This article studies the zero-error feedback capacity of causal discrete channels with memory. First, by extending the classical zero-error feedback capacity concept, a new notion of uniform zero-error feedback capacity $C_{0f} $ for such channels is introduced. Using this notion a tight condition for bounded stabilization of unstable noisy linear systems via causal channels is obtained, assuming no channel state information at either end of the channel. Furthermore, the zero-error feedback capacity of a class of additive noise channels is investigated. It is known that for a discrete channel with correlated additive noise, the ordinary capacity with or without feedback is equal $\log q-\mathcal {H}_{ch} $ , where $\mathcal {H}_{ch} $ is the entropy rate of the noise process and $q $ is the input alphabet size. In this paper, for a class of finite-state additive noise channels (FSANCs), it is shown that the zero-error feedback capacity is either zero or $C_{0f} =\log q -h_{ch} $ , where $h_{ch} $ is the topological entropy of the noise process. A condition is given to determine when the zero-error capacity with or without feedback is zero. This, in conjunction with the stabilization result, leads to a “Small-Entropy Theorem”, stating that stabilization over FSANCs can be achieved if the sum of the topological entropies of the linear system and the channel is smaller than $\log q$ .

中文翻译：

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有界稳定和有限状态加性噪声通道的零误差反馈能力

本文研究了零误差反馈能力具有记忆的因果离散通道。首先，通过扩展经典的零误差反馈容量概念，一个新的概念均匀的零误差反馈能力 $C_{0f} $对于此类渠道进行介绍。使用这个概念，假设通道两端没有通道状态信息，通过因果通道获得了不稳定噪声线性系统的有界稳定的严格条件。此外，研究了一类加性噪声信道的零误差反馈能力。已知对于具有相关附加噪声的离散信道，有或没有反馈的普通容量是相等的 $\log q-\mathcal {H}_{ch} $ ， 在哪里 $\mathcal {H}_{ch} $是噪声过程的熵率， $q$是输入字母大小。在本文中，对于一类有限状态加性噪声通道（FSNCs），表明零误差反馈容量为零或 $C_{0f} =\log q -h_{ch} $ ， 在哪里 $h_{ch} $是个噪声过程的拓扑熵。给出了一个条件来确定有或没有反馈的零误差容量何时为零。这与稳定结果一起导致了“小熵定理”，指出如果线性系统和通道的拓扑熵之和小于 FSANC，则可以实现稳定。 $\log q$ .