Georgian Mathematical Journal 1(1994), No. 4, 353-365

OPTIMAL TRANSMISSION OF GAUSSIAN SIGNALS THROUGH A FEEDBACK CHANNEL

O. GLONTI

Abstract. 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 = (θ_{t}, F_{t}), t ∈ S = {0, Δ, 2Δ, …, T }, Δ > 0, be a discrete Gaussian process given on the basic probability space (Ω, F, P) with the nondecreasing family of σ-algebras (F_{t}), t ∈ S, F_{t} ⊆ F, t < s, satisfying the following finite difference equation:

Δθ_{t} = a(t)θ_{t}Δ + b(t)Δv_{t},  (1)

where v = (v_{t}, F_{t}), t ∈ S, is a Gaussian random sequence (G.R.S.) N(0, t) with independent increments independent of θ_{0}, 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

Δξ_{t} = [A_{0}(t, ξ) + A_{1}(t, ξ)θ_{t}]Δ + Δw_{t}, ξ_{0} = 0,  (2)

353 where w = (w_{t}, F_{t}), t ∈ S, is a G.R.S. N(0, t) with independent increments which is independent of θ_{0} and v. The nonanticipating (with respect to ξ) functionals A_{0}(t, ξ) and A_{1}(t, ξ) define the coding.

1991 Mathematics Subject Classification. 60G35,93E20,94A40.

1072-947X/1994/0700-0353$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1994 Plenum Publishing Corporation

The transmission performed according to the scheme (2) is a transmission of a Gaussian message θ through a noiseless feedback channel which is an analogue of the additive “White noise” channel in the discrete time case. No instantaneous feedback is required here (which is essential in the continuous time), but we assume that the quantization step Δ is equal to the time of signal return.

Suppose that the coding functionals A_{0} and A_{1} satisfy the condition

E[A_{0}(t, ξ) + A_{1}(t, ξ)θ_{t}]^{2} ≤ p,  (3)

where p is a constant characterizing the energetic potential of the transmitter.

Consider the decoding θ̂ = θ̂_{t}(ξ) satisfying for every t the condition

Eθ̂_{t}^{2} < ∞.  (4)

Such a kind of [(A_{0}, A_{1}), θ̂] forms a class of admissible codings and decodings.

The problem is to find the codings (A_{0}^{∗}, A_{1}^{∗}) and the decodings θ̂^{∗} optimal in the sense of the criterion

δ(t) = inf_{A_{0},A_{1},θ̂}E[θ_{t} − θ̂_{t}(ξ)]^{2}  (5)

where inf is taken in the class of all admissible [(A_{0}, A_{1}), θ̂].

Theorem 1. During the transmission a discrete Gaussian process described by the finite difference equation (1) according to the transmission scheme (2) under conditions (3),(4) and the optimal coding functionals A_{0}^{∗} and A_{1}^{∗}, have the form

A_{1}^{∗}(t) = \sqrt{\frac{p}{γ}} (1 + pΔ)^{\frac{t}{2Δ}} , A_{0}^{∗}(t, ξ^{∗}) = −A_{1}^{∗}(t)m_{t}^{∗},  (6)

where the optimal decoding θ̂_{t}^{∗} = m_{t}^{∗} = E[θ_{t}|F_{t}^{ξ^{∗}}] and the transmitted signal are defined by the relations

Δm_{t}^{∗} = a(t)m_{t}^{∗}Δ + \sqrt{pγ_{t}^{∗}} \frac{(1 + a(t)Δ)}{1 + pΔ}Δξ_{t}^{∗}, m_{0}^{∗} = m,  (7)
Δξ_{t}^{∗} = \sqrt{\frac{p}{γ_{t}^{∗}}} (θ_{t} − m_{t}^{∗})Δ + Δw_{t}, ξ_{0}^{∗} = 0.  (8)

The minimal error of message reproduction has the following form:

δ(t) = γ_{t}^{∗} = γ( ∏_{k=0}^{t−Δ}(1 + a(k)Δ)^{2})(1 + pΔ)^{− \frac{t}{Δ}} +

+ ∑_{k=0}^{t−Δ}(b^{2}(k)Δ)( ∏_{s=k+Δ}^{t−Δ}(1 + a(s)Δ)^{2})(1 + pΔ)^{− \frac{t−k−2Δ}{Δ}} .

Proof. For the given A_{0} and A_{1} it is known that θ̂_{t} = m_{t} = E(θ_{t}|F_{t}^{ξ}). Hence δ(t) = inf_{(A_{0},A_{1})} E(θ_{t} − m_{t})^{2} = inf_{(A_{0},A_{1})} Eγ_{t}.

In order to find m_{t} and γ_{t} = E[(θ_{t} − m_{t})^{2}|F_{t}^{ξ}] we shall use filtering equations for the conditionally Gaussian random sequence (see [1] or [2]).

