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首页> 外文期刊>IEEE Transactions on Signal Processing >Extending Classical Multirate Signal Processing Theory to Graphs—Part I: Fundamentals
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Extending Classical Multirate Signal Processing Theory to Graphs—Part I: Fundamentals

机译：将经典的多速率信号处理理论扩展到图形—第一部分：基础知识

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Signal processing on graphs finds applications in many areas. In recent years, renewed interest on this topic was kindled by two groups of researchers. Narang and Ortega constructed two-channel filter banks on bipartitie graphs described by Laplacians. Sandryhaila and Moura developed the theory of linear systems, filtering, and frequency responses for the case of graphs with arbitrary adjacency matrices, and showed applications in signal compression, prediction, etc. Inspired by these contributions, this paper extends classical multirate signal processing ideas to graphs. The graphs are assumed to be general with a possibly nonsymmetric and complex adjacency matrix. The paper revisits ideas, such as noble identities, aliasing, and polyphase decompositions in graph multirate systems. Drawing such a parallel to classical systems allows one to design filter banks with polynomial filters, with lower complexity than arbitrary graph filters. It is shown that the extension of classical multirate theory to graphs is nontrivial, and requires certain mathematical restrictions on the graph. Thus, classical noble identities cannot be taken for granted. Similarly, one cannot claim that the so-called delay chain system is a perfect reconstruction system (as in classical filter banks). It will also be shown that M -partite extensions of the bipartite filter bank results will not work for M-channel filter banks, but a more restrictive condition called M-block cyclic property should be imposed. Such graphs are studied in detail. A detailed theory for M -channel filter banks is developed in a companion paper.
机译：图形上的信号处理在许多领域都有应用。近年来，两组研究人员对这个话题重新产生了兴趣。 Narang和Ortega在拉普拉斯主义者描述的二分图中构造了两个通道的滤波器组。 Sandryhaila和Moura针对具有任意邻接矩阵的图的情况开发了线性系统，滤波和频率响应的理论，并展示了其在信号压缩，预测等方面的应用。受这些贡献的启发，本文将经典的多速率信号处理思想扩展到了图。假设这些图是一般的，带有可能不对称且复杂的邻接矩阵。本文回顾了图多速率系统中的一些概念，例如贵族身份，混叠和多相分解。绘制与经典系统的平行关系可以使人们设计具有多项式滤波器的滤波器组，其复杂度低于任意图形滤波器。结果表明，经典的多速率理论向图的扩展是不平凡的，并且需要对图进行一定的数学约束。因此，经典的贵族身份不能被认为是理所当然的。类似地，不能断言所谓的延迟链系统是一种完美的重建系统（如在经典滤波器组中一样）。还将显示，二分滤波器组结果的M部分扩展不适用于M通道滤波器组，但应施加更具限制性的条件，称为M块循环特性。对这些图进行了详细研究。 M通道滤波器组的详细理论在随附的论文中得到了发展。

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