20072ee132A_1_hwk7_sol

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online