HW04 - University of Illinois Spring 2011 ECE 361 Problem...

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University of Illinois Spring 2011 ECE 361: Problem Set 4: Solutions M -ary Orthogonal and Trans-Orthogonal Signaling 1. [Hadamard matrices] Let h 1 , h 2 , h 3 denote three different rows of the matrix. Let a 0 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 i -th row differs from the other two. Then we have h h 1 , h 2 i = a 0 - a 1 - a 2 + a 3 = 0 h h 1 , h 3 i = a 0 - a 1 + a 2 - a 3 = 0 h h 2 , h 3 i = a 0 + a 1 - a 2 - a 3 = 0 which, when added in pairs, give a 0 = a 1 = a 2 = a 3 , and since a 0 + a 1 + a 2 + a 3 = n , we conclude that n must be a multiple of 4. 2. [Bit Error Probabilities for M -ary 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 output is any of 110 or 100 which has probability 2 p/ 7 6 = (4 p/ 7) 2 except when p = 7 / 8 and all symbols are equally likely to be the output regardless of what was transmitted. Thus, in general,
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