20092ee132A_1_hwk7_sol

20092ee132A_1_hwk7_sol - EE132A Spring 2009 Prof John...

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EE132A, Spring 2009 Communication Systems Prof. John Villasenor Handout # 24 TAs: Pooya Monajemi and Erica Han Homework 7 Solutions 1. The list of messages and the corresponding codewords is given in the following table. Message Codeword 0 0 0 0 0 01 0 01 0 010 010 010 0 011 0111 10 0 10 0 0 101 1011 110 1101 111 1110 2. (a) All rows of [ ] H are linearly independent, then the row rank of [ ] H is 3. Next, let’s find the total number of linearly independent columns to find the column rank. Let’s label the columns of [ ] H as follows 1 2 3 4 5 6 [ ] H c c c c c c | | | | | | = | | | | | | We find that 1 3 2 3 5 4 2 6 c c c c c c c c = = = This means that the column rank of [ ] H is 3. Therefore, rank([ ]) 3 H = (b) 1 2 3 [ ] r H r r − − −− = − − −− − − −−
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The generator matrix [ ] G that satisfies [ ][ ] [0] T G H = also satisfies the condition 2 [ ][ ] [0] T G H = ,where the rows of
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20092ee132A_1_hwk7_sol - EE132A Spring 2009 Prof John...

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