Soln03 - ECE 342 Communication Theory Fall 2005 Solutions...

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Unformatted text preview: ECE 342 Communication Theory Fall 2005, Solutions to Homework 3 Prof. Tiffany J Li www: http://www.eecs.lehigh.edu/ ∼ jingli/teach email: [email protected] Problem 3.9 If we let x ( t ) =- Π t + T p 4 T p 2 + Π t- T p 4 T p 2 then using the results of Problem 2.23, we obtain v ( t ) = m ( t ) s ( t ) = m ( t ) ∞ X n =-∞ x ( t- nT p ) = m ( t ) 1 T p ∞ X n =-∞ X ( n T p ) e j 2 π n Tp t where X ( n T p ) = F - Π t + T p 4 T p 2 + Π t- T p 4 T p 2 f = n Tp = T p 2 sinc( f T p 2 ) e- j 2 πf Tp 4- e j 2 πf Tp 4 f = n Tp = T p 2 sinc( n 2 )(- 2 j ) sin( n π 2 ) Hence, the Fourier transform of v ( t ) is V ( f ) = 1 2 ∞ X n =-∞ sinc( n 2 )(- 2 j ) sin( n π 2 ) M ( f- n T p ) The bandpass filter will cut-off all the frequencies except the ones centered at 1 T p , that is for n = ± 1. Thus, the output spectrum is U ( f ) = sinc( 1 2 )(- j ) M ( f- 1 T p ) + sinc( 1 2 ) jM ( f + 1 T p ) =- 2 π jM ( f- 1 T p ) + 2 π jM ( f + 1 T p ) = 4 π M ( f ) ?...
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Soln03 - ECE 342 Communication Theory Fall 2005 Solutions...

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