Optimal Transmission of Gaussian Signals through a Feedback Channel

Using the methodology and results of the theory of filtering of conditionally Gaussian processes, the optimal schemes of transmission of Gaussian signals through the noisy feedback channel are constructed under the new power conditions. In the present paper, in contrast to our previous work [1], the problem of transmission of Gaussian signals through the noisy feedback channel under new power conditions (see conditions (15),(38) and (40)) is investigated. The obtained results, in our opinion, imply significant simplification and better clarity. In §1 the optimal (in the sense of mean square criterion) linear schemes of transmission in the case of discrete time are constructed. In §2 the optimal linear schemes of transmission in the case of continuous time are investigated and it is proved that these schemes are also optimal in the general class of transmission schemes. § 1. Optimal Transmission in the Discrete Case 1. Let the signal θ = (θt,Ft), t ∈ S = {0, ∆, 2∆, . . . , T}, ∆ > 0, be a discrete Gaussian process given on the basic probability space (Ω,F , P ) with the nondecreasing family of σ-algebras (Ft), t ∈ S, Ft ⊆ F , t < s, satisfying the following finite difference equation: ∆θt = a(t)θt∆ + b(t)∆vt, (1) where v = (vt,Ft), t ∈ S, is a Gaussian random sequence (G.R.S.) N(0, t) with independent increments independent of θ0, which is a Gaussian N(m, γ), γ > 0, random variable; a(t) and b(t) are known functions on S, for every t |a(t)| ≤ k, |b(t)| ≤ k, where k is some constant. Suppose that θ is transmitted according to the scheme ∆ξt = [A0(t, ξ) + A1(t, ξ)θt]∆ + ∆wt, ξ0 = 0, (2) 1991 Mathematics Subject Classification. 60G35,93E20,94A40. 353 1072-947X/1994/0700-0353$12.50/0 c © 1994 Plenum Publishing Corporation