By understanding the local behaviour over one single step channel

By understanding the local behaviour over one single

This preview shows page 19 - 25 out of 32 pages.

By understanding the local behaviour over one single-step channel transformation, conclusions can be reached about the 19
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overall channel transformation from W N to W (1) N , . . . , W ( N ) N . We begin with the rate parameter. Theorem 6.1 Suppose ( W, W ) ( W , W ) for some B- DMC W . Then I ( W ) + I ( W ) = 2 I ( W ) (10) I ( W ) I ( W ) (11) with equality iff I ( W ) is 0 or 1. (10) indicates that the single-step channel transformation pre- serves the symmetric capacity. (11) together with (10) implies that I ( W ) = I ( W ) = I ( W ) iff W is either a noiseless chan- nel (i.e., I ( W ) = 1 ) or a useless channel (i.e., I ( W ) = 0 ). If W is neither noiseless nor useless (so that I ( W ) < I ( W ) ), the single-step channel transformation shifts the symmetric capacity of W and W away from the center in the sense that I ( W ) < I ( W ) < I ( W ) . 20
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Next, we turn to the reliability parameter. Theorem 6.2 Suppose ( W, W ) ( W , W ) for some B- DMC W . Then Z ( W ) = Z ( W ) 2 (12) Z ( W ) 2 Z ( W ) Z ( W ) 2 (13) Z ( W ) Z ( W ) Z ( W ) (14) Equality holds in (13) iff W is a BEC. We have Z ( W ) = Z ( W ) iff Z ( W ) is 0 or 1, or equivalently, if I ( W ) is 1 or 0. (13) together with (12) imply that reliability improves under a single-step channel transformation in the sense that Z ( W ) + Z ( W ) 2 Z ( W ) with equality iff W is a BEC. In the special case where W , W or W is a BEC, we have Theorem 6.3 If W is a BEC with erasure probability , then W and W are BECs with erasure probabilities 2 2 and 2 , respectively. Conversely, if W or W is a BEC, then W is a BEC. 21
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We now turn to the single-step channel transformation of the form (7). Theorem 6.4 follows from Theorems 6.1 and 6.2 as a special case. Theorem 6.4 For any B-DMC W , N = 2 n , n 0 , 1 i N , the transformation W ( i ) N , W ( i ) N W (2 i 1) 2 N , W (2 i ) 2 N is rate-preserving and reliability-improving in the sense that I ( W (2 i 1) 2 N ) + I ( W (2 i ) 2 N ) = 2 I ( W ( i ) N ) (15) Z ( W (2 i 1) 2 N ) + Z ( W (2 i ) 2 N ) 2 Z ( W ( i ) N ) (16) with equality in (16) iff W is a BEC. Channel splitting moves the rate and reliability away from the center in the sense that I ( W (2 i 1) 2 N ) I ( W ( i ) N ) I ( W (2 i ) 2 N ) (17) Z ( W (2 i 1) 2 N ) Z ( W ( i ) N ) Z ( W (2 i ) 2 N ) (18) with equality in (17) and (18) iff I ( W ) equals 0 or 1. The reliability parameter further satisfies Z ( W (2 i 1) 2 N ) 2 Z ( W ( i ) N ) Z ( W ( i ) N ) 2 (19) Z ( W (2 i ) 2 N ) = Z ( W ( i ) N ) 2 (20) 22
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with equality in (19) iff W is a BEC. The cumulative rate and reliability satisfy N i =1 I ( W ( i ) N ) = NI ( W ) (21) N i =1 Z ( W ( i ) N ) NZ ( W ) (22) with equality in (22) iff W is a BEC. The cumulative relations (21) and (22) follow by repeated application of (15) and (16). For example, for N = 8 , we have for the rate parameter, I ( W (7) 8 ) + I ( W (8) 8 ) = 2 I ( W (4) 4 ) I ( W (5) 8 ) + I ( W (6) 8 ) = 2 I ( W (3) 4 ) I ( W (3) 8 ) + I ( W (4) 8 ) = 2 I ( W (2) 4 ) I ( W (1) 8 ) + I ( W (2) 8 ) = 2 I ( W (1) 4 ) I ( W (3) 4 ) + I ( W (4) 4 ) = 2 I ( W (2) 2 ) I ( W (1) 4 ) + I ( W (2) 4 ) = 2 I ( W (1) 2 ) I ( W (1) 2 ) + I ( W (2) 2 ) = 2 I ( W (1) 1 ) = 2 I ( W ) 23
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Combining these equations, we have 8 i =1 I ( W ( i ) 8 ) = 2 4 i =1 I ( W ( i ) 4 ) = 4 2 i =1 I ( W ( i ) 2 ) = 8 I ( W ) as expected. Further, the conditions for equality in Theorem 6.4 are stated in terms of W rather than W ( i ) N , which is possible since by Theorem 6.3, W is a BEC iff W ( i ) N is a BEC, by Theorem 6.1, I ( W ) ∈ { 0 , 1 } iff I ( W ( i ) N ) ∈ { 0 , 1 } .
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