The rest of the proof is similar to that of the theorem on the optimal scheme for the transmission of Gaussian processes through a noiseless feedback channel in continuous time (see [2]).

2. It is natural to investigate the case of Gaussian signal transmission when white noise is imposed on the back signal, i.e., the message is transmitted according to the scheme

Δξ_{t} = A(t, θ, ξ̃)Δ + σ(t)Δw_{t}, ξ_{0} = 0,  (9)

where the back signal ξ̃_{t} has the form, say, ξ̃_{t} = ξ_{t} + η_{t} or ξ̃_{t} = Π(t, ξ) + η_{t}, where Π(t, ξ) is some nonanticipating functional and η_{t} is the noise in the back channel.

We shall specify the problem under consideration.

Let the signal θ = (θ_{t}, F_{t}), t ∈ S, be a discrete Gaussian process described by the equation (1).

Assume that θ is transmitted according to the scheme

Δξ_{t} = [A_{0}(t)ξ̃_{t} + A_{1}(t)θ_{t}]Δ + σ(t)Δw_{t}, ξ_{0} = 0,  (10)

where the functions A_{0} and A_{1} define the coding.

The back signal has the form

ξ̃_{t} = Π(t, ξ) + η_{t}.  (11)

Here Π(t, ξ) is a transformator of the back message and η_{t} is the noise in the back channel governed by the finite difference equation

Δη_{t} = c(t)η_{t}Δ + d(t)Δw̄_{t},  (12)

where w̄ = (w̄_{t}, F_{t}), t ∈ S, is a G.R.S. N(0, t) with independent increments, for every t |c(t)| ≤ L and |d(t)| ≤ L, where L is some constant.

Denote

m_{t}^{(1)} = m_{t} = E[θ_{t}|F_{t}^{ξ}], m_{t}^{(2)} = E[η_{t}|F_{t}^{ξ}],
γ_{t}^{(1)} = γ_{t} = E[(θ_{t} − m_{t})^{2}|F_{t}^{ξ}], γ_{t}^{(2)} = E[(η_{t} − m_{t}^{(2)})^{2}|F_{t}^{ξ}],
γ_{t}^{(1,2)} = γ_{t}^{(2,1)} = E[(θ_{t} − m_{t})(η_{t} − m_{t}^{(2)})|F_{t}^{ξ}],

γ̃_{t} = γ_{t}^{(2)} − \frac{γ_{t}^{^{2}(1,2)}}{γ_{t}} , γ̂_{t} = \frac{γ_{t}^{(1,2)}}{γ_{t}} .

The problem is to find the codings (A_{0}^{∗}, A_{1}^{∗}), decodings θ̂^{∗} = (θ̂_{t}^{∗}(ξ^{∗})), t ∈ S, and the transformator Π^{∗} optimal in the sense of the square criterion

δ(t) = inf_{A_{0},A_{1},Π,θ̂}E[θ_{t} − θ̂_{t}(ξ)]^{2},  (13)

where inf is taken in the class of admissible A_{0}, A_{1}, Π, θ̂ for which the power condition^{1}

E[A_{0}(t)ξ̃_{t} + A_{1}(t)θ_{t}]^{2} ≤ p(t),  (14)
A_{0}^{2}(t)γ̃_{t} ≥ q(t)  (15)

holds where p(t) and q(t) are summable functions on S characterizing the changes of the energetic transmitter potential, and q(t) ≤ p(t), t ∈ S. Let

Eθ̂_{t}^{2} < ∞.  (16)

Theorem 2. When the Gaussian random sequence θ = (θ_{t}, F_{t}), t ∈ S, governed by the finite difference equation (1) is transmitted according to the scheme of transmission (10)–(12) through a noisy feedback channel under conditions (14)–(16), then the optimal [in the sense of square criterion (13)] coding functions A_{0}^{∗}, A_{1}^{∗}, decoding functional, θ̂^{∗} and transformator of the back message Π^{∗} have the form

A_{0}^{∗}(t) = (\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} , A_{1}^{∗}(t) = −(\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} γ̂_{t}^{∗} + (\frac{p(t) − q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} ,

\begin{matrix}Π^{∗}(t, ξ) = −m_{t}^{∗(2)} − \frac{A_{1}^{∗}(t)}{A_{0}^{∗}(t)}m_{t}^{∗}, θ̂_{t}^{∗}(ξ^{∗}) = m_{t}^{∗}, \\ Δm_{t}^{∗} = a(t)m_{t}^{∗}Δ + (1 + a(t)Δ)[γ_{t}^{∗}(p(t) − q(t))]^{\frac{1}{2}} × \\ ×(σ^{2}(t) + p(t)Δ)^{−1}Δξ_{t}^{∗}, m_{0}^{∗} = m. \end{matrix}  (17)

1The fact that condition (15) is the power condition indeed will be shown at the end of the section.

