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A method is developed to realize optimal channel input conditional distributions, which maximize the finite transmission feedback information (FTFI) capacity, often called $n$ -block length feedback capacity, by information lossless randomized strategies. The method is applied to compute closed form expressions for the FTFI capacity and feedback capacity, of nonstationary, nonergodic, unstable, multiple input multiple output Gaussian channels with memory on past channel outputs, subject to average transmission cost constraints of quadratic form in the channel inputs and outputs. It is shown that randomized strategies decompose into two orthogonal parts-an deterministic part, which controls the channel output process, and an innovation part, which transmits new information over the channel. Then a separation principle is shown between the computation of the optimal deterministic part and the random part of the optimal randomized strategies. Finally, the ergodic theory of linear-quadratic-Gaussian stochastic optimal control theory, is applied to identify sufficient conditions, expressed in terms of solutions to matrix difference and algebraic Riccati equations, so that the optimal control part of randomized strategies induces asymptotic stationarity and ergodicity, and feedback capacity is characterized by the per unit time limit of the FTFI capacity. The method reveals an interaction of the control and the information transmission parts of the optimal randomized strategies, and that whether feedback increases capacity, is directly related to the channel parameters and the transmission cost function, through the solutions of the matrix Riccati equations. For unstable channels, it is shown that feedback capacity exists and it is strictly positive, provided the power exceeds a critical threshold. [ABSTRACT FROM AUTHOR]

The Finite Transmission Feedback Information (FTFI) capacity is characterized for any class of channel conditional distributions ${\mathbf{P}}_{B_{i}|B^{i-1}, A_{i}}$ and ${\mathbf{P}}_{B_{i}|B_{i-M}^{i-1}, A_{i}}$ , where $M$ is the memory of the channel, $B^{i} \stackrel {\triangle }{=}\{B^{-1},B_{0}, \ldots, B_{i}\}$ are the channel outputs, and $A^{i} \stackrel {\triangle }{=}\{A_{0}, A_{1}, \ldots, A_{i}\}$ , are the channel inputs, for $i=0, \ldots, n$. The characterizations of FTFI capacity are obtained by first identifying the information structures of the optimal channel input conditional distributions ${\mathscr P}_{[0, n]} \stackrel {\triangle }{=}\big \{ {\mathbf{P}}_{A_{i}|A^{i-1}, B^{i-1}}: i=0, \ldots, n\big \}$ , which maximize directed information $C_{A^{n} \rightarrow B^{n}}^{FB} \stackrel {\triangle }{=}\sup _{\mathscr P_{[0, n]} } I(A^{n} \rightarrow B^{n}), \hspace {.2in} I(A^{n} \rightarrow B^{n}) \stackrel {\triangle }{=}\sum _{i=0}^{n} I(A^{i};B_{i}|B^{i-1})$. The main theorem states that, for any channel with memory $M$ , the optimal channel input conditional distributions occur in the subset satisfying conditional independence $\stackrel {\circ }{\mathscr P}_{[0, n]} \stackrel {\triangle }{=}\big \{ {\mathbf{P}}_{A_{i}|A^{i-1}, B^{i-1}}= {\mathbf{P}}_{A_{i}|B_{i-M}^{i-1}}: i=0, \ldots, n\big \}$ , and the characterization of FTFI capacity is given by $C_{A^{n} \rightarrow B^{n}}^{FB, M} \stackrel {\triangle }{=}\sup _{ \stackrel {\circ }{\mathscr P}_{[0, n]} } \sum _{i=0}^{n} I(A_{i}; B_{i}|B_{i-M}^{i-1})$. Similar conclusions are derived for problems with average cost constraints of the form $\frac {1}{n+1} {\mathbf{E}}\Big \{c_{0,n}(A^{n}, B^{n-1})\Big \} \leq \kappa,\,\, \kappa \geq 0$ , for specific functions $c_{0,n}(a^{n},b^{n-1})$. The feedback capacity is addressed by investigating $\lim _{n \longrightarrow \infty } \frac {1}{n+1}C_{A^{n} \rightarrow B^{n}}^{FB, M} $. The methodology utilizes stochastic optimal control theory, to identify the control process, the controlled process, and often a variational equality of directed information, to derive upper bounds on $I(A^{n} \rightarrow B^{n})$ , which are achievable over specific subsets of channel input conditional distributions ${\mathscr P}_{[0, n]}$ , which are characterized by conditional independence. The main results illustrate a direct analogy, in terms of conditional independence, of the characterizations of FTFI capacity and Shannon’s capacity formulae of memoryless channels. An example is presented to illustrate the role of optimal channel input process in the derivations of the direct and converse coding theorems. [ABSTRACT FROM AUTHOR]