Fourier Transform. Fourier Transform
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1 ECE 307 Z. Aliyziioglu Eleril & Compuer Engineering Dep. Cl Poly Pomon The Fourier rnsform (FT is he exension of he Fourier series o nonperiodi signls. The Fourier rnsform of signl exis if sisfies he following ondiion. x ( d< The Fourier rnsform X( ω = xe ( d The inverse Fourier rnsform (IFT of X(ω is x(nd given by jω x( = X( ω e dω
2 Also, The Fourier rnsform n be defined in erms of frequeny of Herz s j f X( f x( e d = nd orresponding inverse Fourier rnsform is x( X( f e j π f df = Exmple: Deermine he Fourier rnsform of rengulr pulse shown in he following figure x( -/ / h / ω ω j j j h ω X( ω = he d= e e jω / ω h ω sin( = sin( = h ω ω ω = hsin π
3 Exmple: To find in frequeny domin, / f f j j j f h π Xf ( = he d= e e j f / h sin( π f = sin( π f = h πf πf = hsin ( f h =, = ω X( ω = sin π >> h=; >> =; >> f=-3.5:0.0:3.5; >> w=*pi*f; >> x=h**sin(w*/(*pi; >> plo (w,x >> ile ('X(\omeg' >> xlbel('\omeg'; >> h =, = ω X( ω = sin π >> h=; >> =; >> f=-3.5:0.0:3.5; >> w=*pi*f; >> x=bs(h**sin(w*/(*pi; >> subplo (,, >> plo (w,x >> ile (' X(\omeg ' >> xlbel('\omeg' >> xp=phse(h**sin(w*/(*pi; >> subplo (,, >> plo (w,xp >> ile ('phse X(\omeg' >> xlbel('\omeg'
4 Exmple Deermine he Fourier rnsform of he Del funion δ( jω jω0 ( ω = δ( = = X e d e X(ω ω Properies of he We summrize severl imporn properies of he s follows.. Lineriy (Superposiion If x x ( X ( ω ( X( ω nd Then, x( + x ( X( ω + X ( ω jω jω jω ( ( ( ( [ + ] = + x x e d x e d x e d = X( ω + X ( ω
5 Properies of he. Time Shifing If x ( Xω ( Then, x ( X( ω e ω 0 j 0 = + d = dτ Le τ = 0 hen τ 0 nd jωτ ( + 0 x ( 0 e d= x( τ e d = 0 jωτ e x( e d = e 0 τ X( ω τ τ Le y ( = x ( 0 Y( ω = X( ω e = X( ω e e = X( ω e jω0 j X( ω jω0 j( X( ω ω0 j Y( ω j ( X( ω ω0 Y( ω e = X( ω e Therefore, he mpliude sperum of he ime shifed signl is he sme s he mpliude sperum of he originl signl, nd he phse sperum of he ime-shifed signl is he sum of he phse sperum of he originl signl nd liner phse erm.
