# Information Structures of Capacity Achieving Distributions for Feedback Channels with Memory and Transmission Cost: Stochastic Optimal Control & Variational Equalities-Part I

Authors
Type
Preprint
Publication Date
Dec 14, 2015
Submission Date
Dec 14, 2015
Identifiers
arXiv ID: 1512.04514
Source
arXiv
The Finite Transmission Feedback Information (FTFI) capacity is characterized for any class of channel conditional distributions $\big\{{\bf P}_{B_i|B^{i-1}, A_i} :i=0, 1, \ldots, n\big\}$ and $\big\{ {\bf P}_{B_i|B_{i-M}^{i-1}, A_i} :i=0, 1, \ldots, n\big\}$, where $M$ is the memory of the channel, $B^n = \{B_j: j=0,1, \ldots, n\}$ are the channel outputs and $A^n=\{A_j: j=0,1, \ldots, n\}$ are the channel inputs. The characterizations of FTFI capacity, are obtained by first identifying the information structures of the optimal channel input conditional distributions ${\cal P}_{[0,n]} =\big\{ {\bf P}_{A_i|A^{i-1}, B^{i-1}}: i=1, \ldots, n\big\}$, which maximize directed information $C_{A^n \rightarrow B^n}^{FB} = \sup_{ {\cal P}_{[0,n]} } I(A^n\rightarrow B^n), \ \ \ I(A^n \rightarrow B^n) = \sum_{i=0}^n I(A^i;B_i|B^{i-1}) .$ The main theorem states, for any channel with memory $M$, the optimal channel input conditional distributions occur in the subset ${\cal Q}_{[0,n]}= \big\{ {\bf P}_{A_i|B_{i-M}^{i-1}}: i=1, \ldots, n\big\}$, and the characterization of FTFI capacity is given by $C_{A^n \rightarrow B^n}^{FB, M} = \sup_{ {\cal Q}_{[0,n]} } \sum_{i=0}^n I(A_i; B_i|B_{i-M}^{i-1})$ . Similar conclusions are derived for problems with transmission cost constraints. The methodology utilizes stochastic optimal control theory, to identify the control process, the controlled process, and 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. For channels with limited memory, this implies the transition probabilities of the channel output process are also of limited memory.