The optimal transmission has the following form:

Δξ_{t}^{∗} = \{(\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} (η_{t} − m_{t}^{∗(2)}) + [ − (\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} γ̂_{t}^{∗} + (\frac{p(t) − q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} ] ×

×(θ_{t} − m_{t}^{∗})}Δ + σ(t)Δw_{t}, ξ_{0}^{∗} = 0,  (18)

where m_{t}^{∗(2)} satisfies the finite difference equation

Δm_{t}^{∗(2)} = c(t)m_{t}^{∗(2)}Δ + (1 + c(t)Δ)\{γ̂_{t}^{∗}[γ_{t}^{∗}(p(t) − q(t))]^{\frac{1}{2}} + [q(t)γ̂_{t}^{∗}]^{\frac{1}{2}} \} ×

×(σ^{2}(t) + p(t)Δ)^{−1}Δξ_{t}^{∗}, m_{0}^{∗(2)} = m^{(2)} = Eη_{0}^{(2)}

and γ_{t}^{∗} = γ_{t}^{∗(2)} − \frac{(γ_{t}^{∗(1,2)})^{2}}{γ_{t}^{∗}} and γ̂_{t}^{∗} = \frac{γ_{t}^{∗(1,2)}}{γ_{t}^{∗}} are defined by the filtering equations (19)–(23) given below, where A_{0}^{∗} and A_{1}^{∗} are substituted from (17).

The minimal message reproduction error δ(t) has the form

δ(t) = γ_{t}^{∗} = γ ∏_{k=0}^{t−Δ}(1 + a(k)Δ)^{2}(σ^{2}(k) + q(k)Δ)(σ^{2}(k) + p(k)Δ)^{−1} +

+∑_{k=0}^{t−Δ} b^{2}(k)( ∏_{m=k+Δ}^{t−Δ}(1+a(m)Δ)^{2}(σ^{2}(m)+q(m)Δ)(σ^{2}(m)+p(m)Δ)^{−1}).

Corollary. Let a(t) = b(t) ≡ 0, i.e., according to the scheme (10)–(12), a Gaussian N(m, γ), γ > 0, random variable θ is transmitted. Then the reproduction error δ(t) is

δ(t) = γ ∏_{k=0}^{t−Δ} (1 + \frac{q(k)}{σ^{2}(k)}Δ)(1 + \frac{p(k)}{σ^{2}(k)}Δ)^{−1}.

In this case γ_{t}^{∗} and γ̂_{t}^{∗} satisfy the following finite difference equations:

\frac{Δγ_{t}^{∗}}{Δ} = d^{2}(t) + γ_{t}^{∗}(c(t)σ^{2}(t) − q(t))(σ^{2}(t) + q(t)Δ)^{−1}, γ_{0}^{∗} = 0,

\frac{Δγ̂_{t}^{∗}}{Δ} = \{γ̂_{t}^{∗}[c(t)σ^{2}(t) − q(t)γ_{t}^{∗} + q(t)(1 + c(t)Δ)] −

−(1 + c(t)Δ)(\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} (\frac{p(t) − q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} γ_{t}^{∗}\}[σ^{2}(t) + q(t)Δ]^{−1}, γ̂_{0}^{∗} = 0.

Proof of the Theorem. It can be easily seen that

δ(t) = inf_{(A_{0},A_{1},Π)} Eγ_{t} = inf_{(A_{0},A_{1},Π)} γ_{t}.

Rewrite (10) in the following form:

Δξ_{t} = [A_{0}(t)Π(t, ξ) + A_{0}(t)η_{t} + A_{1}(t)θ_{t}]Δ + σ(t)Δw_{t}, ξ_{0} = 0.

Then one can see that the equation of optimal filtering of a partially observable conditionally Gaussian process (θ̃_{t}, ξ_{t}), t ∈ S, with an unobservable component θ̃_{t} = (θ_{t}, η_{t}) (see [1] or [2]) leads to the following closed system of finite difference equations:

Δm_{t} = a(t)m_{t}Δ + (1 + a(t)Δ)(A_{1}(t)γ_{t} + A_{0}(t)γ_{t}^{(1,2)}) ×
×[σ^{2}(t) + (A_{1}^{2}(t)γ_{t} + 2A_{1}(t)A_{0}(t)γ_{t}^{(1,2)} + A_{0}^{2}(t)γ_{t}^{(2)})Δ]^{−1} ×
×[Δξ_{t} − (A_{0}(t)(Π(t, ξ) + m_{t}^{(2)}) + A_{1}(t)m_{t})Δ], m_{0} = m,  (19)
Δm_{t}^{(2)} = c(t)m_{t}^{(2)}Δ + (1 + c(t)Δ)(A_{1}(t)γ_{t}^{(1,2)} + A_{0}(t)γ_{t}^{(2)}) ×
[σ^{2}(t) + (A_{1}^{2}(t)γ_{t} + 2A_{1}(t)A_{0}(t)γ_{t}^{(1,2)} + A_{0}^{2}(t)γ_{t}^{(2)})Δ]^{−1} ×
×[Δξ_{t} − (A_{0}(t)(Π(t, ξ) + m_{t}^{(2)}) + A_{1}(t)m_{t})Δ], m^{(2)} = m, (20)

