8. Transmission of a Gaussian Signal through a Channel with Noiseless Feedback. Optimal Coding and Decoding

We consider X as a signal which has to be transmitted through a noise channel. Assume also that the exit (Yt)t≥0 of the noisy channel is observable. Three parameters: the time t, the signal X, and the observation Y t 0 might me used for creating of a code A(t,X, Y t 0 ) for X. In other words, at the time t, instead of X, we transmit a new signal A(t,X, Y t 0 ) generated by t, X, and observation Y t 0 = {Ys, 0 ≤ s ≤ t}. The transmitted signal is interrupted by Gaussian white noise: A(t,X, Y t 0 ) + “white noise”, i.e. dYt = A(t,X, Y t 0 )dt + σdWt, Y0 = 0, where Wt is Wiener process independent of X and σ is the noise intensity. Our task now is to choose a coding functional A(t,X, Y t 0 ) such that the signal X is transmitted with a good accuracy. It would be noted that without any restrictions such a setting of problem is useless. In fact, with N , being essentially larger than σ2, taking A(t,X, Y t 0 ) = NX and letting the filtering estimate