Optimal shaping properties of the truncated polydisc

induced by the linear feedback scheme X; = BjY " + y " , j = 1,2;.., m, that may be used to characterize the capacity region. The general outer bound of Section 111 ((19)) might be relevant here. This technique may also provide insight in determining the capacity region of general m-user Gaussian white noise multiple access channels with feedback. (0 = h (2 "). (All Here (i) is the chain rule, (ii) merely adds functions of the conditions, (rii) removes the conditionally deterministic constants XIL, (I U) uses the conditional independence ofdent and also independent of the noise. We begin by using Fano's inequality to show that for any (a)-< H(w(s)(w(S))-H(w(s)IY " , ~ (3)) + ne,, = z(w(s); Y " Iw(S)) + nefl = h (y n l ~ (S))-~(Y " Iw(s), ~ (3)) + ne,, (b) = h (y n l ~ (S))-h (Z ") + nefl = h X;-(C) Here (a) follows from the fact that W (S) and W (S) are independent and from the conditioning inequality h(AIB) 2 h(AIB, C) , (b) follows from Lemma 3.1, and (c) follows from Lemma 3.2 with If Z " is multivariate Gaussian, (A3) follows from the expression for the entropy of a Gaussian vector and from the entropy maximizing property of the normal: A coding scheme for additive noise channels with feedback-Part 1. No bandwidth constraints, " IEEE Trans. The capacity region for a multiple access discrete memoryless channel can increase with feedback, " IEEE Trans. An achievable rate region for the multiple access channel with feedback, " IEEE Trans. Inform. The feedback capacity of a class of discrete memoryless multiple access channels, " IEEE Trans. S. Verdh, " Multiple-access channels with memory with and without frame synchronism, " IEEE Trans.