In information transmission over additive white Gaussian channels with feedback, the use of feedback link to improve the performance of communication systems has been studied by a number of authors. It is well known that the error probability in information transmission can be substantially reduced by using feedback, namely, under the average power constraint, the error probability decreases more rapidly than the exponential of any order. Recently, for discrete-time additive white Gaussian channels, Gallager and Nakiboğlu proposed a feedback coding scheme such that the resulting error probability Pe(N) decreases with an exponential order αN which is linearly increasing with block length N, where α is a positive constant. In this paper, we consider continuous-time additive white Gaussian channels with feedback. 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 → ∞.

In this paper, we discuss the coding schemes and error probabilities in information transmission over additive Gaussian noise channels with feedback, where the Gaussian noise processes are stationary but not necessarily white. In the case of the white Gaussian channel it is known that the minimum error probability, under the average power constraint, decreases faster than the exponential of any order. Recently Gallager and Nakiboğlu (2010) proposed a coding scheme for the white Gaussian channel and successfully showed the multiple-exponential decay of the error probability for all rates below capacity. This paper aims to prove that, without any special assumptions on the noise of the stationary Gaussian channel, the minimum error probability decreases multiple-exponentially fast. In general, no explicit formulas are known for the capacity of the stationary Gaussian channel. In this paper, we introduce a lower boundC*on the capacityC. Then we prove that the minimum error probability decreases multiple-exponentially fast for all rates belowC*, but not for all rates belowC. In the process of proving, the scheme proposed by Gallager and Nakiboğlu proves itself to be quite useful, even though the Gaussian channel is not white.

We consider a continuous-time white Gaussian channel with full feedback. The use of feedback link to improve the perfomance of communication systems has been studied by a number of authors. It is well known that the channel capacity is not increased by feedback, however error probability in information transmission can be substantially reduced by using feedback. The first remarkable work was done by Shalkwijk and Kailath (1966) where they proposed a feedback coding scheme and showed that the error probability goes down double exponentially fast. It was pointed out by Pinsker (1968) that one can construct a coding scheme for which the error probability decreases more rapidly than the exponential of any order. In this paper we propose a new feedback encoding scheme and a decoding scheme, and show that, under our coding sheme, the error probability goes down more rapidly than the exponential of any order.

We derive some information-theoretic inequalities to evaluate the channel capacity and mean-squared error. We prove an inequality for the capacity of an additive-noise channel with feedback. We also prove an inequality for mutual information and mean-squared error. The inequality is applied to bound minimum mean-squared transmission errors. The results for the channel capacity and the mean-squared error are expressed in terms of the capacity of the related Gaussian channel, the distortion rate function of the related Gaussian process, the mutual information, and the relative entropy.

The main aim of the paper is to prove a coding theorem for a continuous-time Gaussian channel with feedback, under an average power constraint. In the case of discrete-time, the coding theorem for the feedback Gaussian channel has been shown by Cover and Pombra (1989).>

The averaged information capacity of the mismatched stationary continuous-time Gaussian channel is considered, in the limit as the observation time becomes infinite. It is required only that the signal satisfy a reproducing kernel Hilbert space constraint on expected energy. This requires the, signal energy to be distributed in regions where the noise energy, is not too small, and is the weakest condition that provides finite capacity. In the case when both the noise covariance and the filter on the message are expressed by rational functions, the results complement those previously obtained by Gallager and Holsinger (1968), and the combination gives a complete solution to the capacity problem. The treatment provides a desirable generality on the class of transmitted signals that is not present in previous treatments.>

A continuous time stationary channel with additive noise presented byY(t) = X(t) + Z(t)is considered, where\{X(t)\}and\{Z(t)\}are mutually independent stationary processes representing the channel input and the noise, respectively, and\{Y(t)\}represents the channel output. It is shown that, under some general assumptions, the mutual information between the input\{ X(t); 0 \leq t \leq T \}and the output\{ Y(t); 0 \leq t \leq T \}can be expressed in the formAT + B(T)with a constantAand a functionB(T)given in terms of the conditional mutual informations between the "past" and the "future" given the "present" of the output and the noise. It is also shown that the remainder termB(T)converges to a constant asTtends to infinity.

The maximum entropy spectral analysis for discrete-time stationary processes was first proposed by J. P. Burg. He showed that if a finite number of covariance lag values of a stationary process are known, then an autoregressive (AR) process with the given autocorrelation values best fits the given constraints in the sense of maximizing thc differential entropy rate of the model. A more general type of prior knowledge of the process is considered, and it is shown that the maximum entropy method, subject to our constraints, is equivalent to fitting a mixed autoregressive moving average (ARMA) model.