ECE 8700 Communication System Engineering, Spring 2009
Homework Set # 6: modulation scheme performance & bandwidth; basic MLSE
Suggested Problems from the Text
3.8, 3.9, 3.30, 3.31 (CPM trellis);
4.23, 4.29, 4.39 (performance & bandwidth);
4.35 (simple MLSE)
Homework # 6 (Due Wed., Mar. 25, before class): (Do all. Submit problems
1,3,5,7,8,10.)
1. Problem 4.23 of the Course Text.
2. Problem 4.39 of the Course Text. Use the dimensionality theorem discussed in Section
4.6 of the Course Text.
3. Problem 4.35 of the Course Text.
4. Problem 3.8 of the Course Text.
5. Problem 3.9 of the Course Text.
6. Problem 3.30 of the Course Text. Assume binary CPM (i.e. In=±1).
7. Problem 3.31 of the Course Text.
8. Consider an N= 2 dimensional modulation scheme, with two symbols which have
signal space representations s1= [1,1]Tand s1= [−1,−1]T. Two symbols,
sm(n);n= 1,2 are transmitted. Two observations, rn;n= 1,2, are received. The joint
PDF of the observation vector r= [rT
1, rT
2]T, conditioned on the symbols, is
p(r/sm(1), sm(2)) = 1
(2π)2(σ2
n)2e−P2
n=1 |rn−xn|2/2σ2
n(1)
where x1=sm(1) and x2=sm(1) −sm(2). Given r1= [0.5,−0.2]Tand r2= [1.0,1.1]T,
determine the MLSE estimate of the symbols. You must justify your answer.
9. Consider coherent MLSE of binary DPSK. Assume states s0=√Eb= 1,
s1=−√Eb=−1, and for initialization (at stage n= 0) the state is s0.
(a) Sketch the binary DPSK trellis diagram up to the n= 4 stage.
(b) Given data r1= 1.1, r2= 0.1, r3=−0.3 and r4=−2.1.
i. Label each branch with the branch cost.
ii. Label each state with the minimum path cost into the state.
iii. At stage n= 4, determine the best path, and the corresponding cost and
MLSE.
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