HW1S - ECE 402: Communications Engineering Homework 1...

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ECE 402: Communications Engineering Homework 1 Solutions (Fall 2009) 1. The period is T = 1 / 2 so f 0 = 2. We can evaluate the FS using Poisson’s Sum Formula with central pulse g ( t ) = ( cos(2 πt ) , | t | < 1 / 4 0 , else = Π(2 t )cos(2 πt ) . By the modulation property G ( f ) = (1 / 4)[sinc(( f - 1) / 2) + sinc(( f + 1) / 2)]. From PSF, we then get x n = f 0 G ( nf 0 ) = (1 / 2)[sinc( n - 1 / 2) + sinc( n + 1 / 2)] . The FS is therefore x ( t ) = X n = -∞ x n e 2 πjnf 0 t = 2 π + 2 3 π [ e 4 πjt + e - 4 πjt ] - 2 15 π [ e 8 πjt + e - 8 πjt ] + ··· 2. Expanding the comb, we get Z -∞ Λ( t )comb 1 / 2 ( t ) dt = Z -∞ X n = -∞ Λ( n/ 2) δ ( t - n/ 2) dt = Z -∞ δ ( t ) + (1 / 2)[ δ ( t - 1 / 2) + δ ( t + 1 / 2)] dt = 1 + 1 / 2 + 1 / 2 = 2 3. From the sifting property of δ ( t ), we have x ( t ) = Λ( t ) X n = -∞ δ ( t - n/ 2) = X n = -∞ Λ( n/ 2) δ ( t - n/ 2) = δ ( t ) + (1 / 2)[ δ ( t - 1 / 2) + δ ( t + 1 / 2)] so the transform is F [ x ( t
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HW1S - ECE 402: Communications Engineering Homework 1...

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