} \ | f \ | _ {L ^ {p} (\ mathbb {R } ^ {n})}}$.

The Fourier transform has a continuation to a continuous operator , which is given by ${\ displaystyle {\ mathcal {F}}: L ^ {2} (\ mathbb {R} ^ {n}) \ to L ^ {2} (\ mathbb {R} ^ {n})}$${\ displaystyle {\ mathcal {F}}: L ^ {p} (\ mathbb {R} ^ {n}) \ to L ^ {q} (\ mathbb {R} ^ {n})}$

${\ displaystyle {\ mathcal {F}} (f) (\ xi) = \ lim _ {r \ to \ infty} {\ frac {1} {(2 \ pi) ^ {\ frac {n} {2} }}} \ int _ {B_ {r} (0)} f (x) e ^ {- \ mathrm {i} x \ xi} \ mathrm {d} x}$

is described. The limit value is to be understood here in the sense of . ${\ displaystyle L ^ {q}}$

### Differentiation rule

If the function is weakly differentiable, there is a differentiation rule analogous to that for black functions. So let us be a k-times weakly differentiable L 2 -function and a multi-index with . Then applies ${\ displaystyle f}$${\ displaystyle f \ in W ^ {k, 2} (\ mathbb {R} ^ {n}) = H ^ {k} (\ mathbb {R} ^ {n})}$${\ displaystyle \ alpha}$${\ displaystyle | \ alpha | \ leq k}$

${\ displaystyle {\ mathcal {F}} (D ^ {\ alpha} f) (\ xi) = \ mathrm {i} ^ {| \ alpha |} \ xi ^ {\ alpha} {\ mathcal {F}} (f) (\ xi)}$.

### Unitary illustration

The Fourier transform is a unitary operator with respect to the complex -scalar product , that is, it holds ${\ displaystyle L ^ {2}}$

${\ displaystyle \ langle {\ mathcal {F}} (f), g \ rangle _ {L ^ {2}} = \ int _ {\ mathbb {R} ^ {n}} {\ overline {{\ mathcal { F}} (f)}} (x) g (x) \ mathrm {d} x = \ int _ {\ mathbb {R} ^ {n}} {\ overline {f}} (x) {\ mathcal { F}} ^ {- 1} (g) (x) \ mathrm {d} x = \ langle f, {\ mathcal {F}} ^ {- 1} (g) \ rangle _ {L ^ {2}} .}$

The spectrum of the Fourier transformation is thus on the unit circle . In the one-dimensional case ( ), the Hermite functions in space also form a complete orthonormal system of eigenfunctions to the eigenvalues . ${\ displaystyle n = 1}$ ${\ displaystyle \ left (h_ {n} \ right) _ {n \ in \ mathbb {N} _ {0}}}$${\ displaystyle L ^ {2} \ left (\ mathbb {R} \ right)}$${\ displaystyle \ left (- \ mathrm {i} \ right) ^ {n}}$

## Fourier transformation in the space of tempered distributions

Be a tempered distribution, the Fourier transform is defined by for all${\ displaystyle u \ in S '(\ mathbb {R} ^ {n})}$${\ displaystyle {\ mathcal {F}} (u)}$${\ displaystyle \ phi \ in S (\ mathbb {R} ^ {n})}$

${\ displaystyle {\ mathcal {F}} (u) (\ phi): = u ({\ mathcal {F}} (\ phi))}$.

If one equips the room with the weak - * - topology , then the Fourier transformation is a continuous, bijective mapping . Your reverse mapping is ${\ displaystyle S '(\ mathbb {R} ^ {n})}$${\ displaystyle S '(\ mathbb {R} ^ {n})}$

${\ displaystyle u (\ phi) (- x) = {\ frac {1} {(2 \ pi) ^ {n}}} {\ mathcal {F}} ({\ mathcal {F}} (u)) (\ phi) (x)}$.

## Fourier transformation of measures

The Fourier transform is generally for finite Borel measures on defined: ${\ displaystyle \ mathbb {R} ^ {n}}$

${\ displaystyle {\ check {\ mu}} (x) = \ int e ^ {\ mathrm {i} xy} \ mu (\ mathrm {d} y)}$

is called the inverse Fourier transform of the measure. The characteristic function is then the inverse Fourier transform of a probability distribution .