\frac{Δγ_{t}^{(1)}}{Δ} = b^{2}(t) + a^{2}(t)γ_{t}^{(1)}Δ + 2a(t)γ_{t}^{(1)} −

−(1 + a(t)Δ)^{2}(A_{1}(t)γ_{t}^{(1)} + A_{0}(t)γ_{t}^{(1,2)})^{2}[σ^{2}(t) +

+(A_{1}^{2}(t)γ_{t}^{(1)} + 2A_{1}(t)A_{0}(t)γ_{t}^{(1,2)} + A_{0}^{2}(t)γ_{t}^{(2)})Δ]^{−1},

γ_{0}^{(1)} = γ,  (21)

\frac{Δγ_{t}^{(2)}}{Δ} = d^{2}(t) + c^{2}(t)γ_{t}^{(2)}Δ + 2c(t)γ_{t}^{(2)} −

−(1 + c(t)Δ)(A_{1}(t)γ_{t}^{(1,2)} + A_{0}(t)γ_{t}^{(2)})^{2}[σ^{2}(t) +

+(A_{1}^{2}(t)γ_{t}^{(1)} + 2A_{1}(t)A_{0}(t)γ_{t}^{(1,2)} + A_{0}^{2}(t)γ_{t}^{(2)})Δ]^{−1},

γ_{0}^{(2)} = γ_{2},  (22)

\frac{γ_{t}^{(1,2)}}{Δ} = a(t)γ_{t}^{(1,2)} + c(t)γ_{t}^{(1,2)} − (1 + a(t)Δ)(1 + c(t)Δ) ×

×(A_{1}(t)γ_{t}^{(1)} + A_{0}(t)γ_{t}^{(1,2)})(A_{1}(t)γ_{t}^{(1,2)} + A_{0}(t)γ_{t}^{(2)})[σ^{2}(t) +

+A_{1}^{2}(t)γ_{t}^{(1)} + 2A_{1}(t)A_{0}(t)γ_{t}^{(1,2)} + A_{0}^{2}(t)γ_{t}^{(2)})Δ]^{−1}.  (23)

Equation (21) can be reduced to the form

γ_{t} = ( ∏_{k=0}^{t−Δ}(1 + a(k)Δ)^{2}(σ^{2}(k) + A_{0}^{2}(k)γ̃_{k}Δ)[σ^{2}(k) +

+(A_{1}^{2}(k)γ_{k} + 2A_{1}(k)A_{0}(k)γ_{k}^{(1,2)} + A_{0}^{2}(k)γ_{k}^{(2)})Δ]^{−1}) ×

×[ ∑_{l=0}^{t−Δ} b^{2}(l)( ∏_{m=0}^{l}(1 + a(m)Δ)^{2}(σ^{2}(m) + A_{0}^{2}(m)γ̃_{m}Δ) ×

×[σ^{2}(m) + (A_{1}^{2}(m)γ_{m} + 2A_{1}(m)A_{0}(m)γ_{m}^{(1,2)} +

+A_{0}^{2}(m)γ_{m}^{(2)})Δ]^{−1})^{−1} + γ].  (24)

Using inequality (15) and the inequality resulting from (14)

A_{1}^{2}(t)γ_{t} + 2A_{1}(t)A_{0}(t)γ_{t}^{(1,2)} + A_{0}^{2}(t)γ_{t}^{(2)} ≤ p  (25)

we obtain from (24) E(θ_{t} − m_{t})^{2} = γ_{t} ≥ ψ(t), where

ψ(t) = ( ∏_{k=0}^{t−Δ}(1 + a(k)Δ)^{2}(σ^{2}(k) + q(k)Δ)(σ^{2}(k) + p(k)Δ)^{−1}) ×

×[ ∑_{l=0}^{t−Δ} b^{2}(l)( ∏_{m=0}^{l}(1 + a(m)Δ)^{2}(σ^{2}(m) + q(m)Δ) ×

×(σ^{2}(m) + p(m)Δ)^{−1})^{−1} + γ]

and since ψ(t) is a known function, we have

δ(t) ≥ ψ(t)  (26)

for all t ∈ S.

