We investigate the coding scheme and error probability in information transmission over continuous-time additive Gaussian noise channels with feedback. As is known, the error probability can be substantially reduced by using feedback, namely, under the average power constraint, the error probability may decrease more rapidly than the exponential of any order. Recently Gallager and Nakibolu proposed, for discrete-time additive white Gaussian noise channels, a feedback coding scheme such that the resulting error probability Pe(N) at time N decreases with an exponential order αN which is linearly increasing with N. The multiple-exponential decay of the error probability has been studied mostly for white Gaussian channels, so far. In this paper, we treat continuous-time Gaussian channels, where the Gaussian noise processes are not necessarily white nor stationary. The aim is to prove a stronger result on the multiple-exponential decay of the error probability. More precisely, for any positive constant α, there exists a feedback coding scheme such that the resulting error probability Pe(T) at time T decreases more rapidly than the exponential of order αT as T→∞.

We investigate the error exponent in the lossy source coding with a fidelity criterion. Marton (1974) established a formula of the reliability function for the stationary memoryless source with finite alphabet. In this paper, we consider a stationary memoryless source assuming that the alphabet space is a metric space and not necessarily finite nor discrete. Our aim is to prove that Marton's formula for the reliability function remains true even if the alphabet is general.

We are interesting in the error exponent for source coding with fidelity criterion. For each fixed distortion level Δ, the maximum attainable error exponent at rate R, as a function of R, is called the reliability function. The minimum rate achieving the given error exponent is called the minimum achievable rate. For memoryless sources with finite alphabet, Marton (1974) gave an expression of the reliability function. The aim of the paper is to derive formulas for the reliability function and the minimum achievable rate for memoryless Gaussian sources.

We study the reliability functions or the minimum r-achievable rates of the lossless coding for the general sources in the sense of Han-Verdu, where r means the exponent of the error probability. Han has obtained formulas for the minimum r-achievable rates of the general sources. Our aim is to give alternative expressions for the minimum r-achievable rates. Our result seems to be a natural extension of the known results for the stationary memoryless sources and Markov sources.