Fourier transforms

 


Fourier transform pairs

x(t) Δt=1/ Δf x( t )X( f )Δf=1/Δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iDaiabg2da9maalyaabaGaaGymaaqaaiabfs5aejaadAgaaaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaamiEamaabmaabaGaamiDaaGaayjkaiaawMcaaiaaykW7caaMc8Ua eSigI8MaeyOeI0IaeyOeI0IaeyOeI0IaeyOeI0IaeyOiGCRaaGPaVl aaykW7caWGybWaaeWaaeaacaWGMbaacaGLOaGaayzkaaGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeu iLdqKaamOzaiabg2da9iaaigdacaGGVaGaeuiLdqKaamiDaaaa@72D6@ X(f)
1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@36B1@
DC component
Spectral line at frequency f=0
δ( f ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH0oazpaWaaeWaaeaacaWGMbaacaGLOaGaayzkaaaaaa@3A39@
δ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH0oazpaWaaeWaaeaapeGaamiDaaWdaiaawIcacaGLPaaaaaa@3A66@
Time signal - dirac delta impulse
Constant spectrum
1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaaGymaaaa@36AC@
cos(2π f 0 t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaaiikaiaackdacqaHapaCcaaMc8UaamOzamaaBaaaleaa caGGWaaabeaakiaacshacaGGPaaaaa@40E8@
Time signal - cosine
Spectral lines
1 2 [δ(f f 0 )+δ(f+ f 0 )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaaaacaGGBbaeaaaaaaaaa8qacqaH0oazcaGGOaGa amOzaiabgkHiTiaadAgadaWgaaWcbaGaaGimaaqabaGccaGGPaGaey 4kaSIaeqiTdqMaaiikaiaadAgacqGHRaWkcaWGMbWaaSbaaSqaaiaa icdaaeqaaOGaaiykaiaac2faaaa@4791@
1 2 [δ(t t 0 )+δ(t+ t 0 )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaaaacaGGBbaeaaaaaaaaa8qacqaH0oazcaGGOaGa amiDaiabgkHiTiaadshadaWgaaWcbaGaaGimaaqabaGccaGGPaGaey 4kaSIaeqiTdqMaaiikaiaadshacqGHRaWkcaWG0bWaaSbaaSqaaiaa icdaaeqaaOGaaiykaiaac2faaaa@47C9@
Time shifted diracs
Cosine spectrum
cos(2π t 0 f) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbiqaaWvgciGGJb Gaai4BaiaacohacaGGOaGaaGOmaiabec8aWjaaykW7caWG0bWaaSba aSqaaiaaicdaaeqaaOGaaiOzaiaacMcaaaa@416C@
1für Δt 2 <t< Δt 2 0sonst }=( t Δt )
Time signal ? rectangular pulse
Sinc frequency spectrum
Δt sin(πΔtf) πΔtf =Δtsi(πΔtf) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeuiLdqKaam iDaiaaykW7cqGHflY1caaMc8+aaSaaaeaaciGGZbGaaiyAaiaac6ga caGGOaGaeqiWdaNaaGPaVlabfs5aejaadshacaaMc8UaamOzaiaacM caaeaacqaHapaCcaaMc8UaeuiLdqKaamiDaiaaykW7caWGMbaaaiaa ykW7caaMc8Uaeyypa0JaaGPaVlaaykW7cqqHuoarcaWG0bGaaGPaVl aaykW7caWGZbGaamyAaiaacIcacqaHapaCcaaMc8UaeuiLdqKaamiD aiaaykW7caWGMbGaaiykaaaa@6A2E@
Δf sin(πΔft) πΔft MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeuiLdqKaam OzaiaaykW7cqGHflY1caaMc8+aaSaaaeaaciGGZbGaaiyAaiaac6ga caGGOaGaeqiWdaNaaGPaVlabfs5aejaadAgacaaMc8UaamiDaiaacM caaeaacqaHapaCcaaMc8UaeuiLdqKaamOzaiaaykW7caWG0baaaiaa ykW7caaMc8oaaa@5533@
Sinc-shaped pulse
Rectangular spectrum
1für Δf 2 <f< Δf 2 0sonst }=( f Δf )
t Δt +1fürΔtt0 t Δt +1für0tΔt 0sonst }=Λ( t Δt )
Triangular pulse
sinc^2 spectrum
Δt [ sin(πΔtf) πΔtf ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeuiLdqKaam iDamaadmaabaWaaSaaaeaaciGGZbGaaiyAaiaac6gacaGGOaGaeqiW daNaaGPaVlabfs5aejaadshacaaMc8UaamOzaiaacMcaaeaacqaHap aCcaaMc8UaeuiLdqKaamiDaiaaykW7caWGMbaaaaGaay5waiaaw2fa amaaCaaaleqabaGaaGOmaaaaaaa@4FA6@
Δf [ sin(πΔft) πΔft ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeuiLdqKaam OzamaadmaabaWaaSaaaeaaciGGZbGaaiyAaiaac6gacaGGOaGaeqiW daNaaGPaVlabfs5aejaadAgacaaMc8UaamiDaiaacMcaaeaacqaHap aCcaaMc8UaeuiLdqKaamOzaiaaykW7caWG0baaaaGaay5waiaaw2fa amaaCaaaleqabaGaaGOmaaaaaaa@4F98@
Time signal ? sinc
Triangular spectrum
f Δf +1fürΔff0 f Δf +1für0fΔf 0sonst }=Λ( f Δf )
e | 2t Δt | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaeyOeI0YaaqWaaeaadaWcaaqaaiaaikdacaWG0baabaGa euiLdqKaamiDaaaaaiaawEa7caGLiWoaaaaaaa@3F3B@
Exponential pulse
1 / (1+f) spectrum
Δt 1 1+ (πΔtf) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeuiLdqKaam iDaiabgwSixpaalaaabaGaaGymaaqaaiaaigdacqGHRaWkcaGGOaGa eqiWdaNaaGPaVlabfs5aejaadshacaaMc8UaamOzaiaacMcadaahaa Wcbeqaaiaaikdaaaaaaaaa@4761@
Δf 1 1+ (πΔft) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeuiLdqKaam OzaiabgwSixpaalaaabaGaaGymaaqaaiaaigdacqGHRaWkcaGGOaGa eqiWdaNaaGPaVlabfs5aejaadAgacaaMc8UaamiDaiaacMcadaahaa Wcbeqaaiaaikdaaaaaaaaa@4753@
1 / (1+t) pulse
Exponential spectrum
e | 2f Δf | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaaGPaVlaadw gadaahaaWcbeqaaiabgkHiTmaaemaabaWaaSaaaeaacaaIYaGaamOz aaqaaiabfs5aejaadAgaaaaacaGLhWUaayjcSdaaaaaa@40AA@
e π ( t Δt ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaeyOeI0IaeqiWda3aaeWaaeaadaWcaaqaaiaadshaaeaa cqqHuoarcaWG0baaaaGaayjkaiaawMcaamaaCaaameqabaGaaGOmaa aaaaaaaa@3F8D@
Gaussian pulse
Gaussian spectrum
Δt e π ( f Δf ) 2 ;Δt= 1 Δf MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeuiLdqKaam iDaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiabec8aWnaabmaa baWaaSaaaeaacaWGMbaabaGaeuiLdqKaamOzaaaaaiaawIcacaGLPa aadaahaaadbeqaaiaaikdaaaaaaOGaaGPaVlaacUdacaaMc8UaaGPa VlaaykW7cqqHuoarcaGG0bGaeyypa0ZaaSaaaeaacaaIXaaabaGaeu iLdqKaamOzaaaaaaa@518F@
n= δ(tn T 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaaabCaeaacq aH0oazcaGGOaGaamiDaiabgkHiTiaad6gacaWGubWaaSbaaSqaaiaa icdaaeqaaaqaaiaad6gacqGH9aqpcqGHsislcqGHEisPaeaacqGHEi sPa0GaeyyeIuoakiaacMcaaaa@4590@
Dirac series time
Dirac series spectrum
f 0 n= δ(fn f 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIWaaabeaakiabgwSixpaaqahabaGaeqiTdqMaaiikaiaa cAgacqGHsislcaWGUbGaamOzamaaBaaaleaacaaIWaaabeaaaeaaca WGUbGaeyypa0JaeyOeI0IaeyOhIukabaGaeyOhIukaniabggHiLdGc caGGPaaaaa@49B8@
Table of Fourier transform pairs

