Exam1Solutions

# Exam1Solutions - ECEN 661-Modulation Theory Spring 2006...

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ECEN 661 -Modulation Theory Spring 2006 Midterm Exam #1 Solutions Problem 1. (15 points) Consider a 4-ary two-dimensional signalling for- mat that uses the basis functions and as shown. Note that while the basis functions are orthogonal, they are not normalized. (a) Suppose the 4 possible transmitted signals are as shown. Write each of the four possible signals in its vector representa- tion based on the two basis functions given. (b) Carefully sketch the signal constellation. Be sure to carefully label the axes in your sketch. (c) If is a white noise process with a PSD of , find the variance of the components of , where is the representation of based on the basis functions given. Answers: (a) , , , . (b) See drawing below. (c) . . T s T s 2 ---- T s 2 ---- T s 1 1 t t f 1 t ( ) f 2 t ( ) f 1 t ( ) f 2 t ( ) T s T s 2 ---- A t s 1 t ( ) T s T s 2 ---- A t s 3 t ( ) T s T s 2 ---- A t s 2 t ( ) T s T s 2 ---- A t s 4 t ( ) A A s m t ( ) ; m =1 2 3 4 , , , { } n t ( ) N 0 2 n n 1 n 2 , ( ) = n n t ( ) s 1 A A , ( ) = s 2 A A , ( ) = s 3 A A , ( ) = s 4 A A , ( ) = n 1 n t ( ) f 1 t ( ) t d f 1 t ( ) 2 t d ---------------------------- 2 T s ---- n t ( ) t d 0 T s 2 = = A A A A * * * * s 1 s 4 s 3 s 2 E n 1 2 [ ] 4 T s 2 ----- E n t ( ) n u ( ) t d [ ] 0 T s 2 u d 0 T s 2 = 2 N o T s 2 --------- t u ( ) δ 0 T s 2 u d 0 T s 2 N o T s ----- = =

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Problem 2. (20 points)
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