Fourierova transformacija

Naj bo ${\displaystyle f\in L^{1}(\mathbb {R} ^{n})}$ integrabilna funkcija, kjer ${\displaystyle L^{1}}$ imenovan Lebesgueov prostor. (Zvezna) Fourierova transformacija ${\displaystyle {\mathcal {F}}f}$ od ${\displaystyle f}$ je definirana z

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

in ustrezna inverzna transformacija je:

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

Pri tem sta ${\displaystyle \mathrm {d} x}$ in ${\displaystyle \mathrm {d} y}$ so ${\displaystyle n}$ -dimenzionalna prostorska elementa, ${\displaystyle \mathrm {i} }$ imaginarna enota, ${\displaystyle y\cdot x}$ pa standardni produkt vektorjev ${\displaystyle y}$ in ${\displaystyle x}$ .

Normalizacijska konstanta v literaturi ni dosledna. V teoriji psevdodiferencialnih operatorjev in pri obdelavi signalov se faktor ${\displaystyle 1/(2\pi )^{n/2}}$ v transformaciji pogosto izpušča, tako da inverzna transformacija ustrezno dobi na začetek faktor ${\displaystyle 1/(2\pi )^{n}}$ . Transformacija je tako:

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

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

Faktor na začetku faktor preprečuje neposredno uporabo Plancherelovega izreka, ker potem Fourierova transformacija ne daje več unitarne preslikave ${\displaystyle L^{1}(\mathbb {R} ^{n})\cap L^{2}(\mathbb {R} ^{n})}$ in tako se spremeni moč signala. Vendar, kot pri vseh ortogonalnih transformacijah, je to mogoče zlahka kompenzirati s substitucijo (ponovno skaliranje abscise) in zato ne predstavlja temeljne težave. Točno to predlaga literatura o obdelavi signalov in sistemski teoriji: prehod od naravne frekvence h kotni frekvenci ${\displaystyle \omega =2\pi y}$ (ki vključuje faktor):

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

Realna inačica Fourierove transformacije se imenuje Hartleyjeva transformacija. Za realne funkcije ${\displaystyle f}$ Fourierovo transformacijo lahko nadomestimo s sinusno in kosinusno transformacijo.

Stiskanje podatkov za digitalno komunikacijo

Stiskanje digitalnih podatkov s pomočjo Fourierove transformacije je osrednja tehnologija na področju komunikacije, izmenjave podatkov in pretakanja medijev na (mobilnem) internetu. ^[1]

Tabele pomembnih Fourierovih transformacij

Naslednje tabele prikazujejo nekaj zaprtih Fourierovih transformacij. Funkciji f(x) in g(x) označimo njuni Fourierovi transformaciji s f̂ in ĝ. Vključene so samo tri najpogostejše konvencije. Morda bi bilo koristno opaziti, da podaja vnos 105 razmerje med Fourierovo transformacijo funkcije in izvirno funkcijo, ki jo je mogoče razumeti kot povezavo Fourierjeve transformacije in njenega obrata.

Funkcionalne relacije, enodimenzionalne

Fourierove transformacije v tej tabeli lahko najdete pri Erdélyi (1954) or Kammler (2000, appendix).

