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problem4

# problem4 - UNIVERSITY OF CALIFORNIA SAN DIEGO Electrical...

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UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 259B - Winter Quarter 2011 Probabilistic Coding Problem Set #4 Due Thursday, February 17, 2011 1. In this problem, we consider a parallel concatenation of two systematic block encoders through a uniform interleaver. The block encoder we use is for the (7,4)-Hamming code, and is given in the table below. The parallel turbo architecture is shown in the figure below, where u 1 u 2 u 3 u 4 represent the input data, p 1 p 2 p 3 are the parity bits generated by the first encoder, and q 1 q 2 q 3 are the parity bits generated by the second encoder. The resulting code has parameters ( n,k ) = (10 , 4) . Information Parity 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 2 3 4 u u u u 1 2 3 4 u u u u S Systematic block encoder Systematic block encoder 1 2 3 p p p 1 2 3 q q q

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(a) For any input-weight w , let A w ( Z p ) = 3 j =0 A w,j Z j p be the conditional weight enumerator (CWEF). Using the encoder table, fill in the missing CWEF expressions for the Hamming encoder: A 1 ( Z p ) = 3 Z 2 p + Z 3 p A 2 ( Z p ) = A 3 ( Z p ) = A 4 ( Z p ) = Z 3 p (b) Assume the use of a uniform interleaver. Let
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