Sequential Necessary and Sufficient Conditions for Capacity Achieving Distributions of Channels with Memory and Feedback

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We derive sequential necessary and sufficient conditions for any channel input conditional distribution P₀,ₙ{Pₓₜ|ₓᵗ⁻¹,Yᵗ⁻¹: t=0,,n} to maximize the finite-time horizon directed information defined by CFBₓₙ → Yₙ P₀,ₙ I(Xⁿ→Yⁿ), I(Xⁿ → Yⁿ) =∑ₜ₌₀ⁿI(Xᵗ;Yₜ|Yᵗ⁻¹) for channel distributions {PYₜ|Yᵗ⁻¹,ₓₜ: t=0,,n} and {PYₜ|Yₜ₋ₘᵗ⁻¹,ₓₜ: t=0,,n}, where Yᵗ{Y₀,,Yₜ} and Xᵗ{X₀,,Xₜ} are the channel input and output random processes, and M is a finite nonnegative integer. We apply the necessary and sufficient conditions to application examples of time-varying channels with memory and we derive recursive closed form expressions of the optimal distributions, which maximize the finite-time horizon directed information. Further, we derive the feedback capacity from the asymptotic properties of the optimal distributions by investigating the limit Cₓ∞ → Y∞FB ∞ 1/n+1 Cₓₙ → YₙFB without any á priori assumptions, such as, stationarity, ergodicity or irreducibility of the channel distribution. The necessary and sufficient conditions can be easily extended to a variety of channels with memory, beyond the ones considered in this paper.