On the feedback capacity of the first-order moving average Gaussian channel

Abstract

In this paper, we study the discrete-time additive Gaussian noise channel given by \(Y_i=X_i+Z_i\), \(i=1,2,\ldots ,\) where the input signal \(\{X_i\}\) satisfies an average power constraint and the noise \(\{Z_i\}\) is a stationary Gaussian process. We are interested in the capacity of the channel with feedback. Despite numerous lower and upper bounds having been reported, except for some special cases, it is an open problem to find the closed form for the feedback capacity. In the paper, we consider the case where the Gaussian noise \(\{Z_i\}\) is a first-order moving average process defined by \(Z_i=W_i+\alpha W_{i-1}\), \(|\alpha |\le 1\), with white Gaussian innovations \(W_i\), \(i=0,1,2,\ldots\). For this channel, Kim (IEEE Trans Inf Theory 52(7):3063–3079, 2006; IEEE Trans Inf Theory 56(1):57–85, 2010) showed that a modified Schalkwijk–Kailath scheme achieves the feedback capacity and obtained the closed form of the feedback capacity. The main aim of the paper is to give a new proof of the optimality of the coding scheme given by Kim. In our proof, decompositions of the input signal \(X_i\) and the Gaussian innovation \(W_i\) into independent components play important roles. For this channel, the minimum decoding error probability decreases with an exponential order which linearly increases with block length.