Hermite域和Fourier域稀疏信号重构算法研究
On reconstruction algorithms for signals sparse in Hermite and Fourier domains
摘要：
本论文是在数字信号处理领域的原创性贡献。在不同变换域上稀疏（高度集中）信号的重构是本文分析的主要问题。所考虑的域包括傅立叶，离散厄米，一维和二维离散余弦变换，以及各种时频表示。稀疏信号重建使用稀疏性措施，是，事实上，在所考虑的域中的信号浓度的措施。本文分析了压缩感知重构算法，并提出了新的解决方法。详细研究了丢失样本对分析变换域的影响，建立了与一般压缩感知理论的联系。这项研究提供了新的见解所产生的现象，由于信号样本的数量减少。理论上的贡献包括新的精确的数学表达式，描述重建算法的性能和结果，还包括研究的影响，加性噪声，稀疏水平和可用的测量的重建性能，重建误差和错误概率的精确表达式。研究了离散Hermite变换的参数优化问题，以及加性噪声对Hermite系数的影响，提出了新的参数优化和去噪算法。此外，还介绍了一种多变量多分量信号的分解算法，以及一种基于Wigner分布的瞬时频率估计算法。大量的数值算例和实验验证了所提出的理论，并揭示了新的实际应用的结果。
英文摘要：
This thesis consists of original contributions in the area of digital signal processing. The reconstruction of signals sparse (highly concentrated) in various transform domains is the primary problem analyzed in the thesis. The considered domains include Fourier, discrete Hermite, one-dimensional and two-dimensional discrete cosine transform, as well as various time-frequency representations. Sparse signals are reconstructed using sparsity measures, being, in fact, the measures of signal concentration in the considered domains. The thesis analyzes the compressive sensing reconstruction algorithms and introduces new approaches to the problem at hand. The missing samples influence on analyzed transform domains is studied in detail, establishing the relations with the general compressive sensing theory. This study provides new insights on phenomena arising due to the reduced number of signal samples. The theoretical contributions involve new exact mathematical expressions which describe performance and outcomes of reconstruction algorithms, also including the study of the influence of additive noise, sparsity level and the number of available measurements on the reconstruction performance, exact expressions for reconstruction errors and error probabilities. Parameter optimization of the discrete Hermite transform is also studied, as well as the additive noise influence on Hermite coefficients, resulting in new parameter optimization and denoising algorithms. Additionally, an algorithm for the decomposition of multivariate multicomponent signals is introduced, as well as an instantaneous frequency estimation algorithm based on the Wigner distribution. Extensive numerical examples and experiments with real and synthetic data validate the presented theory and shed a new light on practical applications of the results.
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