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

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

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