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: q t ( ) 0 = t 1.5 T < q t ( ) 1 2 = 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) G M S K p h a s e p u ls e , B T= 0 .3 G M S K 2 R E C nT t n 1 + ( ) T < φ t I ; ( ) θ n θ t I n I n 1 I n 2 , , ; ( ) + = θ t I n I n 1 I n 2 , , ; ( ) π I n q t nT ( ) I n 1 q t n 1 ( ) T ( ) I n 2 q t n 2 ( ) T ( ) + + [ ] = θ n π 2 -- I n 3 θ n 1 + = 0 t T < p 1 t ( ) j q t ( ) q t T + ( ) q t 2 T + ( ) + + ( ) ( ) exp = p 2 t ( ) j q t ( ) q t T + ( ) q t 2 T + ( ) + ( ) ( ) exp = p 3 t ( ) j q t ( ) q t T + ( ) q t 2 T + ( ) + ( ) ( ) exp = p 4 t ( ) j q t ( ) q t T + ( ) q 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 t nT ( ) e j θ n p m * t nT ( ) ; 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 r t ( ) e j θ
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