This experiment illustrates the Fourier transform. Demo examples of time signals and corresponding spectra are simulated. Click the play button to open an oscilloscope and spectrum analyzer. Vary the time signal (amplitude, frequency or pulse witdth) and watch the impact on the spectrum!

Time domainFrequency domain Start
DC component
DC component
Spectral line at frequency f=0
Spectral line at frequency f=0
Start
Time signal - dirac delta impulse
Dirac delta impulse
Constant spectrum
Constant spectrum
Start
Time signal ? cosine
Cosine
Spectral lines
Spectral lines
Start
Time signal ? cosine with frequency fmax
Rectangular pulse
Spectral line at frequency fmax
Sinc Function
Start
Time signal ? sinc
Sinc pulse
Rectangular spectrum (approximation)
Rectangular spectrum (approximation)
Start
Time signal ? sinc
Sinc^2 pulse
Triangle spectrum (approximation)
Triangle spectrum (approximation)
Start
Exponential pulse
Exponential pulse
1 / (1+f) spectrum
1 / (1+f) spectrum
Start
Gaussian pulse
Gaussian pulse
Gaussian spectrum
Gaussian spectrum
Start
Example demonstrations of time signals and corresponding frequency signals.


1. Fourier transforms. Time signals are shown on the left. Assign the corresponding spectra on the right per drag & drop!

Time domainFequency domain