Lecture2

# Lecture2 - optical comm ch2

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Review - Linear Systems Shift Invariant Systems y ( t ) = Z -∞ h ( t - τ ) x ( τ ) = x ( t ) ~ h ( t ) Output is convolution of input x ( t ) and impulse response function h ( t ) ECE2443b Lightwave Communictions Lecture 2 – p.1/26

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Fourier Transform Fourier transform X ( f ) = Z -∞ x ( t ) e - j 2 πft dt. e j 2 πft = cos(2 πft ) + j sin(2 πft ) Period T and frequency f f = 1 T = 1 2 π ( t ) dt where φ ( t ) = 2 πft = ωt. Inverse Fourier transform x ( t ) = Z -∞ X ( f ) e j 2 πft df. ECE2443b Lightwave Communictions Lecture 2 – p.2/26
Example -Rect and Sinc Functions Define rect function rect t T « = 8 < : 1 , ˛ ˛ t T ˛ ˛ < 1 2 0 , ˛ ˛ t T ˛ ˛ > 1 2 where T is the width of the pulse. Then X ( f ) = Z -∞ x ( t ) e - j 2 πft dt = AT sin( πfT ) πfT = AT sinc( fT ) where the sinc function is defined as sinc( x ) = sin( πx ) πx ECE2443b Lightwave Communictions Lecture 2 – p.3/26

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Symmetry Properties If x ( t ) is real, 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 is the magnitude φ ( f ) = arg X ( f ) = tan - 1 X i ( f ) X r ( f ) is the phase. ECE2443b Lightwave Communictions Lecture 2 – p.4/26
Complex Analytic Signals For x ( t ) real define new frequency function Z ( f ) Z ( f ) = 2 X ( f ) for f > 0 X (0) for f = 0 0 for f < 0 One sided frequency function No loss of information about x ( t ) Inverse transform of Z ( f ) is e z ( t ) and is called complex analytic signal ECE2443b Lightwave Communictions Lecture 2 – p.5/26

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Complex Analytic Signals-cont. Using X ( f ) = X * ( - f ) , e z ( t ) and x ( t ) are related by x ( t ) = 1 2 ( e z ( t ) + e z * ( t )) = Re { e z ( t ) } For example if x ( t ) = cos(2 πft ) , then e z ( t ) = e j 2 πft = cos(2 πft ) + j sin(2 πft ) . ECE2443b Lightwave Communictions Lecture 2 – p.6/26
Eigenfunctions (Modes) of Linear Systems Letting x ( t ) = e j 2 πft . Output y ( t ) is convolution y ( t ) = Z -∞ h ( τ ) e j 2 πf ( t - τ ) = e j 2 πft Z -∞ h ( τ ) e - j 2 πfτ = H ( f ) e j 2 πft where H ( f ) is the Fourier transform of h ( t ) .

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