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Soln02 - ECE 342 Communication Theory Fall 2005 Solutions...

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Unformatted text preview: ECE 342 Communication Theory Fall 2005, Solutions to Homework 2 Prof. Tiffany J Li www: http://www.eecs.lehigh.edu/ ∼ jingli/teach email: [email protected] Problem 2.10 1) Using the Fourier transform pair e- α | t | F-→ 2 α α 2 + (2 πf ) 2 = 2 α 4 π 2 1 α 2 4 π 2 + f 2 and the duality property of the Fourier transform: X ( f ) = F [ x ( t )] ⇒ x (- f ) = F [ X ( t )] we obtain 2 α 4 π 2 F " 1 α 2 4 π 2 + t 2 # = e- α | f | With α = 2 π we get the desired result F 1 1 + t 2 = πe- 2 π | f | 2) F [ x ( t )] = F [Π( t- 3) + Π( t + 3)] = sinc( f ) e- j 2 πf 3 + sinc( f ) e j 2 πf 3 = 2sinc( f ) cos(2 π 3 f ) 3) F [ x ( t )] = F [Λ(2 t + 3) + Λ(3 t- 2)] = F [Λ(2( t + 3 2 )) + Λ(3( t- 2 3 )] = 1 2 sinc 2 ( f 2 ) e jπf 3 + 1 3 sinc 2 ( f 3 ) e- j 2 πf 2 3 4) T ( f ) = F [sinc 3 ( t )] = F [sinc 2 ( t )sinc( t )] = Λ( f ) ? Π( f ). But Π( f ) ? Λ( f ) = Z ∞-∞ Π( θ )Λ( f- θ ) dθ = Z 1 2- 1 2 Λ( f- θ ) dθ = Z f + 1 2 f- 1 2 Λ( v ) dv 1 For f ≤ -...
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Soln02 - ECE 342 Communication Theory Fall 2005 Solutions...

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