Finite-State Channels With Feedback and State Known at the Encoder

We consider finite-state channels (FSCs) with feedback and state information known causally at the encoder. This setting is quite general and includes: a memoryless channel with i.i.d. state (the Shannon strategy), Markovian states that include look-ahead (LA) access to the state and energy harvesting. We characterize the feedback capacity of the general setting as the directed information between auxiliary random variables with memory to the channel outputs. We also propose two methods for computing the feedback capacity: (i) formulating an infinite-horizon average-reward dynamic program; and (ii) a single-letter lower bound based on auxiliary directed graphs called
$Q$
-graphs. We demonstrate our computation methods on three examples. In the first example, we introduce a channel with LA and establish a closed-form, analytic lower bound on its feedback capacity. Furthermore, we extend the channel with general parameters, and derive numerical lower bounds for each parameter. In the second example, we show that the mentioned methods achieve the feedback capacity of known unifilar FSCs such as the Ising channel. Finally, in the last example, we generalize the Ising channel such that the state is stochastically dependent on the input, and investigate its feedback capacity.