HW9Solutions

HW9Solutions - ECEN 661 Modulation Theory Homework#9...

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ECEN 661 - Modulation Theory Homework #9 Solutions 1. (a) Starting from the expression developed in class, we solve for the phase estimate according to , where . Let and . Then the above equation becomes . (b) In this case the starting point is: . The addition of the sgn function makes this equation difficult to solve analytically. To develop a procedure for solving this equation, define . Note that for a given set of samples , there is a continuum of values of that will lead to the same set . Hence for the purposes of finding the we only need a value of which is close to the ML value. Once we get a sufficiently close value of , then we can determine the correctly. At that point, we can then solve for the ML value of according to Re y n e j φ [] Im y n e j φ n 0 = y n r l t () p * tn T s t d nT s n 1 + T s = y nr , n = y ni , n = y , φ cos y , φ sin + y , φ cos y , φ sin n 0 = y , y , cos 2 φ sin 2 φ n y , 2 y , 2 cos φ φ sin n +0 = y , y , cos 2 φ n 1 2 -- y , 2 y , 2 2 φ sin n = φ 1 2 --tan 1 2 y , y , n y , 2 y , 2 n -----------------------------------    = n e j φ sgn n e j φ n 0 = L n n e j φ sgn = y n {} φ L n L n φ φ L n φ
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HW9Solutions - ECEN 661 Modulation Theory Homework#9...

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