HW2S - ECE 402: Communications Engineering Homework 2...

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Unformatted text preview: ECE 402: Communications Engineering Homework 2 Solutions (Fall 2009) 1. From the transform table, X ( f ) = 1 2+2 jf and so E x ( f ) = | X ( f ) | 2 = 1 4+4 2 f 2 . The autocorre- lation is therefore R x ( ) = F- 1 [ E x ( f )] = (1 / 4) e- 2 | t | . Using the hint, we can find the energy in the band- 1 / < f < 1 / by Z 1 /- 1 / E x ( f ) df = Z 1 /- 1 / df 4 + 4 2 f 2 = 1 4 2 Z 1 /- 1 / df a 2 + f 2 = 1 4 2 a (tan- 1 (1)- tan- 1 (- 1)) = 1 / 8 . where a = 1 / . 2. For periodic signals, to find the PSD we first need to find the Fourier series coefficients. Note the signal is of the form n =- g ( t- nT ) where g ( t ) = sinc 2 (2 t ) and T = 2. Thus f = 1 / 2 and G ( f ) = (1 / 2)( f/ 2). From Poissons Sum Formula, we then obtain x n = f G ( nf ) = (1 / 4)( n/ 4). The PSD is therefore P x ( f ) = X n =- | x n | 2 ( f- nf ) = (1 / 16) ( f ) + (9 / 256)[ ( f- 1 / 2) + ( f + 1 / 2)] +(4 / 256)[ ( f- 1) + ( f + 1)] + (1...
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HW2S - ECE 402: Communications Engineering Homework 2...

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