	Funkcija	Fourierova transformacija asbscisa je unitarna običajna frekvenca	Fourierova transformacija unitarna, kotna frekvenca	Fourierova transformacija< ne-unitarna, kotna frekvenca	pripombe
	${\displaystyle f(x)\,}$	${\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}$	${\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}$	${\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}$	Definiticije
101	${\displaystyle a\,f(x)+b\,g(x)\,}$	${\displaystyle a\,{\hat {f}}(\xi )+b\,{\hat {g}}(\xi )\,}$	${\displaystyle a\,{\hat {f}}(\omega )+b\,{\hat {g}}(\omega )\,}$	${\displaystyle a\,{\hat {f}}(\omega )+b\,{\hat {g}}(\omega )\,}$	Linearnost
102	${\displaystyle f(x-a)\,}$	${\displaystyle e^{-i2\pi \xi a}{\hat {f}}(\xi )\,}$	${\displaystyle e^{-ia\omega }{\hat {f}}(\omega )\,}$	${\displaystyle e^{-ia\omega }{\hat {f}}(\omega )\,}$	Pomik v časovni domeni
103	${\displaystyle f(x)e^{iax}\,}$	${\displaystyle {\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)\,}$	${\displaystyle {\hat {f}}(\omega -a)\,}$	${\displaystyle {\hat {f}}(\omega -a)\,}$	Pomik v frekvenčni domeni, dualno vrstici 102
104	${\displaystyle f(ax)\,}$	${\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)\,}$	${\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}$	${\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}$	Skalirano v časovni domeni. Če je ${\displaystyle {|a|}}$ velik, se ${\displaystyle f(ax)\,}$ koncentrira na 0 in ${\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}$ razširi in zniža.
105	${\displaystyle {\hat {f_{n}}}(x)\,}$	${\displaystyle {\hat {f}}_{1}(x)\ {\stackrel {{\mathcal {F}}_{1}}{\longleftrightarrow }}\ f(-\xi )\,}$	${\displaystyle {\hat {f}}_{2}(x)\ {\stackrel {{\mathcal {F}}_{2}}{\longleftrightarrow }}\ f(-\omega )\,}$	${\displaystyle {\hat {f}}_{3}(x)\ {\stackrel {{\mathcal {F}}_{3}}{\longleftrightarrow }}\ 2\pi f(-\omega )\,}$	Ista transformacija dvakrat zapored, x po prvi transformaciji nastopi namesto frekvence (ξ or ω).
106	${\displaystyle {\frac {d^{n}f(x)}{dx^{n}}}\,}$	${\displaystyle (i2\pi \xi )^{n}{\hat {f}}(\xi )\,}$	${\displaystyle (i\omega )^{n}{\hat {f}}(\omega )\,}$	${\displaystyle (i\omega )^{n}{\hat {f}}(\omega )\,}$	Odvod n-tega reda. f je Schwartzova funkcija
	${\displaystyle \int _{-\infty }^{x}f(\tau )d\tau }$	${\displaystyle {\frac {{\hat {f}}(\xi )}{i2\pi \xi }}+C\,\delta (\xi )}$			Integracija.[2] Opomba: ${\displaystyle \delta }$ je Diracova delta funkcija in ${\displaystyle C}$ povprečna (DC) vrednost ${\displaystyle f(x)}$ , tako da je ${\displaystyle \int _{-\infty }^{\infty }(f(x)-C)\,dx=0}$
107	${\displaystyle x^{n}f(x)\,}$	${\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}{\hat {f}}(\xi )}{d\xi ^{n}}}\,}$	${\displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\omega )}{d\omega ^{n}}}}$	${\displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\omega )}{d\omega ^{n}}}}$	To je dualno vrstici 106
108	${\displaystyle (f*g)(x)\,}$	${\displaystyle {\hat {f}}(\xi ){\hat {g}}(\xi )\,}$	${\displaystyle {\sqrt {2\pi }}{\hat {f}}(\omega ){\hat {g}}(\omega )\,}$	${\displaystyle {\hat {f}}(\omega ){\hat {g}}(\omega )\,}$	Izraz f ∗ g pomeni konvolucijo f in g — to pravilo je t.i. konvolucijski teorem
109	${\displaystyle f(x)g(x)\,}$	${\displaystyle \left({\hat {f}}*{\hat {g}}\right)(\xi )\,}$	${\displaystyle {\frac {1}{\sqrt {2\pi }}}\left({\hat {f}}*{\hat {g}}\right)(\omega )\,}$	${\displaystyle {\frac {1}{2\pi }}\left({\hat {f}}*{\hat {g}}\right)(\omega )\,}$	To je dualno 108
110	For f(x) purely real	${\displaystyle {\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}}\,}$	${\displaystyle {\hat {f}}(-\omega )={\overline {{\hat {f}}(\omega )}}\,}$	${\displaystyle {\hat {f}}(-\omega )={\overline {{\hat {f}}(\omega )}}\,}$	Hermitska simetrija z zahteva kompleksni konjugat.
113	Za imaginarno f(x)	${\displaystyle {\hat {f}}(-\xi )=-{\overline {{\hat {f}}(\xi )}}\,}$	${\displaystyle {\hat {f}}(-\omega )=-{\overline {{\hat {f}}(\omega )}}\,}$	${\displaystyle {\hat {f}}(-\omega )=-{\overline {{\hat {f}}(\omega )}}\,}$	z pomeni kompleksni konjugat.
114	${\displaystyle {\overline {f(x)}}}$	${\displaystyle {\overline {{\hat {f}}(-\xi )}}}$	${\displaystyle {\overline {{\hat {f}}(-\omega )}}}$	${\displaystyle {\overline {{\hat {f}}(-\omega )}}}$	[[<kompleksna konjugacija]], generalizacija 110 in 113
115	${\displaystyle f(x)\cos(ax)}$	${\displaystyle {\frac {{\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)+{\hat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2}}}$	${\displaystyle {\frac {{\hat {f}}(\omega -a)+{\hat {f}}(\omega +a)}{2}}\,}$	${\displaystyle {\frac {{\hat {f}}(\omega -a)+{\hat {f}}(\omega +a)}{2}}}$	Slledi iz 101 in 103 s pomočjo Eulerjeve formule: ${\displaystyle \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.}$
116	${\displaystyle f(x)\sin(ax)}$	${\displaystyle {\frac {{\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)-{\hat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}$	${\displaystyle {\frac {{\hat {f}}(\omega -a)-{\hat {f}}(\omega +a)}{2i}}}$	${\displaystyle {\frac {{\hat {f}}(\omega -a)-{\hat {f}}(\omega +a)}{2i}}}$	Sledi iz 101 in 103 s pomočjo Eulerjeve formule: ${\displaystyle \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.}$

kvadratno integrabilne funkcije, enodimenzionalne

Fourierove transformacije v tej tabeli je najti v Campbell & Foster (1948), Erdélyi (1954), ali Kammler (2000, priloga).

