Achievable Rate Sample Clauses

Achievable Rate. Extension to Continuous Valued Inputs In this section, we further analyze the achievable rate in the limit that the quantization error due to discretization in section IV-A approaches zero. In this limit, the rate required over the public discussion channel will also increase to infinity. However we note that there is no rate constraint imposed over the discussion channel and thus the proposed rate can be approached arbitrarily closely. In our analysis we make use of the following fact: Fact 1: [36, pg. 23] Let u and v be two arbitrary random variables (discrete or continuous) with a joint probability measure ∫ µ(u, v ) and marginal measures µ(u) and µ(v ). The mutual information between u and v , defined as, I(u; v ) = log dµ(u, v ) dµ(u, v ) (63) d(µ(u) × µ(v )) × where dµ(u, v )/d(µ(u) µ(v )) denotes the Radon-Xxxxxxx derivative of the joint probability measure with respect to the product measure, can be equivalently expressed as: I(u; v ) = lim j,k→∞ I .[u]j ; [v ]kΣ (64) where [u]j and [v ]k are any sequence of fine quantizations of u and v respectively such that .u − [u]j . and |v − [v ]k| tend to zero as pointwise as j → ∞, and k → ∞, respectively ˆ ˆ Upon examining (61),(62) and taking j, l → ∞, with k fixed, we have that with u = (hAB, hBA): lim l,j→∞ I .[yB,k]l ; [xA]k, [u]j Σ = I(yB,k; [xA]k, u) (65) . Σl xxx I [yB,k] ; xXX,k, gAE, hAB = I(yB,k; xXX,k, gAE, hAB) (66) l→∞ lim l,j→∞