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ISBN-10: 0120121212

ISBN-13: 9780120121212

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Correspondingly, for another BSC V on the same X and Y we denote V (0 |1) = V (1 |0) = v1 , V (0 |0) = V (1 |1) = v2 . It is clear that w1 + w2 = 1, v1 + v2 = 1. The maximal value of the mutual information IP,V (X ∧ Y ) in the definition of Rsp (E, W ) is obtained when p∗ (0) = p∗ (1) = 1/2 because of symmetry of the channel, therefore IP ∗ ,V (X ∧ Y ) = 1 + v1 log v1 + v2 log v2 . The condition D(V W |P ∗ ) ≤ E will take the following form: v1 log v1 v2 + v2 log ≤ E. 15))   −(1 + v1 log v1 + v2 log v2 ) = max − v1 log wv11 + v2 log wv22 − E = 0   v1 + v2 = 1.

6, it repeats all steps of analogous demonstration for reliability function E(R, W ), made by Csisz´ar and K¨orner by the method of graph decomposition [50]. 10) and Rx (E, W ) = max Rx (P, E, W ). 7. For DMC W for any E > 0 the following bound holds R(E, W ) ≥ max(Rr (E, W ), Rx (E, W )). In the next theorem the region, where the upper and the lower bounds coincide, is pointed out. Let Ecr (P, W ) = min E : ∂Rsp (P, E, W ) ≥ −1 . 8. For DMC W and PD P , for E ∈ [0, Ecr (P, W )] we have R(P, E, W ) = Rsp (P, E, W ) = Rr (P, E, W ), and, particularly, for E = 0 Rsp (P, 0, W ) = Rr (P, 0, W ) = IP,W (X ∧ Y ).

In other words Rsp (P, E, W ) − Rsp (P, E , W ) < −1, E−E when E < E ≤ Ecr , from where Rsp (P, E, W ) + E < Rsp (P, E , W ) + E , and consequently min E :E ≤E

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Advances in Computers, Vol. 21


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