Lecture2

Lecture2 - Review-LinearSystems y t)= Z ∞-∞ h t τ x τ...

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Unformatted text preview: Review-LinearSystems ShiftInvariantSystems y ( t )= Z ∞-∞ h ( t- τ ) x ( τ ) dτ = x ( t ) ~ h ( t ) Outputisconvolutionofinput x ( t ) and impulse responsefunction h ( t ) ECE2443bLightwaveCommunictionsLecture2–p.1/26 FourierTransform Fouriertransform X ( f )= Z ∞-∞ x ( t ) e- j 2 πft dt. e j 2 πft =cos(2 πft )+ j sin(2 πft ) Period T andfrequency f f = 1 T = 1 2 π dφ ( t ) dt where φ ( t )=2 πft = ωt. InverseFouriertransform x ( t )= Z ∞-∞ X ( f ) e j 2 πft df. ECE2443bLightwaveCommunictionsLecture2–p.2/26 Example-RectandSincFunctions Define rectfunction rect „ t T « = 8< : 1 , ˛ ˛ t T ˛ ˛ < 1 2 , ˛ ˛ t T ˛ ˛ > 1 2 where T isthewidthofthepulse.Then X ( f ) = Z ∞-∞ x ( t ) e- j 2 πft dt = AT sin( πfT ) πfT = AT sinc( fT ) wherethe sincfunction isdefinedas sinc( x )= sin( πx ) πx ECE2443bLightwaveCommunictionsLecture2–p.3/26 SymmetryProperties If x ( t ) isreal,then X ( f )= X r ( f )+ jX i ( f ) and X ( f )= X * (- f ) X r ( f )= X r (- f ) X i ( f )=- X i (- f ) | X ( f ) | = | X (- f ) | φ ( f )=- φ (- f ) where | X ( f ) | = p X r ( f ) 2 + X i ( f ) 2 isthemagnitude φ ( f )=arg X ( f )=tan- 1 X i ( f ) X r ( f ) isthephase. ECE2443bLightwaveCommunictionsLecture2–p.4/26 ComplexAnalyticSignals For x ( t ) realdefinenewfrequencyfunction Z ( f ) Z ( f )= 2 X ( f )for f> X (0)for f =0 for f< Onesidedfrequencyfunction Nolossofinformationabout x ( t ) Inversetransformof Z ( f ) is e z ( t ) andiscalled complex analyticsignal ECE2443bLightwaveCommunictionsLecture2–p.5/26 ComplexAnalyticSignals-cont. Using X ( f )= X * (- f ) , e z ( t ) and x ( t ) arerelatedby x ( t )= 1 2 ( e z ( t )+ e z * ( t )) = Re { e z ( t ) } Forexampleif x ( t )=cos(2 πft ) ,then e z ( t )= e j 2 πft =cos(2 πft )+ j sin(2 πft ) . ECE2443bLightwaveCommunictionsLecture2–p.6/26 Eigenfunctions(Modes)ofLinearSystems Letting x ( t )= e j 2 πft...
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Lecture2 - Review-LinearSystems y t)= Z ∞-∞ h t τ x τ...

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