solution7

solution7 - UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical...

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UNIVERSITY OF CALIFORNIA, SAN DIEGO ECE 259B - Winter Quarter 2011 Probabilistic Coding Solutions to Problem Set #7 1. (a) To simplify notation, denote p ( j, k )by p . For the ensemble of ( j, k )-regular LDPC codes, the condition (i) in the problem statement translates to p (1 (1 x ) k 1 ) j 1 x =0 (1) or p = x (1 (1 x ) k 1 ) j 1 . (2) In terms of the function p ( x ) de±ned in the problem statement, this can also be written as p = p ( x ) . (3) De±ning y =1 x ,wehave p = p (1 y )= 1 y (1 y k 1 ) j 1 . (4) We now want to show that y is the largest positive real root of the polynomial q ( x )=(( j 1)( k 1) 1) x k 2 k 3 X i =0 x i . Condition (ii) in the problem statement implies that ∂x g ( p ,x ) | x = x =0 which means that p ( j 1)(1 (1 x ) k 1 ) j 2 (( k 1)(1 x ) k 2 ) 1=0 .

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This note was uploaded on 04/02/2011 for the course ECE 264 taught by Professor Song during the Spring '11 term at UCSB.

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solution7 - UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical...

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