## Partial differential equations

In the theory of partial differential equations , the Fourier transform plays an important role. With their help, one can find solutions to certain differential equations. The differentiation rule and the convolution theorem are of essential importance. Using the example of the heat conduction equation, it is now shown how to solve a partial differential equation with the Fourier transformation. The initial value problem of the heat equation is

${\ displaystyle \ left \ {{\ begin {array} {rcll} {\ frac {\ partial u} {\ partial t}} (x, t) & = & \ Delta _ {x} u (x, t) & {\ text {in}} \ mathbb {R} ^ {n} \ times] 0, \ infty [\\ u (x, t) & = & g (x, t) & {\ text {on}} \ mathbb {R} ^ {n} \ times \ {t = 0 \} \,. \ end {array}} \ right.}$

Here is the Laplace operator , which only affects the variables. Applying the Fourier transform to both equations with respect to the variable and applying the differentiation rule gives ${\ displaystyle \ Delta _ {x}}$${\ displaystyle x}$${\ displaystyle x}$

${\ displaystyle \ left \ {{\ begin {array} {rcll} {\ mathcal {F}} \ left ({\ frac {\ partial u} {\ partial t}} \ right) (\ xi, t) & = & - | \ xi | ^ {2} {\ mathcal {F}} (u) (\ xi, t) & {\ text {in}} \ mathbb {R} ^ {n} \ times] 0, \ infty [\\ {\ mathcal {F}} (u) (\ xi, t) & = & {\ mathcal {F}} (g) (\ xi, t) & {\ text {on}} \ mathbb { R} ^ {n} \ times \ {t = 0 \} \,. \ End {array}} \ right.}$

This is now an ordinary differential equation that gives the solution

${\ displaystyle {\ mathcal {F}} (u) (\ xi, t) = e ^ {- t | \ xi | ^ {2}} {\ mathcal {F}} (g) (\ xi, t) }$

Has. It follows from this and, due to the convolution theorem, applies ${\ displaystyle \ textstyle u (x, t) = {\ mathcal {F}} ^ {- 1} \ left (\ exp (-t | \ xi | ^ {2}) {\ mathcal {F}} (g ) \ right) (x, t)}$

${\ displaystyle u (x, t) = {\ frac {g (x, t) * F (x, t)} {(2 \ pi) ^ {\ frac {n} {2}}}}}$

with it follows ${\ displaystyle {\ mathcal {F}} (F) (\ xi, t) = \ exp (-t | \ xi | ^ {2}).}$

${\ displaystyle F (x, t) = {\ frac {1} {(2 \ pi) ^ {\ frac {n} {2}}}} \ int _ {\ mathbb {R} ^ {n}} e ^ {\ mathrm {i} x \ cdot yt | y | ^ {2}} \ mathrm {d} y = {\ frac {1} {(2t) ^ {\ frac {n} {2}}}} e ^ {\ frac {- | x | ^ {2}} {4t}} \ ,.}$

This is the fundamental solution to the heat equation. The solution of the initial value problem considered here has the representation

${\ displaystyle u (x, t) = {\ frac {g (x, t) * F (x, t)} {(2 \ pi) ^ {\ frac {n} {2}}}} = {\ frac {1} {(4 \ pi t) ^ {\ frac {n} {2}}}} \ int _ {\ mathbb {R} ^ {n}} e ^ {- {\ frac {| x- \ xi | ^ {2}} {4t}}} g (\ xi) \ mathrm {d} \ xi \ ,.}$

## Table of important Fourier transform pairs

This chapter contains a list of important Fourier transform pairs.

