HW7Solutions

HW7Solutions - ECEN 661 Modulation Theory Homework#7...

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ECEN 661 - Modulation Theory Homework #7 Solutions GMSK: The GMSK phase pulse shape for BT =0.3 is shown in the figure below. Note that for all practical purposes, for and for . Hence, with insignificant loss in receiver performance, we can assume that the frequency pulse shape lasts for only bit intervals. In the following, I will induce a delay of so that the pulse shape is causal. In that way, the time varying part of the pulse shape is limited to the time interval . During the time interval , the phase waveform is of the form , where , . Define the following functions (over the interval ): , , , . The transmitted signal during the n th bit interval can take on one of the following forms: qt () 0 = t 1.5 T < 12 = t 1.5 T > L 3 = 1.5 T 0 t 3 T < -4 -3 -2 -1 0 1 2 3 4 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 t/T b q(t) GMSK phase pulse, BT=0.3 GMSK 2REC nT t n 1 + T < φ t I ; () θ n θ tI n I n 1 I n 2 ,, ; + = θ n I n 1 I n 2 ; π I n qt nT I n 1 n 1 T I n 2 n 2 T ++ [] = θ n π 2 -- I n 3 θ n 1 + = 0 tT < p 1 t jqt qt T + 2 T + exp = p 2 t + 2 T + + exp = p 3 t + 2 T + + exp = p 4 t + 2 T + exp =
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. The path metric associated with a data sequence is given by . This suggests the following receiver structure: The MLSE uses the Viterbi algorithm to find the path through the trellis which maximizes as defined above. The trellis consists of 16 states defined by . The transition from the state at time n to the state at time n +1 will be labeled with the branch metric . The trellis structure is shown in the figure s l t () e j θ n p m tn T e j θ n p m * T ; m =1, 2, 3, 4, θ n = 0 π 2 -- π 3 π 2 ------ ,,, ,    I λ I Re r t s l * t I ; t d [] Re e j θ n rt e j θ tI n I n 1 I n 2 ,, ; t d nT n 1 + T n == j θ n λ n I
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This note was uploaded on 02/21/2012 for the course ECEN 661 taught by Professor Miller during the Spring '11 term at Texas A&M.

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HW7Solutions - ECEN 661 Modulation Theory Homework#7...

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