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HW3Solutions - ECEN 661 Modulation Theory Homework#3...

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ECEN 661 - Modulation Theory Homework #3 Solutions 1. (a) Since the shifted pulses in each series are non-overlapping, all cross terms will be zero and hence this simplifies to: . Taking the data symbols to be , this simplifies to . Focus attention on the interval , and we get . Note, an identical result is obtained for any other interval of length we look at. Hence, for all . (b) Write , where , . Note that since both and are zero-mean and independent, then . u t ( ) 2 a n g t 2 nT ( ) n 2 b n g t 2 nT T ( ) n 2 + = u t ( ) 2 a n 2 g 2 t 2 nT ( ) n b n 2 g 2 t 2 nT T ( ) n + = a n b n , 1 ± { } u t ( ) 2 g 2 t 2 nT ( ) g 2 t 2 nT T ( ) + [ ] n = 0 t T < u t ( ) 2 g 2 t ( ) g 2 t T + ( ) + sin 2 π t 2 T ------ sin 2 π t T + ( ) 2 T ------------------- + = = sin 2 π t 2 T ------ cos 2 π t 2 T ------ + 1 = = T u t ( ) 2 1 = t u t ( ) x t ( ) jy t ( ) + = x t ( ) a n g t 2 nT ( ) n = y t ( ) b n g t 2 nT T ( ) n = x t ( ) y t ( ) φ uu τ ( ) 1 2 -- E x t ( ) jy t ( ) ( ) x t τ + ( ) jy t τ + ( ) + ( ) [ ] = 1 2 -- E x t ( ) x t τ + ( ) [ ] 1 2 -- E y t ( ) y t τ + ( ) [ ] + = 1 2 -- φ xx τ ( ) 1 2 -- φ yy τ ( ) + φ xx τ ( ) = =
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