The equality in (26) is obtained when

A_{0}(t)γ̃_{t} = q(t),  (27)
A_{1}^{2}(t)γ_{t} + 2A_{1}(t)A_{0}(t)γ_{t}^{(1,2)} + A_{0}^{2}(t)γ_{t}^{(2)} = p(t).  (28)

Since

p(t) ≥ E(A_{0}(t)ξ̃_{t} + A_{1}(t)θ_{t})^{2} = E[A_{0}(t)Π(t, ξ) +

+A_{0}(t)m_{t}^{(2)} + A_{1}(t)m_{t}]^{2} + A_{1}^{2}(t)γ_{t} + 2A_{1}(t)A_{0}(t)γ_{t}^{(1,2)} + A_{0}^{2}(t)γ_{t}(2),

(28) implies (P -a.s.)

A_{0}(t)Π(t, ξ) + A_{0}(t)m_{t}^{(2)} + A_{1}(t)m_{t} = 0.  (29)

Consequently, (27),(28) and (29) are the relations from which optimal codings (A_{0}^{∗}, A_{1}^{∗}) and the transformator Π^{∗} are obtained. This completes the proof of the theorem.

Remark. As can be seen from relation (17), the optimal transformator of the back message Π_{0}^{∗} is constructed in such a way that the back message is multiplied by the value of some deterministic function of time at the moment t and m_{t}^{(2)} with a negative sign, i.e., the noise η_{t} is compensated by the best (in the sense of the square criterion) estimate m_{t}^{(2)} = E[η_{t}|F_{t}^{ξ}] (see optimal transmission scheme (18)).

To conclude this section we can show that condition (15) is indeed a power-type condition.

It can be easily seen that

E(A_{0}(t)ξ̃_{t} + A_{1}(t)θ_{t})^{2} = E[A_{0}(t)Π(t, ξ) + A_{0}(t)m_{t}^{(2)} + A_{1}(t)m_{t}]^{2} +

+A_{0}^{2}(t)γ_{t}^{(2)} + 2A_{0}(t)A_{1}(t)γ_{t}^{(1,2)} + A_{1}^{2}(t)γ_{t} = E[A_{0}(t)Π(t, ξ) + A_{0}(t)m_{t}^{(2)} +

+A_{1}(t)m_{t}]^{2} + A_{0}^{2}(t)γ̃_{t} + (A_{0}^{2}(t)γ_{t}^{(1,2)} + A_{1}(t)γ_{t})^{2}γ_{t}^{−1}  (30)

and since A_{0}^{2}(t)γ̃_{t} ≥ q(t), (30) implies

E[A_{0}(t)ξ̃_{t} + A_{1}(t)θ_{t}]^{2} ≥ q(t).  (31)

Consequently (15) implies power condition (31).

§ 2. Optimal transmission in a continuous case

Consider the following problem. A Gaussian message θ is transmitted through an additive white noise instantaneous feedback channel described by the following stochastic differential equation:

dξ_{t} = A(t, θ, ξ̃)dt + σ(t)dw_{t}  (32)

where w = (w_{t}, F_{t}) is a Wiener process. In contrast to the traditional schemes (see, e.g., [2]–[6]), the feedback here is not assumed to be noiseless.

The functional A in (32) defines the coding, and the back signal ξ̃ has the following form:

ξ̃_{t} = Π(t, ξ) + η_{t}  (33)

where Π is the transformator of the back signal, η is a noise in the back channel.

In this section optimal transmission schemes are constructed under certain power restrictions in a linear case when A(t, θ, ξ̃) = A_{0}(t)ξ̃_{t} + A_{1}(t)θ_{t} and it is proved that these particular linear schemes are also optimal in the general class given by (32),(33).

1. Let the transmitted message θ = (θ_{t}, F_{t}), t ∈ [0, T], be a Gaussian process described by the stochastic differential equation

dθ_{t} = a(t)θ_{t}dt + b(t)dv_{t},  (34)

where v = (v_{t}, F_{t}) is a Wiener process independent of the Gaussian N(m, γ), γ > 0, random variable θ_{0}, and |a(t)| ≤ k, |b(t)| ≤ k, where k is some constant.

Suppose that θ is transmitted according to the following linear scheme:

dξ_{t} = [A_{0}(t)ξ̃_{t} + A_{1}(t)θ_{t}]dt + σ(t)dw_{t}, ξ_{0} = 0,  (35)

where w = (w_{t}, F_{t}) is a Wiener process independent of v; A_{0}(t) and A_{1}(t) are the coding functions, σ(t) > 0. The back signal ξ̃ has the form (33). The noise in the back channel η admits the stochastic differential

dη_{t} = ā(t)η_{t}dt + b̄(t)dw̄_{t}  (36)

where w̄ = (w̄_{t}, F_{t}) is a Wiener process independent of w and v and of the Gaussian N(m_{2}, γ_{2}), γ_{2} > 0, random variable η_{0}, |ā(t)| ≤ k, |b̄(t)| ≤ k.

