20072ee132A_1_hwk7_sol

# 20072ee132A_1_hwk7_sol - EE132A Spring 2007 Prof John...

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EE132A, Spring 2007 Communication Systems Prof. John Villasenor Handout # 24 TA: Choo Chin (Jeffrey) Tan 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. ( ) C n k , can be described by any k orthogonal basis vectors chosen from the set of all possible codewords in ( ) C n k , . If G is such a basis, we can find all other basis by counting all possible 2 G that satisfy the following: 2 0 T G H × = k G rank = ) ( 2 . This means that all rows of 2 G are linearly independent. These two conditions can be satisfied by choosing 2 G as follows Y G G × = 2 Where Y is a full rank k k × matrix. Clearly, Y has linearly independent rows. Our goal is to find the total number of all possible matrices. Let us label the rows of Y as follows - - - - - - - - - - - - = k r r r Y . . . . . . . . . 2 1

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For 1 r , we have 2 1 k - valid selections for 1 r which are all possible k –bit vectors except for the all zero vector. 1 2 1 - = k r N For 2 r , we can use as any k –bit vector except for the all zero vector and the one that is used for 1 r . 1 ) 1 2 ( 2 - - = k r N For 3 r , we can use as any k –bit vector except for the all zero vector and any linear combination of 1 r and 2 r
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20072ee132A_1_hwk7_sol - EE132A Spring 2007 Prof John...

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