6 Exmple: Deermine he Fourier rnsform of he following ime shifed rengulr pulse. x( h 0 ω j X( ω = hsin e ω π >> h=; >> =; >> f=-3.5:0.0:3.5; >> w=*pi*f; >> x=bs(h**sin(w*/(*pi.*exp(- j*w*/; >> subplo (,, >> plo (w,x >> ile (' X(\omeg ' >> xlbel('\omeg' >> xp=phse(h**sin(w*/(*pi.*exp(- j*w.*/; >> subplo (,, >> plo (w,xp >> xlbel('\omeg' >> ile ('phsex(\omeg' 3. Time Sling If x ( Xω ( ω x ( X( hen Le τ = hen τ / nd = d = (/ dτ If, >0 hen ω j τ x( e d = x( τ e dτ ω = X ( If, <0 hen ω j τ xe ( d= x( τ e dτ ω j τ x( e ω = τ dτ X( =
7 Exmple. if, x ( Xω ( hen find he Fourier rnsform of he following signls. b.. x( X( ω x (/5 5 X(5 ω ω x( 5( X( e ω 5 5 j Exmple: Find he Fourier rnsform of he following signl.. b.. ω x( = ( X( ω = sin π x( (5 X( X( ω sin ω = ω = = π ω x3( = ( /5 X3( ω = 5 X(5 ω = 5sin 0.4 π 4. Duliy (Symmery If x ( Xω ( hen X ( π x( ω or X( x( f Sine nd ω re rbirry vribles in he inverse Fourier rnsform jω x( = X( ω e dω we n reple ω wih nd wih - ω o ge x( ω = X( e d Therefore, F{ } X ( = Xe ( d= π x( ω
8 Similrly, if we n reple f wih nd wih -f in he inverse Fourier rnsform x( X( f e j πf df = o ge j f x( f X( e df = Therefore, { } F X ( = x( f Exmple: x ( = δ ( X( ω = Applying symmery propery, x ( = Xω ( = πδ ( ω = πδ ( ω ( δ ( ω is even funion or x ( = Xf ( = δ ( f = δ ( f Exmple: ω x ( = re X( ω sin = π ω ω x ( = sin X( ω re re = = Le = hen = ω ω x( = sin ( X( ω = re re =
9 Time Reversl If x ( Xω ( hen x( X( ω Le = τ. Then = τ nd d = dτ jω j( ω τ ( = ( τ τ = ( ω x e d x e d X Frequeny Shifing If x ( Xω ( hen jω xe ( X( ω ω ( ( j ω j ω ( j ω ω = = ( ω ω xe e d xe d X
10 Exmple: Deermine he Fourier rnsform of osω nd sinω ( os j ω j ω x = ω = e + e X( ω = π δ( ω ω + δ( ω + ω or [ ] x( os j j e ω ω = ω = + e X( f = δ( f f + δ( f + f X(f / [ ] -f The phse sperum is zero everywhere. f f ( sin j ω j ω x = ω = e e X( ω = jπ δ( ω ω δ( ω + ω j j [ ] ( sin j ω j ω j x ω e e = = X( f = δ( f f δ( f + f j j X(f / [ ] -f θ(f π/ f f f -f f -π/
11 7. Modulion If x ( Xω ( hen x ( os( ω X( ω ω + X( ω + ω [ ] jω jω jω jω x (os( ωe d= x ( e + e e d = + = + + j( ω ω j( ω+ ω xe ( d xe ( d [ X( ω ω X( ω ω ] 8. Time Differeniion: If x ( Xω ( hen dx( jωx( ω d Generl se n d x( n ( jω X( ω n d we obin Therefore Tking he derivive of he inverse Fourier rnsform jω x( = X( ω e dω dx( jω = jωx( ω e dω d dx( jωx( ω d
12 9. Time Differeniion: If x ( Xω ( hen Generl se x( dx( jω j dω n x( j n n d X( ω n dω Tking derivive of X( ω = xe ( d wih respe o ω, we obin dx( ω = ( j xe ( dω dx( jω Therefore x( j dω d 0 Conjuge If x ( Xω ( hen * * x ( X ( ω If x( is rel * * jω j( ω * x ( e d = x( e d = X ( ω x * ( x( * = X ω X so h ( = ( ω
13 . Convoluion If x ( Xω (, h ( Hω (, nd y ( Yω ( y ( = h (* x ( = h( τ x ( τ dτ Y( ω = h( τ x( τ dτ e d Inerhnging he order of inegrion, we obin Y( ω = H( ω X( ω Y( ω = h( τ x( τ e d dτ jωτ jωτ Y( ω = h( τ X( ω e dτ = X( ω h( τ e dτ = X( ω H( ω. Mulipliion If x ( X ( ω, nd x ( X ( ω x x( X( ω* X( ω = X( vx ( ω vdv or x x( X(* f X( f = X( vx ( f vdv
14 3. Prsevl s Theorem If x( X( ω, hen ol normlized(bsed on one ohms resisor energy E of nd x( is given by Proof E = x( d = X( ω dω = X( f df x( d = x( x ( d = x( X ( ω e dω d * * Inerhnging he order of inegrion, we obin Proof (on x( d = X ( ω x( e d dω * = = * X X d ( ω ( ω ω X( ω dω
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