	Funkcija	Fourierova transformacija asbscisa je unitarna običajna frekvenca	Fourierova transformacija unitarna, kotna frekvenca	Fourierova transformacija< ne-unitarna, kotna frekvenca	pripombe
	${\displaystyle f(x)\,}$	${\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}$	${\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}$	${\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}$	Definicije
201	${\displaystyle \operatorname {rect} (ax)\,}$	${\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\xi }{a}}\right)}$	${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}$	${\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}$	Pravokotni pulz in normalizirana sinc funkcija, definirana kot sinc(x) = sin(πx)/πx
202	${\displaystyle \operatorname {sinc} (ax)\,}$	${\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\xi }{a}}\right)\,}$	${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}$	${\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}$	Dualno vrstici 201.
203	${\displaystyle \operatorname {sinc} ^{2}(ax)}$	${\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\xi }{a}}\right)}$	${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}$	${\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}$	tri(x) je trikotna funkcija
204	${\displaystyle \operatorname {tri} (ax)}$	${\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,}$	${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}$	${\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}$	Dualno vrstici 203.
205	${\displaystyle e^{-ax}u(x)\,}$	${\displaystyle {\frac {1}{a+i2\pi \xi }}}$	${\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}}$	${\displaystyle {\frac {1}{a+i\omega }}}$	Funkcija u(x) je Heavisidov prag in a > 0.
206	${\displaystyle e^{-\alpha x^{2}}\,}$	${\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}$	${\displaystyle {\frac {1}{\sqrt {2\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}$	${\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}$	Za unitarne Fourierove tranasformacije je torej e−αx2 svoja lastna transformacija - do multiplikatorja α. Da to velja, mora biti Re(α) > 0.
208	${\displaystyle e^{-a|x|}\,}$	${\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}\xi ^{2}}}}$	${\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {a}{a^{2}+\omega ^{2}}}}$	${\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}}$	Za Re(a) > 0. Fourierova transformacija eksponentne padajoče funkcije je Lorentzova funkcija .
209	${\displaystyle \operatorname {sech} (ax)\,}$	${\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}\xi \right)}$	${\displaystyle {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}$	${\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}$	Hyperbolična funkcija si je lastna Fourtierova transformacija
210	${\displaystyle e^{-{\frac {a^{2}x^{2}}{2}}}H_{n}(ax)\,}$	${\displaystyle {\frac {{\sqrt {2\pi }}(-i)^{n}}{a}}e^{-{\frac {2\pi ^{2}\xi ^{2}}{a^{2}}}}H_{n}\left({\frac {2\pi \xi }{a}}\right)}$	${\displaystyle {\frac {(-i)^{n}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}$	${\displaystyle {\frac {(-i)^{n}{\sqrt {2\pi }}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}$	Hn je Hermiteov polinom ntega reda. Če velja a = 1, so Gauss-Hermiteove funkcije lastne funkcije Fourierovega operatorja. Enačba se poenostavi na 206 za n = 0.

Dvodimenzionalne funkcije

	Function	Fourier transform unitary, ordinary frequency	Fourier transform unitary, angular frequency	Fourier transform non-unitary, angular frequency	Remarks
400	${\displaystyle f(x,y)}$	${\displaystyle {\begin{aligned}&{\hat {f}}(\xi _{x},\xi _{y})\triangleq \\&\iint f(x,y)e^{-i2\pi (\xi _{x}x+\xi _{y}y)}\,dx\,dy\end{aligned}}}$	${\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&{\frac {1}{2\pi }}\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}$	${\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}$	Spremenljivke ξx, ξy, ωx, ωy so realna števila. Področje integrala je celotna ravnina.
401	${\displaystyle e^{-\pi \left(a^{2}x^{2}+b^{2}y^{2}\right)}}$	${\displaystyle {\frac {1}{|ab|}}e^{-\pi \left({\frac {\xi _{x}^{2}}{a^{2}}}+{\frac {\xi _{y}^{2}}{b^{2}}}\right)}}$	${\displaystyle {\frac {1}{2\pi \,|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}$	${\displaystyle {\frac {1}{|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}$	Obe funkciji sta gaussovi, lahko da brez prostornine = 1
402	${\displaystyle \operatorname {circ} \left({\sqrt {x^{2}+y^{2}}}\right)}$	${\displaystyle {\frac {J_{1}\left(2\pi {\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}\right)}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}$	${\displaystyle {\frac {J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}$	${\displaystyle {\frac {2\pi J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}$	Funkcija je definirana circ(r) = 1 za 0 ≤ r ≤ 1, sicer pa = 0. Rezultat je porazdelitev amplitud za Airyjev disk

• Diskretna Fourierjeva transformacija
• Fourierjeva transformacija za signale z diskretnim časom
• Hitra Fourierjeva transformacija
• Inverzna hitra Fourierjeva transformacija

• Weisstein, Eric Wolfgang. »Fourier Transform«. MathWorld.
• Was ist eine Fourier-Transformation? Eine visuelle Einführung na YouTubu