A class of admissible codings, transformator, and decodings is formed by such [(A_{0}, A_{1}), Π, θ̂] for which the following conditions are satisfied:

1) the stochastic differential equation (35) has a unique strong solution, sup_{t∈[0,T]} |A_{i}(t)| < ∞, i = 0, 1;

2) E[A_{0}(t)ξ̃_{t} + A_{1}(t)θ_{t}]^{2} ≤ p(t), (37)

A_{0}^{2}(t)γ̃_{t} ≥ q(t), (38) where p(t) and q(t) are some functions integrable on [0, T] and, for every t, q(t) ≤ p(t).

Let

Δ(t) = inf E[θ_{t} − θ̂_{t}(ξ)]^{2},  (39)

where inf is taken in the class of admissible [(A_{0}, A_{1}), Π, θ̂].

Theorem 3. When a Gaussian random process θ_{t} governed by a stochastic differential equation (34) is transmitted through a noisy feedback channel (35),(33),(36) under conditions 1) and 2) optimal in the sense of the square criterion (39), the coding functions A_{0}^{∗}, A_{1}^{∗} and the transformator of the back message Π^{∗} have the following form^{2}

A_{0}^{∗}(t) = (\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} , A_{1}^{∗}(t) = −(\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} γ̂_{t} + (\frac{p(t) − q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} ,

Π^{∗}(t, ξ^{∗}) = −m_{t}^{∗(2)} + \frac{A_{0}^{∗}(t)}{A_{1}^{∗}(t)}m_{t}^{∗},

where γ_{t}^{∗} is equal to the minimal message reproduction error

Δ(t) = γ_{t}^{∗} = γ exp\{2 ∫_{0}^{t} a(s)ds − ∫_{0}^{t} \frac{(p(s) − q(s))}{σ^{2}(s)}ds\} +

+ ∫_{0}^{t} b(s) exp\{2 ∫_{s}^{t} a(u)ds − ∫_{s}^{t} \frac{(p(u) − q(u))}{σ^{2}(u)}du\}ds,

and γ_{t}^{∗} and γ̂_{t}^{∗} are defined by the following equations:

\frac{dγ_{t}^{∗}}{dt} = b̄^{2}(t) + 2γ_{t}^{∗}(ā(t) − \frac{q(t)}{σ^{2}(t)}) + γ̂_{t}^{∗}(b^{2}(t) + 2a(t)γ_{t}^{∗}), γ_{0}^{∗} = γ_{2},

2The notation of §1 is used.

\frac{dγ̂_{t}^{∗}}{dt} = γ̂_{t}^{∗}[ā(t) − \frac{q(t)}{σ^{2}(t)} − b^{2}(t) − 2a(t)γ_{t}^{∗}] − A_{1}^{∗}(t)A_{0}^{∗}(t)\frac{γ̃_{t}}{σ^{2}(t)}, γ̂_{0}^{∗} = 0.

The optimal decoding m_{t}^{∗} satisfies the following stochastic differential equation:

dm_{t}^{∗} = a(t)m_{t}^{∗}dt + [γ_{t}^{∗}(p(t) − q(t))]^{\frac{1}{2}} σ^{−2}(t)dξ_{t}^{∗}, m_{t}^{∗} = m.

The optimal transmission is

dξ_{t}^{∗} = \{(\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} ξ̃_{t}^{∗} + [ − (\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} γ̂_{t} + (\frac{p(t) − q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} ]θ_{t}\} + σ(t)dw_{t}

or

dξ_{t}^{∗} = \{(\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} (η_{t} − m_{t}^{∗(2)}) + [ − (\frac{q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} γ̂_{t}^{∗} +

+(\frac{p(t) − q(t)}{γ_{t}^{∗}})^{\frac{1}{2}} ](θ_{t} − m_{t}^{∗})\}dt + σ(t)dw_{t}, ξ_{0}^{∗} = 0,

where m_{t}^{∗(2)} is defined by the equation

dm_{t}^{∗(2)} = ā(t)m_{t}^{∗(2)}dt + \{γ̂_{t}^{∗}[γ_{t}^{∗}(p(t) − q(t))]^{\frac{1}{2}} +
+[q(t)γ_{t}^{∗}]^{\frac{1}{2}} σ^{−2}(t)dξ_{t}^{∗}, m_{0}^{∗(2)} = m_{2}.

The proof of the theorem is similar to that of an analogous theorem (Theorem 2, §1) for the discrete case, and the equations of nonlinear filtering of conditionally Gaussian type processes (see [2]) are used.

