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FinalSpring08

# FinalSpring08 - EE 231E Channel Coding Instructor Rick...

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EE 231E Spring 2008 Final Channel Coding Thursday, June 12, 2008 Instructor: Rick Wesel 11:30 a.m. - 2:30 p.m. 116 pts, 180 minutes Your Name: Your ID Number: Problem Score Possible 1 24 2 20 3 16 4 21 5 5 6 18 7 12 Total 116 1

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1. (24 pts) T ( W, I ). Analyze the convolutional encoder G ( D ) = 1 + D 1 as follows: (a) (4 points) Draw the state diagram for the encoder and label each branch with ( i, x 1 x 2 ) where i is the input bit associated with that state transition, x 1 is the output bit produced by the 1 + D in G ( D ), and x 2 is the output bit produced by the 1 in G ( D ). You will use this diagram in both this problem and the next one. Prepare it carefully. (b) (4 pts) Draw the split-state diagram. 2
(c) (4 pts) Find the matrices A , b , c , and d for this convolutional encoder. (d) (4 pts) Compute T ( W, I ), and express it both as a ratio of polynomials and as an infinite sum. 3

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(e) (4 pts) Compute an upper bound on the probability of error for this code on the binary symmetric channel with probability of error p = 10 4 . You may assume that 1 10 4 1 (f) (4 pts) Would this be a good constituent code for a turbo code? Why or why not? 4
2. (20 pts) Asymmetric 4-QAM. Consider the following asymmetric 4-QAM constellation: 1 3 9 5.6 10 01 11 00 Figure 1: An asymmetric 4-QAM constellation. The constellation does have symmetry between the top two and bottom two points so that all distances can be inferred from those indicated. The distances shown are squared Euclidean distances. The following table gives the squared Euclidean distance mapping for each point. Point Label Binary symbol errors 00 01 10 11 00 0.0 9.0 3.0 5.6 01 0.0 9.0 3.0 5.6 10 0.0 1.0 3.0 5.6 11 0.0 1.0 3.0 5.6 Suppose that the convolutional encoder used with this constellation is G ( D ) = 1 + D 1 . (This is the encoder you just examined in the previous problem, and you should use your results from part (a) of that problem as a starting point for this one.) For this problem, please take the leftmost polynomial to be the most significant bit so that the polynomial position matches the bit position in the constellation label.

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