Capacity of the gaussian channel without feedback

Consider a communication channel with stochastic input message X and independent additive zero-mean Gaussian noise N. Let I(X, AX + N) denote the average mutual information of X and AX + N. Almost all paths of the process X are assumed to belong to a real separable Banach space B, almost all paths of N are assumed to belong to a real separable Hilbert space H, and A is a measurable function from B into H. AX is the signal process, and can be assumed to have zero mean. R_N is the covariance operator of N, če:italic>λ_n, n ⩾ 1⩽ce:italic> its set of strictly positive eigenvalues with associated orthonormal eigenvectors če:italic>e_n, n ⩾ 1⩽ce:italic>. For X Gaussian and A linear, I(X, AX +N) < ∞ if and only if almost all paths of AX belong to range (R_N^1/2). If range (R_N^1/2) is infinite-dimensional and contains almost all paths of AX, and one requires that E[Σ_n λ_n⁻¹<AX, e_n>²] ⩽ P₀ (a generalized average energy constraint), then the capacity is shown to be P₀/2. The capacity cannot be attained by any AX. If X is subject to an average energy constraint, E″>X 〈/ce:section-title>² ⩽ S, then the capacity is finite if and only if A is constrained so that ‖R_N^−½A‖²⩽K. Under these constraints, thecapacity is SK/2, and cannot be attained. If one also constrains thedimension of the signal space to be no greater than M, M < ∞ then thechannel capacity is (M/2) log(1 + P₀/M), where P₀ is theconstraint on generalized average energy. This supremum can be attained by aGaussian X and a continuous linear A.