Corollary. When a Gaussian N(m, γ), γ > 0, random variable θ is transmitted through the channel (35),(33),(36), the minimal message reproduction error is

Δ(t) = γ exp [ − ∫_{0}^{t} \frac{(p(s) − q(s))ds}{σ^{2}(s)}].

Now we shall consider the simplest case of a Gaussian N(m, γ) random variable θ transmission through a noisy feedback channel (32),(33) with A(t, θ, ξ̃) = ξ̃_{t} + A(t)θ. Let σ(t) ≡ 1.

For simplicity we assume that η_{t} = b̄w(t), where b̄ is some constant. Then the optimization problem is simplified and instead of obtaining optimal A_{0}, A_{1}, Π, θ̂ as in Part 1 of Section 1 we must find optimal A, Π, θ̂.

The necessity of condition (38) is eliminated but it should be required that p(t) ≥ γ̃_{t}, t ∈ [0, T].

The optimal A^{∗}, Π^{∗}, and θ̂^{∗} will have the following form:

A^{∗}(t) = −γ̂_{t}^{∗} + (\frac{p(t) − γ̃_{t}}{γ_{t}^{∗}} )^{\frac{1}{2}} ,

Π^{∗}(t, ξ^{∗}) = −m_{t}^{∗(2)} + [γ̂_{t}^{∗} − (\frac{p − γ̃_{t}}{γ_{t}^{∗}} )^{\frac{1}{2}} ]m_{t}^{∗}, θ̂_{t}^{∗} = θ̂_{t}^{∗}(ξ^{∗}) = m_{t}^{∗},

where m_{t}^{∗} admits the representation

dm_{t}^{∗} = [γ_{t}^{∗}(p(t) − γ̃_{t})]^{\frac{1}{2}} dξ_{t}^{∗}, m_{0}^{∗} = m

and γ̃_{t} and γ̂_{t} are found from the relations

\frac{dγ̃_{t}}{dt} = b̄^{2} − γ_{t}^{2}, γ̃_{0} = 0, γ̂_{t} = − ∫_{0}^{t} (\frac{p(s) − γ̃_{s}}{γ_{s}} )^{\frac{1}{2}} γ̃_{s}ds,

while m_{t}^{∗(2)} is found from the stochastic differential equation

dm_{t}^{∗(2)} = \{γ̂_{t}^{∗}[γ_{t}^{∗}(p(t) − γ̃_{t})]^{\frac{1}{2}} + γ̃_{t}\}dξ_{t}^{∗}, m_{0}^{∗(2)} = 0.

The minimal message reproduction error is

Δ(t) = γ exp [ − ∫_{0}^{t}(p(s) − γ̃_{s})ds].

In the case b̄ = 1 we have

γ̃_{t} = \frac{e^{2t} − 1}{e^{2t} + 1},

Δ(t) = γ exp [ − ∫_{0}^{t} p(s)ds]ch t,

where ch t is the hyperbolic cosine.

In the case b̄ = 0, i.e., when the noise η in the back channel is absent, we have

Δ(t) = γ exp ( − ∫_{0}^{t} p(s)ds),

which coincides with the transmission through a feedback noiseless channel (see [2]) and our optimal Π and A coincide with the optimal codings A_{0} and A_{1} by using the notation of [2], i.e., in this case the transformator Π can be placed in the coding device by virtue of a noiseless feedback.

2. Consider the transmission of the Gaussian process described by the stochastic differential equation (34) through the channel (32),(33). Let

Ã(t, θ_{t}, ξ) = E[A(t, θ_{t}, ξ̃)|F_{t}^{θ_{t},ξ}], F^{θ_{t},ξ} = σ{θ_{t}, ξ_{s}, s ≤ t},
Ā(t, ξ) = E[A(t, θ_{t}, ξ̃)|F_{t}^{ξ}].

Assume that the following conditions are satisfied:

1) Equation (1) has a unique strong solution,

2) EA^{2}(t, θ_{t}, ξ̃) ≤ p(t), E[A(t, θ_{t}, ξ̃) − Ã(t, θ_{t}, ξ)]^{2} ≥ q(t) (40) where p(t) and q(t) are some functions integrable on [0, T] and for every t p(t) ≥ q(t).

Let I_{T} (θ, ξ) be the mutual information of signals θ and ξ and let I_{T} (θ̃, ξ) be the mutual information of θ̃ = (θ_{t}, η_{t}) and ξ, t ∈ [0, T].

