This paper investigates the capacity of finite-state channels (FSCs) with feedback. We derive an upper bound on the feedback capacity of FSCs by extending the duality upper bound method from mutual information to the case of directed information. The upper bound is expressed as a multi-letter expression that depends on a test distribution on the sequence of channel outputs. For any FSC, we show that if the test distribution is structured on a $Q$-graph, the upper bound can be formulated as a Markov decision process (MDP) whose state being a belief on the channel state. In the case of FSCs and states that are either unifilar or have a finite memory, the MDP state simplifies to take values in a finite set. Consequently, the MDP consists of a finite number of states, actions, and disturbances. This finite nature of the MDP is of significant importance, as it ensures that dynamic programming algorithms can solve the associated Bellman equation to establish analytical upper bounds, even for channels with large alphabets. We demonstrate the simplicity of computing bounds by establishing the capacity of a broad family of Noisy Output is the State (NOST) channels as a simple closed-form analytical expression. Furthermore, we introduce novel, nearly optimal analytical upper bounds on the capacity of the Noisy Ising channel.

翻译：本文研究了带有反馈的有限状态信道（FSC）的容量。我们通过将对偶上界方法从互信息扩展至有向信息，推导了FSC反馈容量的一个上界。该上界表示为依赖于信道输出序列上测试分布的多字母表达式。对于任意FSC，我们证明若测试分布在$Q$图上具有结构，则该上界可表述为一个马尔可夫决策过程（MDP），其状态为信道状态的置信度。在FSC及其状态为单值性或具有有限记忆的情况下，MDP状态简化为有限集合中的取值。因此，该MDP由有限数量的状态、动作和扰动构成。这一有限性质至关重要，因为它确保动态规划算法能够求解相应的贝尔曼方程，从而为即使是大字母表的信道建立解析上界。我们通过将一大类噪声输出状态（NOST）信道的容量表示为简洁的闭式解析表达式，展示了计算上界的简便性。此外，我们针对噪声伊辛模型的容量提出了新颖且近乎最优的解析上界。