HW05 - University of Illinois Spring 2011 ECE 361 Problem...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
University of Illinois Spring 2011 ECE 361: Problem Set 5: Solutions Linear Binary Coding; Gaussian Channel Capacity 1. [Distances and Weights] (a) For bits a,b ∈ { 0 , 1 } , a b = a + b - 2 ab . Since the Hamming weight of a binary vector is the arithmetic sum of its coordinates, the result follows. (b) From the result of part (a), we have that if w H ( x y ) is an even number, then w H ( x ) + w H ( y ) is also an even number. Therefore, it must be that both w H ( x ) and w H ( y ) are even numbers, or both are odd numbers. On the other hand, if w H ( x y ) is an odd number, then w H ( x ) + w H ( y ) is also an odd number. Therefore, it must be that one of w H ( x ) and w H ( y ) is an even number and the other is an odd number. 2. [Odds and Evens] (a) Let C 0 denote the set of all codewords that have a 0 in the i -th coordinate and C 1 denote the set of all codewords that have a 1 in the i -th coordinate. Note that C 0 ∩C 1 = and C 0 ∪C 1 = C , and that 0 ∈ C 0 . If C 1 = , then C 0 = C and we are done: all the codewords have a 0 in the i -th coordinate. Otherwise, C 1 6 = , and so there is at least one codeword y ∈ C 1 . But then, for each x ∈ C 0 , the codeword x y ∈ C 1 since x y obviously has a 1 in the i -th coordinate. In particular, note that 0 y = y ∈ C 1 is a codeword of this type. Since each x ∈ C 0 corresponds to a x y ∈ C 1 (and there may be other codewords in C 1 not obtained by this procedure), we conclude that |C 0 | ≤ |C 1 | . Conversely, for each
Background image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 2

HW05 - University of Illinois Spring 2011 ECE 361 Problem...

This preview shows document pages 1 - 2. Sign up to view the full document.

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