In this paper, we investigate the capacity of finite-state channels (FSCs) in presence of delayed feedback. We show that the capacity of a FSC with delayed feedback can be computed as that of a new FSC with instantaneous feedback and an extended state. Consequently, graph-based methods to obtain computable upper and lower bounds on the delayed feedback capacity of unifilar FSCs are proposed. Based on these methods, we establish that the capacity of the trapdoor channel with delayed feedback of two time instances is given by $\log_2(3/2)$. In addition, we derive an analytical upper bound on the delayed feedback capacity of the binary symmetric channel with a no consecutive ones input constraint. This bound also serves as a novel upper bound on its non-feedback capacity, which outperforms all previously known bounds. Lastly, we demonstrate that feedback does improve the capacity of the dicode erasure channel.

翻译：本文研究了有限状态信道在存在延迟反馈时的容量问题。我们证明，带有延迟反馈的有限状态信道的容量可通过构建一个带有瞬时反馈及扩展状态的新有限状态信道来计算。据此，提出了基于图论的方法，用于获取单义有限状态信道在延迟反馈下可计算的容量上界与下界。基于这些方法，我们确定了锁孔信道在延迟两个时刻的反馈下的容量为$\log_2(3/2)$。此外，针对受无连续“1”输入约束的二进制对称信道，推导了其在延迟反馈下的解析上界。该上界同时作为其无反馈容量的新上界，优于所有已知的先前结果。最后，我们证明了反馈确实能提升双码擦除信道的容量。