signal Fourier transformed
angular frequency
Fourier transformed
frequency
Hints
${\ displaystyle g (t)}$ ${\ displaystyle G (\ omega) = {\ frac {1} {\ sqrt {2 \ pi}}} \ int _ {- \ infty} ^ {\ infty} g (t) e ^ {- \ mathrm {i } \ omega t} \ mathrm {d} t}$ ${\ displaystyle G (f) = \ int _ {- \ infty} ^ {\ infty} g (t) e ^ {- \ mathrm {i} 2 \ pi ft} \ mathrm {d} t}$
${\ displaystyle g (ta)}$ ${\ displaystyle e ^ {- \ mathrm {i} a \ omega} G (\ omega)}$ ${\ displaystyle e ^ {- \ mathrm {i} 2 \ pi af} G (f)}$ time shift
${\ displaystyle e ^ {\ mathrm {i} at} g (t)}$ ${\ displaystyle G (\ omega -a)}$ ${\ displaystyle G \ left (f - {\ frac {a} {2 \ pi}} \ right)}$ Frequency shift
${\ displaystyle g (at)}$ ${\ displaystyle {\ frac {1} {| a |}} G \ left ({\ frac {\ omega} {a}} \ right)}$ ${\ displaystyle {\ frac {1} {| a |}} G \ left ({\ frac {f} {a}} \ right)}$ Frequency scaling
${\ displaystyle g ^ {(n)} (t)}$ ${\ displaystyle (\ mathrm {i} \ omega) ^ {n} G (\ omega)}$ ${\ displaystyle (\ mathrm {i} 2 \ pi f) ^ {n} G (f)}$ Here is a natural number and g is a Schwartz function . denotes the -th derivative of g. ${\ displaystyle n}$${\ displaystyle g ^ {(n)}}$${\ displaystyle n}$

### Functions that can be integrated in a square

signal Fourier transformed
angular frequency
Fourier transformed
frequency
Hints
${\ displaystyle g (t)}$ ${\ displaystyle G (\ omega) = {\ frac {1} {\ sqrt {2 \ pi}}} \ int _ {- \ infty} ^ {\ infty} g (t) e ^ {- \ mathrm {i } \ omega t} \ mathrm {d} t}$ ${\ displaystyle G (f) = \ int _ {- \ infty} ^ {\ infty} g (t) e ^ {- \ mathrm {i} 2 \ pi ft} \ mathrm {d} t}$
${\ displaystyle \ exp \ left (- {\ frac {at ^ {2}} {2}} \ right)}$ ${\ displaystyle {\ frac {1} {\ sqrt {a}}} \ cdot \ exp \ left (- {\ frac {\ omega ^ {2}} {2a}} \ right)}$ ${\ displaystyle {\ sqrt {\ frac {2 \ pi} {a}}} \ exp \ left (- {\ frac {2 \ pi} {a}} \ cdot \ pi f ^ {2} \ right)}$ The Gaussian function gives the same function again after a Fourier transform. Must be for integrability . ${\ displaystyle \ exp (-t ^ {2} / 2)}$${\ displaystyle \ mathrm {Re} (a)> 0}$
${\ displaystyle \ operatorname {rect} (at)}$ ${\ displaystyle {\ frac {1} {{\ sqrt {2 \ pi}} | a |}} \ cdot \ operatorname {sinc} \ left ({\ frac {\ omega} {2 \ pi a}} \ right )}$ ${\ displaystyle {\ frac {1} {| a |}} \ cdot \ operatorname {sinc} \ left ({\ frac {f} {a}} \ right)}$ The rectangle function and the sinc function ( ). ${\ displaystyle \ operatorname {sinc} (x) = \ sin (\ pi x) / (\ pi x)}$
${\ displaystyle \ operatorname {si} (at) \ equiv {\ frac {\ sin (at)} {at}}}$ ${\ displaystyle {\ frac {1} {| a |}} {\ sqrt {\ frac {\ pi} {2}}} \ cdot \ operatorname {rect} \ left ({\ frac {\ omega} {2a} } \ right)}$ ${\ displaystyle {\ frac {\ pi} {| a |}} \ cdot \ operatorname {rect} \ left ({\ frac {\ pi} {a}} f \ right)}$ The square wave function is an idealized low pass filter and the si function is the acausal impulse response of such a filter.
${\ displaystyle \ exp \ left (-a | t | \ right)}$ ${\ displaystyle {\ sqrt {\ frac {2} {\ pi}}} {\ frac {a} {\ omega ^ {2} + a ^ {2}}}}$ ${\ displaystyle {\ frac {2a} {(2 \ pi f) ^ {2} + a ^ {2}}}}$ ${\ displaystyle a> 0.}$The FT of the function falling exponentially around the origin is a Lorentz curve .
${\ displaystyle {\ frac {1} {t ^ {2} + a ^ {2}}}}$ ${\ displaystyle {\ sqrt {\ frac {\ pi} {2}}} {\ frac {1} {a}} \ exp \ left (-a | \ omega | \ right)}$ ${\ displaystyle {\ frac {\ pi} {a}} \ exp \ left (-2 \ pi a | f | \ right)}$

