Upper Bounds on the Capacities of Noncontrollable Finite-State Channels with/without Feedback

Noncontrollable finite-state channels (FSCs) are FSCs in which the channel inputs have no influence on the channel states, i.e., the channel states evolve freely. Since single-letter formulae for the channel capacities are rarely available for general noncontrollable FSCs, computable bounds are usually utilized to numerically bound the capacities. In this paper, we take the delayed channel state as part of the channel input and then define the {\em directed information rate} from the new channel input (including the source and the delayed channel state) sequence to the channel output sequence. With this technique, we derive a series of upper bounds on the capacities of noncontrollable FSCs with/without feedback. These upper bounds can be achieved by conditional Markov sources and computed by solving an average reward per stage stochastic control problem (ARSCP) with a compact state space and a compact action space. By showing that the ARSCP has a uniformly continuous reward function, we transform the original ARSCP into a finite-state and finite-action ARSCP that can be solved by a value iteration method. Under a mild assumption, the value iteration algorithm is convergent and delivers a near-optimal stationary policy and a numerical upper bound.

💡 Research Summary

The paper tackles the long‑standing difficulty of characterizing the capacity of noncontrollable finite‑state channels (FSCs), where the channel input does not influence the evolution of the channel state. Because a single‑letter capacity expression is rarely available for such channels, researchers typically rely on numerically computed bounds. The authors introduce a novel perspective: they treat the delayed channel state as part of the channel input, thereby constructing an extended input sequence ((X_t, S_{t-d})). With this extension they define the directed information rate from the new input sequence to the output sequence, a quantity that naturally captures causal information flow even in the presence of feedback.

Using the directed information rate, the authors derive a hierarchy of upper bounds on the capacity of noncontrollable FSCs, applicable both with and without feedback. The key technical step is to restrict attention to conditional Markov sources, i.e., input processes whose distribution at time (t) depends only on a finite memory of past inputs, outputs, and the delayed state. Under this restriction the problem of maximizing the directed information rate can be reformulated as an average‑reward stochastic control problem (ARSCP). In this control formulation the system state consists of the delayed channel state together with the past output history, the control action is the conditional input distribution, and the stage reward is the instantaneous contribution to directed information.

A central contribution of the work is the proof that the reward function of the ARSCP is uniformly continuous with respect to both state and action. Uniform continuity enables the authors to approximate the original infinite‑dimensional problem by a finite‑state, finite‑action ARSCP through the construction of an (\varepsilon)-net (a discretization of the state and action spaces). The resulting finite problem can be solved by a standard value‑iteration algorithm, which iteratively applies the Bellman optimality equation. Under a mild assumption—namely that the reward is bounded and the transition kernel is continuous—the value‑iteration algorithm is shown to converge to a fixed point. The stationary policy obtained at convergence is a near‑optimal policy for the original problem, and the associated average reward provides a computable numerical upper bound on the channel capacity.

The framework naturally accommodates feedback: when feedback is present, the control action (the conditional input distribution) may depend on past outputs, and this dependence is already captured in the directed information formulation. Consequently, the same value‑iteration machinery yields upper bounds for both feedback and no‑feedback scenarios, allowing a direct quantitative comparison of the feedback gain.

From a practical standpoint, the proposed method offers several advantages. First, it transforms a notoriously intractable information‑theoretic optimization into a well‑studied stochastic control problem, for which a rich set of algorithmic tools exists. Second, the discretization error can be made arbitrarily small by refining the (\varepsilon)-net, giving designers explicit control over the trade‑off between computational complexity and bound tightness. Third, because the delayed state is treated as part of the input, the approach can be applied to a wide variety of channel models where the state evolves independently of the transmitted symbols—such as mobile fading channels, channels with environmental interference, or any scenario where the state dynamics are driven by external processes.

In summary, the paper makes three major contributions: (1) a new directed‑information‑based upper‑bound formulation for noncontrollable FSCs that works with or without feedback; (2) a rigorous reduction of the bound‑optimization to an average‑reward stochastic control problem with provably uniformly continuous rewards; and (3) a practical algorithmic solution via finite‑state value iteration that yields convergent near‑optimal policies and tight numerical capacity upper bounds. The work bridges information theory and stochastic control, opening the door for future extensions to more complex multi‑user or multi‑antenna finite‑state channel models.