constrained Systems and Coding for Recording Channels," to Appear as Book Chapter in Handbook for Coding Theory, Symbolic Dynamics to Coding, Automata and System Theory 8. Symbolic Dynamics and System Theory
constrained Systems and Coding for Recording Channels," to Appear as Book Chapter in Handbook for Coding Theory, Symbolic Dynamics to Coding, Automata and System Theory 8. Symbolic Dynamics and System Theory
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约束系统和记录通道编码”将作为书籍章节出现在《编码理论手册》、《编码符号动力学》、《自动机和系统理论》第 8 章《符号动力学和系统理论》中
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发表时间：
2007
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通讯作者：
J. Wolf
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作者：
J. Wolf
滑动分组码的算法|符号动力学在信息论中的一个应用，IEEE trans. then（w; x）j（？1;0）（w; x）j 0;1）2 B s。状态实现的外部行为是系统（Z; W; B），其中对于某个xg，B = fw：（w; x）2 B s。状态实现可以看作是其外部行为的一种表现，X的元素是状态。一个系统的状态实现与一个带标签的图的表示密切相关;事实上，任何带标签的图都可以被看作是一个状态实现。如果（粗略地说）通过合并”状态47]不能获得相同外部行为的其他实现，则外部行为的实现是最小的。Willems利用期货F（xj（？）1;0）），我们在第4节中描述的。我们在下面的定理8.1中说明这一点。时不变系统（Z; W; B）的过去诱导正则实现是状态实现p =。定理8.1（Willems 47]和其中的参考文献）。一个时不变系统有唯一的最小实现当且仅当它的未来诱导和过去诱导的正则实现在从一个实现的状态到另一个实现的状态的双射将一个实现转换为另一个的意义上是等价的。唯一极小实现是过去诱导的规范实现。注意这个结果与具有唯一的I-极小表示的不可约的π-移位的特征非常相似（定理4.2（1，2））。词典给出了系统论中的许多概念，这些概念与符号动力学中的概念密切相关。例如，一个输入-状态-输出系统被认为是一个受某些自然限制的状态实现（Z; U Y; X; B）（见本词典中的定义VI-1）。在这里，U、Y和X扮演输入、输出和状态的角色。这样的系统与符号动力学中的尼特等价和自动机理论中的转换器密切相关（如第7节所讨论的）。此外，可观测性的概念（定义II-3和II-4）非常类似于nite型移位的概念，可控性的概念（定义II-7和II-8）非常类似于nite型移位的不可约性的概念;参见字典中的注释。我们已经介绍了...
Algorithms for sliding block codes | an application of symbolic dynamics to information theory, IEEE Trans. then (w; x)j (?1;0) (w; x)j 0;1) 2 B s. The external behavior of a state realization is the system (Z; W; B) where B = fw : (w; x) 2 B s for some xg. The state realization may be regarded as a presentation of its external behavior, with the elements of X as states. A state realization of a system is closely related to a presentation of a sooc shift; in fact, any labeled graph may be regarded as a state realization. A realization of an external behavior is minimal if (roughly speaking) no other realization of the same external behavior can be obtained by merging" states 47]. Willems characterized the time-invariant systems with unique minimal realization, using the notion of futures F(xj (?1;0)) that we described in Section 4. We state this in Theorem 8.1 below. The past induced canonical realization of a time-invariant system (Z; W; B) is the state realization p = Likewise there is the notion of future induced canonical realization. Theorem 8.1 (Willems 47], and references therein). A time-invariant system has a unique minimal realization if and only if its future induced and past induced canonical realizations are equivalent in the sense that there is a bijection from states of one realization to states of the other that transforms one realization to the other. The unique minimal realization is the past induced canonical realization. Note how similar this result is to the characterization of irreducible sooc shifts with unique I-minimal presentation (Theorem 4.2 (1 , 2)). The Dictionary gives many concepts in system theory which are closely related to concepts in symbolic dynamics. For instance, an input-state-output system is deened to be a state realization (Z; U Y; X; B) subject to certain natural restrictions (see Deenition VI-1 in the Dictionary). Here, U; Y; and X play the role of input, output and state. Such systems are closely related to nite equivalences in symbolic dynamics and transducers in automata theory (as discussed in Section 7). Also, the notions of observability (Deenitions II-3 and II-4) are very similar to the idea of shift of nite type, and the notions of controllability (Deenitions II-7 and II-8) are very similar to the idea of irreducibility for sooc shifts; see the remarks in the Dictionary. We have given an introduction to …