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problem7

# problem7 - UNIVERSITY OF CALIFORNIA SAN DIEGO...

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Unformatted text preview: UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 259B - Winter Quarter 2011 Probabilistic Coding Problem Set #7 Due Thursday, March 10, 2011 1. Consider the ensemble of LDPC codes with degree distribution pair λ ( x ) , ρ ( x ). On the binary erasure channel BEC( p ), the ensemble message-passing threshold probability p ∗ = p ∗ ( λ, ρ ) can be found graphically, as described in lecture. Recall the function f ( p, x ) = pλ (1 − ρ (1 − x )) , for p, x ∈ [0 , 1] derived from the density evolution recursion. For p < p ∗ , we saw that g ( p, x ) = f ( p, x ) − x < for all x ∈ (0 , 1) . At the threshold value p ∗ , we saw that there was at least one critical point x ∗ satisfying the following properties: (i) The pair ( p, x ) = ( p ∗ , x ∗ ) represents a solution to the equation g ( p, x ) = 0 . (1) (ii) The graph of g ( p ∗ , x ) is tangent to the horizontal axis h ( x ) = 0 , x ∈ [0 , 1] at the critical point x = x ∗ ....
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problem7 - UNIVERSITY OF CALIFORNIA SAN DIEGO...

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