基于线性正则变换的信号采样理论与重构方法研究

The Fourier transform (FT) plays a fundamental role in classical stationary signal analysis and processing. However, with the increasing extension of application scope and the deepening of research, the FT gradually exposed its limitations on non-stationary signal analysis and processing. These limitations have, in return, become the engine to generalize or even modify the FT and promote the continuous development of signal processing. A series of novel signal transforms also came into being. Among of them, the linear canonical transform (LCT), which is developed on the basis of the FT, cannot only include many classical "old" transforms in signal processing, but also reveal phenomena that "old" transforms cannot explain, and it leads to varies of new applications. Sampling theory as the basis of the LCT's engineering applications, has drawn much attention in recent years. In order to solve the problems of the existing LCT-based sampling theory, based on the practical application scenario, this project has revealed the sampling and reconstruction mechanism of the LCT from four aspects: signal structure, sampling strategy, reconstruction mechanism and noise influence. The theory of non-uniform sampling and local average sampling without band-limiting constraints is established, and the sampling theory under noisy background is also developed. Moreover, the theory of compressive sensing for analog signals based on the LCT is proposed. Our derived results laid the theoretical foundation for enhancing the breadth and depth of the LCT's applications in engineering. So, the project has great theoretical significance and application value.

傅里叶变换奠定了经典平稳信号分析与处理的基础。然而，随着应用的扩展和研究的深入，傅里叶变换逐渐暴露出其在非平稳信号分析与处理上的局限性。这些局限性又反过来成为推广乃至变革傅里叶变换的引擎，带动信号处理技术不断更新发展，一系列新型信号变换应运而生。其中，在傅里叶变换基础上发展起来的线性正则变换作为信号处理中许多经典“旧”变换的广义形式，能够揭示“旧”变换不能解释的现象，牵引出许多新的应用。采样理论作为线性正则变换工程应用的基础，近年来备受关注。本项目针对线性正则变换现有采样理论存在的问题，立足实际应用场景，从信号结构、采样策略、重构机制及噪声影响四方面揭示了线性正则变换采样与重构机理，构建了无带限条件约束的非均匀和局部平均采样理论，建立了噪声背景下采样理论，提出了基于线性正则变换的模拟信号压缩感知理论，为推进线性正则变换工程应用的广度和深度奠定了理论基础，具有重要的理论意义和应用价值。

傅里叶变换是平稳信号分析和处理的强有力的工具，在科学研究与工程技术的各个领域发挥着重要的作用。然而，随着研究对象和应用范围的不断扩展，如何解决非平稳信号的分析和处理问题成为限制电子信息系统进一步提升性能的瓶颈。为此，人们对傅里叶变换进行推广乃至变革，提出并发展了一系列新的信号变换，包括短时傅里叶变换、时频分布、小波变换、分数阶傅里叶换、线性正则变换等。其中，线性正则变换作为傅里叶变换和分数阶傅里叶变换等常用信号变换的广义形式，能够解释“旧”变换不能解释的现象，牵引出许多新的应用，近年来受到越来越多的关注。在信号分析与处理中，引入变换的目的是使信号中蕴含的信息变得显然。然而，模拟是自然界的本质，实际的物理信号通常都是模拟的。因此，在深化和推进线性正则变换的实际工程应用中，需要解决的首要问题便是模拟信号的数字化问题。鉴于此，本项目发现并揭示了线性正则变换域一般函数空间的数学结构，构建了无带限约束的线性正则变换域采样理论与重构方法，并从信号结构、采样策略、重构机制和噪声影响四个方面揭示了信号采样与重构机理，得到了采样参数和重构误差的解析表达及理论界。首先，针对实际应用中线性正则变换非均匀采样理论仅适合带限信号的问题，建立了无带限约束的线性正则变换非均匀采样理论。其次，考虑到实际应用中因物理上的限制只能获取信号局部平均采样值的问题，提出了线性正则变换函数空间中的局部平均采样理论。此外，构建了噪声背景下线性正则变换采样理论，解决了实际应用中利用含噪采样值进行信号重构的问题。最后，为克服经典模拟信号压缩感知理论仅适合傅里叶变换域多音和多带信号的不足，构建了线性正则变换域模拟信号压缩感知理论。截止目前，基于项目部分研究成果，发表学术论文12篇，其中SCI检索10篇，EI检索11篇；出版学术专著1部；申请国家发明专利9项，已授权6项。获得黑龙江省自然科学二等奖1项。通过凝练和升华项目科学研究问题，新增科研项目2项，获得科研经费136万元。