基于矩阵增强的复杂噪声背景中谐波参数的超分辨率估计

Parameter estimation of harmonics in noise is a classic problem in the field of signal processing and is also an important content of the study of statistical signal processing. It has a very wide range of applications in many areas of signal processing. This project studies the super-resolution parameter estimation method of harmonics in complex noise and its statistical performance. We investigate the statistical characters of one-dimensional and two-dimensional harmonic signal in the case of different noise and study the super-resolution estimation method of harmonic component number and frequency of one-dimensional harmonics, harmonic component number and frequency pair of two-dimensional harmonics. We will give the computational complexity of the proposed methods, and also study the statistical performance of the proposed methods from the convergence and asymptotic distribution. The proposed the super-resolution parameter estimation method of one-dimensional and two-dimensional harmonics in complex noise have small amount of computation and high accuracy in the small sample cases, and good statistical performance in large sample cases. This study not only gives a practical method for the applications of one-dimensional and two-dimensional harmonic signal model, but also has important theoretical significance and application value for signal analysis and processing.

噪声中谐波参数估计问题是信号处理领域的一个经典问题，也是统计信号处理研究的一个重要内容，它在信号处理的许多领域里有着非常广泛的应用。本项目研究复杂噪声中谐波参数的超分辨率估计方法及其统计性能。针对复杂噪声中一维和二维谐波信号，深入分析它们在不同噪声情形下的统计特征，分别研究基于矩阵增强的一维谐波分量个数和频率、二维谐波分量个数和频率对的超分辨率估计方法，并对所提方法进行计算复杂度分析，同时还从收敛性和渐近分布两个方面对所提方法进行统计性能分析。以期提出复杂噪声中一维和二维谐波参数的既在小样本情形具有计算量小且精度高，又在大样本情形具有良好统计性能的超分辨率估计方法。本项目的研究不仅对于一维和二维谐波信号模型的应用提供切实可行的方法，而且对于信号分析与处理具有重要的理论意义和应用价值。

本项目主要研究复杂噪声背景中一维和二维谐波参数的超分辨率估计。运用概率论与数理统计、随机过程、高阶统计量以及循环平稳等理论，研究了复杂噪声中一维和二维谐波信号的统计特征。在此基础上，利用矩阵增强技术，对于复杂噪声背景中的一维谐波，提出了基于ESPRIT、基于奇异值分解和基于求根MUSIC的三种超分辨率参数估计算法；对于复杂噪声背景中的二维谐波，提出了基于二维ESPRIT和二维求根MUSIC的两种超分辨率参数估计算法。所提算法不但突破了现有基于Fourier 变换估计方法的瑞利限的限制，而且还具有计算复杂度低，估计精度高，且对噪声的颜色无限制等特点。对所提的五种超分辨率参数估计算法进行了统计性能分析，给出了其在大样本情况下的统计性能。在本项目执行期内，我们较好地完成了预定计划中的各项工作，达到了预期目标。本项目为今后进一步的研究工作打下了坚实基础，同时也为实践中复杂噪声背景中一维和二维谐波信号的参数估计提供了切实可行的方法，对于谐波恢复有一定的应用价值。

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