On a Property of the Soft Bit Density associated with Log- Likelihood Ratios of a Binary AWGN Channel

Inhalt:
We derive a relation between the unconditional and conditional (conditioned on the input bit X to be either 0 or 1) densities of the expected bit value, the so-called ’soft bit’ Λ, given the log-likelihood ratios L(X) at the output of an AWGN channel. Both quantities are connected via the relation Λ = tanh(L/2). Denoting the unimodal, conditional soft bit density as f Λ|X,uni(λ|x) and the bimodal unconditional density as f Λ,bi(λ) we prove the following property: the mean of the conditional soft bit distribution is equal to the variance of the unconditional soft bit σ 2 Λ,bi, i.e. μ Λ|X,uni = σ 2 Λ,bi. This property is rooted in the consistency property of the associated L-value density.