基于再生核理论的非线性自适应滤波器

Adaptive filter is widely used to solve many nonlinear processing problems in the real world owe to its adaption to changes of environment. However, the a priori knowledge about the processed systems and noise is required in adaptive filter, and its filtering performance may have a big difference in different nonlinear systems. To improve the universality and signal processing ability of nonlinear adaptive filter on unknown environments, this project mainly carries out researches on kernel adaptive filters and nonlinear Kalman filters by means of the reproducing kernel theory, which are shown as follows. (1) The operation of inner product is introduced into the filter feedback structure, and then novel feedback kernel adaptive filters (FKAFs) are designed by combing appropriate cost functions with universal kernel function. (2) To improve the online processing ability of FKAFs, bilateral online sparsification methods are constructed by utilizing the information hidden in input data and desired responses of training data. (3) The methods for reconstructing nonlinear functions and calculating moments are explored using the reproducing characteristic of kernel function. (4) The Bayesian rule in the reproducing kernel Hilbert spaces is designed using kernel function, and then the signal processing of unknown environment is implemented with the help of the pre-image techniques. The achievements of this project will effectively enhance the abilities of signal processing and intelligent learning of nonlinear adaptive filters, and provide important theoretical and practical foundations for developing novel intelligent information processing systems.

自适应滤波器能够适应环境变化，在解决客观世界的诸多非线性信号处理问题上具有较大优势。但它需要较多关于所处理系统和噪声的先验知识，存在不同非线性系统下滤波性能可能截然不同的问题。本项目拟利用再生核理论提高非线性自适应滤波器的普适性和对未知环境的信号处理能力，主要开展核自适应滤波器和非线性卡尔曼滤波器两类非线性自适应滤波器新方法的研究。在引入内积运算的反馈结构中构造新的代价函数和具有普适性的再生核函数，进而设计出新型反馈核自适应滤波器；为提高实时处理能力，结合训练数据中输入和期望响应两方面信息构建双边在线稀疏化方法。通过核函数的再生特性，探寻非线性函数的重构方法和矩计算的数值方法；利用核函数构建贝叶斯准则在再生核希尔伯特空间的数值表示方法，结合原像技术实现对未知环境的信号处理。项目成果将有效增强非线性自适应滤波器的信号处理和智能学习能力，为开发新型智能信息处理系统提供重要的理论支撑和实践依据。

本项目探索了核自适应滤波器和非线性卡尔曼滤波器的设计方法及其在非线性智能信号处理领域中的应用。项目研究成果主要包括以下四个方面。（1）鲁棒核自适应滤波器研究：从代价函数，优化方法及多核结构等方面探索了影响鲁棒核自适应滤波器滤波精度、收敛速度和环境适应性的关键因素，找到了构造具有鲁棒特性的核自适应滤波器的新方法。（2）核自适应滤波器的反馈结构设计：利用反馈结构实现对历史信息的信息挖掘，从而有效地去除噪声，进一步提升了核自适应滤波器的滤波精度，拓展了核自适应滤波器的应用范围。（3）固定维数的核自适应滤波研究：稀疏化虽然可以降低核自适应滤波器的网络复杂度，但是不能彻底解决网络结构线性增长的问题。因此提出了随机傅立叶特征和Nyström两类非线性映射方法固定了滤波网络的尺寸大小，在保证滤波精度的同时，大大降低了计算复杂度。（4）非线性卡尔曼滤波器设计：在观测噪声统计量未知的环境中，通过检测该噪声方差估计的变化趋势，并利用连续映射函数构建了方差估计的动态模型，进而设计出一类未知环境下的非线性卡尔曼滤波器。本项目的研究成果能够进一步推进基于再生核理论的非线性自适应滤波理论的发展，为自适应滤波器在时间序列预测、系统辨识、目标定位与跟踪等实际应用领域提供了有力的技术支撑。