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Unformatted text preview: University of Illinois Spring 2011 ECE 361: Problem Set 4: Solutions Mary Orthogonal and TransOrthogonal Signaling 1. [Hadamard matrices] Let h 1 ,h 2 ,h 3 denote three different rows of the matrix. Let a denote the number of coordinates where the three rows are identical, and let a i ,i { 1 , 2 , 3 } denote the number of coordinates in which the ith row differs from the other two. Then we have h h 1 ,h 2 i = a a 1 a 2 + a 3 = 0 h h 1 ,h 3 i = a a 1 + a 2 a 3 = 0 h h 2 ,h 3 i = a + a 1 a 2 a 3 = 0 which, when added in pairs, give a = a 1 = a 2 = a 3 , and since a + a 1 + a 2 + a 3 = n , we conclude that n must be a multiple of 4. 2. [Bit Error Probabilities for Mary orthogonal signaling] (a) The first bit is in error if the output is any of 100, 101, 110, 111. Hence we have that P ( A 1  a = 011) = 4 p/ 7. Similarly, the last bit is is in error if the output is any of 000, 010, 100, 110 and hence P ( A 3  a = 011) = 4 p/ 7 also. Note that both the first and last bits are in error if the...
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 Spring '09

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