### Distributions

signal Fourier transformed
angular frequency
Fourier transformed
frequency
Hints
${\ displaystyle g (t)}$ ${\ displaystyle G (\ omega) = {\ frac {1} {\ sqrt {2 \ pi}}} \ int _ {- \ infty} ^ {\ infty} g (t) e ^ {- \ mathrm {i } \ omega t} \ mathrm {d} t}$ ${\ displaystyle G (f) = \ int _ {- \ infty} ^ {\ infty} g (t) e ^ {- \ mathrm {i} 2 \ pi ft} \ mathrm {d} t}$
${\ displaystyle e ^ {\ mathrm {i} at}}$ ${\ displaystyle {\ sqrt {2 \ pi}} \ cdot \ delta (\ omega -a)}$ ${\ displaystyle \ delta \ left (f - {\ frac {a} {2 \ pi}} \ right)}$
${\ displaystyle \ cos (at)}$ ${\ displaystyle {\ sqrt {2 \ pi}} {\ frac {\ delta (\ omega -a) + \ delta (\ omega + a)} {2}}}$ ${\ displaystyle {\ frac {\ delta (f - {\ frac {a} {2 \ pi}}) + \ delta (f + {\ frac {a} {2 \ pi}})} {2}}}$
${\ displaystyle \ sin (at)}$ ${\ displaystyle {\ sqrt {2 \ pi}} {\ frac {\ delta (\ omega -a) - \ delta (\ omega + a)} {2 \ mathrm {i}}}}$ ${\ displaystyle {\ frac {\ delta (f - {\ frac {a} {2 \ pi}}) - \ delta (f + {\ frac {a} {2 \ pi}})} {2 \ mathrm {i }}}}$
${\ displaystyle t ^ {n}}$ ${\ displaystyle \ mathrm {i} ^ {n} {\ sqrt {2 \ pi}} \ delta ^ {(n)} (\ omega)}$ ${\ displaystyle \ left ({\ frac {\ mathrm {i}} {2 \ pi}} \ right) ^ {n} \ delta ^ {(n)} (f)}$ Here is a natural number and the -th derivative of the delta distribution . ${\ displaystyle n}$${\ displaystyle \ delta ^ {(n)}}$${\ displaystyle n}$
${\ displaystyle {\ frac {1} {t ^ {n}}}}$ ${\ displaystyle - \ mathrm {i} {\ sqrt {\ frac {\ pi} {2}}} \ cdot {\ frac {(- \ mathrm {i} \ omega) ^ {n-1}} {(n -1)!}} \ Operatorname {sgn} (\ omega)}$ ${\ displaystyle - \ mathrm {i} \ pi {\ frac {(- \ mathrm {i} 2 \ pi f) ^ {n-1}} {(n-1)!}} \ operatorname {sgn} (f )}$
${\ displaystyle \ operatorname {sgn} (t)}$ ${\ displaystyle {\ sqrt {\ frac {2} {\ pi}}} \ cdot {\ frac {1} {\ mathrm {i} \ \ omega}}}$ ${\ displaystyle {\ frac {1} {\ mathrm {i} \ pi f}}}$
${\ displaystyle \ Theta (t)}$ ${\ displaystyle {\ sqrt {\ frac {\ pi} {2}}} \ left ({\ frac {1} {\ mathrm {i} \ pi \ omega}} + \ delta (\ omega) \ right)}$ ${\ displaystyle {\ frac {1} {2}} \ left ({\ frac {1} {\ mathrm {i} \ pi f}} + \ delta (f) \ right)}$ ${\ displaystyle \ Theta (t)}$is the unit jump ( Heaviside function ).
${\ displaystyle \ sum _ {n = - \ infty} ^ {\ infty} \ delta (t-nT)}$ ${\ displaystyle {\ frac {\ sqrt {2 \ pi}} {T}} \ sum _ {k = - \ infty} ^ {\ infty} \ delta \ left (\ omega -k {\ frac {2 \ pi } {T}} \ right)}$ ${\ displaystyle {\ frac {1} {T}} \ sum _ {k = - \ infty} ^ {\ infty} \ delta \ left (f - {\ frac {k} {T}} \ right)}$ The signal is called the Dirac comb .