SolvedFinalSpring08

# SolvedFinalSpring08 - (e) (4 pts) Compute an upper bound on...

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(e) (4 pts) Compute an upper bound on the probability of error P e for this code on the binary symmetric channel with probability of error p = 10 - 4 . You may assume that 1 - 10 - 4 1 P e dT ( W,I ) dI I =1 ,W = 4 p (1 - p ) (8) = W 3 (1 - IW ) + 4 (1 - ) 2 I =1 ,W = 4 p (1 - p ) (9) (10) = W 3 - 4 + 4 1 - 2 + 2 I =1 ,W = 4 p (1 - p ) (11) = W 3 1 - 2 + 2 I =1 ,W = 4 p (1 - p ) (12) W 3 1 - 2 + 2 I =1 ,W =2 × 10 - 2 (13) = 8 × 10 - 6 1 - 4 × 10 - 2 + 4 × 10 - 4 (14) = 8 . 33 × 10 - 6 (15) (f) (4 pts) Would this be a good constituent code for a turbo code? Why or why not? Solution: This would not be a good constituent code for a turbo code since the 3 error event indicates that α > 0 would occur. Also acceptable would be to say that it cannot be a good constituent code fora turbo code because this is a feedforward encoder. 2
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. 3

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(a) (5 pts) The trellis code product state diagram has four states. Construct the 4 × 4 transition matrix G . You need not draw the product state diagram. First label each product state by ( e i ,s j ) where e j is the error state and s j is the encoder state. Then assign indices e i * 2+ s j to each product-state. Those with indices less than 2 are correct states, and the rest are erroneous states. Now let us ﬁll up the matrix G . Each element G ij is the label of the edge from state j to state i . Since each state has 4 outgoing edges, every element in the matrix is nonzero. All the elements are of the form pI i W w , where p = 1 / 2 in this example, i is the number of information bits in error and w is the squared Euclidean distance between the transmitted and received constellation points. Solution: G = 1 2 × 1 1 W 3 W 3 1 1 W 3 W 3 IW 5 . 6 5 . 6 9 5 . 6 5 . 6 9 (16) (b) (3 pts) Perform the ﬁrst step of the iterative FSM approach. That is, combine all correct states.
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## This note was uploaded on 03/29/2011 for the course EECS 100 taught by Professor Dun during the Spring '11 term at CSU Channel Islands.

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SolvedFinalSpring08 - (e) (4 pts) Compute an upper bound on...

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