Lemma 1. The mutual informations I_{T} (θ, ξ) and I_{T} (θ̃, ξ) have the following forms (see [7]):

I_{T} (θ, ξ) = \frac{1}{2} ∫_{0}^{T} E[Ã^{2}(t, θ_{t}, ξ) − Ā^{2}(t, ξ)]σ^{−2}(t)dt,  (41)
I_{T} (θ̃, ξ)−I_{T} (θ, ξ)= \frac{1}{2} ∫_{0}^{T} E[A(t, θ_{t}, ξ̃)−Ã(t, θ_{t}, ξ)]^{2}σ^{−2}(t)dt. (42)

Corollary. Under conditions 1) and 2) we have

I_{T} (θ̃, ξ) ≤ \frac{1}{2} ∫_{0}^{T} \frac{p(t)}{σ^{2}(t)}dt, I_{T} (θ̃, ξ) − I_{T} (θ, ξ) ≥ \frac{1}{2} ∫_{0}^{T} \frac{q(t)}{σ^{2}(t)}dt.

Let I_{t} = sup I_{t}(θ, ξ), where sup is taken in the class of all admissible transmission schemes (32),(33), i.e. schemes for which conditions 1) and 2) are satisfied. Then from the corollary of the lemma we have

I_{t} ≤ \frac{1}{2} ∫_{0}^{t} \frac{p(s) − q(s)}{σ^{2}(s)}ds.

For the linear case A(t, θ_{t}, ξ̃) = A_{0}(t)ξ̃_{t} + A_{1}(t)θ_{t}, since Ā(t, ξ) = 0 we have

I_{t}(θ, ξ^{∗}) = \frac{1}{2} ∫_{0}^{t} EÃ^{2}(s, θ_{s}, ξ^{∗})ds = \frac{1}{2} ∫_{0}^{t} \frac{p(s) − q(s)}{σ^{2}(s)}ds.

Hence the following theorem is true.

Theorem 4. The optimal codings A_{0}^{∗}, A_{1}^{∗}, the decoding m_{t}^{∗} and the transformator Π^{∗} constructed in Theorem 3 are also optimal in the sense of maximum of mutual information.

3. Finally we prove the following

Theorem 5. When a Gaussian N(m, γ), γ > 0, random variable θ is transmitted according to the transmission scheme (32),(33) under conditions 1) and 2), the minimal reproduction error δ(t) = inf_{(A,Π,θ̂)} E(θ −θ̂_{t}(ξ))^{2} has the following form:

δ(t) = Δ(t) = γ exp { − ∫_{0}^{t}(p(s) − q(s))σ^{−2}ds},

where Δ(t) is the minimal message reproduction error for the optimal linear transformation constructed in Theorem 3, i.e. among all admissible schemes the transmission constructed in Theorem 3 is optimal in the sense of the square criterion.

Proof. Since δ(t) ≤ Δ(t), the theorem will be proved if we show that

δ(t) ≥ γ exp { − ∫_{0}^{t}(p(s) − q(s))σ^{−2}ds}.  (43)

Let θ̂ = θ̂_{t}(ξ) be some decoding. Then by Lemma 16.8 from [2] we have

E(θ − θ̂_{t}(ξ))^{2} ≥ γe^{−2I(θ,θ̂_{t}(ξ))}.

But I(θ, θ̂_{t}(ξ)) ≤ I_{t}(θ, ξ), and according to Theorem 4

I_{t}(θ, ξ) ≤ I_{t}(θ, ξ^{∗}) = \frac{1}{2} ∫_{0}^{t}(p(s) − q(s))σ^{−2}(s)ds.

Hence inequality (43) holds and Theorem 5 is true.

References

1. O.A.Glonti, Investigations in the theory of conditionally Gaussian processes.(Russian) Metsniereba, Tbilisi, 1985.

2. R.Sh.Liptser and A.N.Shiryayev, Statistics of random processes: I. General Theory, II. Applications. Springer, Berlin, 1977, 1978 (transl. from Russian, Nauka, Moscow, 1974).

3. S.Ihara, Coding theory in white Gaussian channel with feedback. J. Multivariate Anal. 4(1974), 74-87.

4. S.Ihara, Coding theory in Gaussian channel with feedback. II. Evaluation of the filtering error. Nagoya Math. J. 58(1975), 127-147.

5. P.K.Katyshev, Transmission of Gaussian process over a feedback channel. (Russian) Problemi Peredachi Informatsii 12(1976), No. 1, 112-118.

6. R.S.Liptser, Optimal coding and decoding in transmission of Gaussian Markov message through channel with noiseless feedback. (Russian) Problemi Peredachi Informatsii 10(1974), No. 4, 3-15.

7. O.A.Glonti, On mutual information of signals in the case of transmission over a “noisy” feedback channel. (Russian) Teor. Veroyatnost. i primenen. 23(1978), No. 2, 395-397; English translation: Theory Probab. Appl. 23(1978), 377-379.

(Received 25.12.1992)

Author’s address:

Laboratory of Probabilistic and Statistical Methods,

I.Javakhishvili Tbilisi State University,

150a David Agmashenebeli Ave., Tbilisi 380012

Republic of Georgia