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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (1): 88-110.doi: 10.3969/j.issn.1005-3085.2024.01.006

• • 上一篇    下一篇

局部平均随机过程随机场采样定理

张  硕   

  1. 天津财经大学理工学院，天津 300222
  • 收稿日期:2022-11-22 接受日期:2023-03-20 出版日期:2024-02-15 发布日期:2024-04-15
  • 基金资助:
    国家自然科学基金 (61901294).

Sampling Theorem for Stochastic Process and Random Field from Local Average

ZHANG Shuo   

  1. School of Science and Technology, Tianjin University of Finance and Economics, Tianjin 300222
  • Received:2022-11-22 Accepted:2023-03-20 Online:2024-02-15 Published:2024-04-15
  • Supported by:
    The National Natural Science Foundation of China (61901294).

PDF (PC)

147

摘要：

随机信号采样定理的相关研究几十年来一直是采样定理主流方向之一，在理论和实际应用两方面都受到信息领域和数学领域的广泛关注。近年来，采样理论在随机过程和随机场的局部平均采样方面得出了一系列研究成果。首先，分析了采样定理在应用中的缺陷和局限性。其次，对近几年在随机过程、高维随机过程、随机场、时空随机场局部平均采样定理方面的发展和相关结论做了系统介绍。最后，对基于混合范数的时空随机场采样逼近、采样形式、信号形式、范数形式等问题进行了展望。结果表明：局部平均的采样方式提高了采样的精度和逼近效果，随机场局部平均采样理论在海浪参数反演和海洋污染监测方面有重要应用。特别是基于混合范数的时空随机场采样理论的研究不仅具有重要的理论价值，而且对于工程领域也有应用价值。

关键词: 随机过程, 采样定理, 局部平均采样, 随机场, 时空随机场

Abstract:

The research of random signal sampling has become one of the mainstream directions in sampling theory in the past few decades, it has been widely concerned in the field of information and mathematics in both theory and practical application. In recent years, a series of research results on local average sampling of random processes and random fields have been obtained in the sampling theory. Firstly, the defects and limitations of sampling theorem in its application are analyzed. Secondly, the development and related results for the local average sampling theorem of stochastic process, multidimensional stochastic process,random field and spatio-temporal random field are introduced systematically. Finally, the further research directions of the spatiotemporal random field sampling problem are pointed out. The results show that the local average sampling method improves the sampling accuracy and approximation effect, and the random field local average sampling theory has important applications in wave parameter inversion and marine pollution monitoring. In particular, the research of the spatiotemporal random field sampling theory based on mixed norm not only has important theoretical value, but also has application value in engineering field.

Key words: stochastic process, sample theorem, local average sampling, random field, spatiotemporal random field

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