基于输入信号方向和滤波结构自正交化的自适应方法研究

Ideally, one would like to have a computationally-simple and numerically-robust adaptive filter with high rate of convergence and small misadjustment that can be implemented easily on a computer. As in any engineering problem, these desirable characteristics, in most cases, are incompatible with each other and some kind of trade-off is needed. Algorithms such as self-orthogonalizing adaptation algorithms attempt to reduce the complexity, by trading-off on convergence rate. We aim at improving the performance (achieving better trade-off) by developing new adaptation algorithms based input vectors and by using "unconventional" structures for adaptive filters..The first part of this project attempts to improve the adaptive filter performance by refining the adaptation algorithm. The normalized least mean square is a very popular algorithm. Unfortunately, for highly colored input signals - with a covariance matrix that exhibits a large dynamic range of eigenvalues - this algorithm suffers from slow convergence. The affine projection algorithms have been proposed to ameliorate this problem. However, for the affine projection algorithms, the iteration direction is the direction vector, and the iteration error of the adaptive filter is caused by the input vector. These two directions are not the same, which leads more deviation for the iteration error. In order to solve this problem, we aim at developing new adaptation algorithm by redefining the iteration error and analyzing the statistical model for the affine projection algorithms and the new achieved adaptation algorithm..The second part of this work looks at improving the convergence rate by using different structures. The input-output characteristic of linear systems is classically described by a ratio of polynomials in shift operator notation. However, there exist an infinite number of equivalent descriptions with the same external behavior, but different internal behaviors. These different realizations have different sensitivity and different numerical properties under finite-word-length conditions. A balanced realization is known to minimize the parameter sensitivity as well as the ratio of maximum-to-minimum eigenvalues of the Grammian matrices. This suggests that the balanced realization will have good noise rejection characteristics. These properties of the balanced realization make it an attractive candidate as a structure for adaptive filtering. This work is aimed at using balanced realization for adaptive filters and reducing the structure complexity..In order to improve the adaptive filter performance, the third part considers the relations between the input vectors driven by Brownian motion and the balanced structures. The research of this project will further expand the direction concept of the input signal and the design method of the system structure. It will achieve the development of the system identification and equalization.

工程问题中自适应滤波算法的复杂度、鲁棒性、收敛性和失调量经常不相容，各性能之间需要权衡，自正交化的自适应方法提供了一种"权衡"的方法。本项目将基于自正交化的自适应方法，从输入信号方向、滤波结构、综合考虑输入信号和滤波结构三方面进行研究，以提高自适应滤波性能并实现有效的权衡。为了解决仿射投影算法迭代方向与引起估计输出误差的方向不一致问题，分析估计权值在迭代方向引起的误差，提出并建立新算法及其收敛性和跟踪性的随机统计模型；通过研究参数化离散时间系统的平衡实现，建立新的自适应滤波结构，使得系统可控与可观Grammian矩阵最大与最小特征值的比值和迭代参数的敏感度最小化，并有效降低计算复杂度；通过分析"布朗运动"激励的输入信号方向同平衡实现结构之间的联系，建立优化的自适应滤波算法。本项目的研究成果将进一步拓展输入信号方向的概念和系统结构的设计方法，对系统辨识、信道均衡等理论的发展具有重要意义。

工程问题中自适应滤波算法的复杂度、鲁棒性、收敛性和失调量经常不相容，各性能之间需要权衡，自正交化的自适应方法提供了一种"权衡“的方法，它能够改善收敛性等性能，但这种改善是以增加计算复杂度为代价。本项目基于自正交化的自适应方法，从输入信号方向、滤波结构等方面进行了研究，在提高自适应滤波器性能的同时，以实现有效的性能权衡。. 为了解决仿射投影算法迭代方向与引起估计输出误差的方向不一致问题，通过分析估计权值在自适应滤波器迭代方向引起的误差，提出了两种不同的新算法，使得自适应滤波器的迭代方向与迭代误差的引起方向完全一致，避免了自适应滤波器其它方向的干扰，并分析了信号方向的自由度等微观因素同宏观信号之间的联系，建立了权值误差和权值均方误差的递归迭代模型，获得了其均方误差收敛性和跟踪性的随机统计模型。同时通过分析仿射投影算法的迭代步长，获得了一种优化的迭代步长，促进了自适应滤波器的收敛性能，并分析了其收敛性的随机统计模型；进而通过分析所建立的两种新算法的迭代不长，获得了优化的迭代步长。. 平衡实现具有最小参数的敏感度，所以具有非常好的鲁棒性；即使由于噪声导致了参数的错误估计，它仍然能够描述接近实际系统的模型，所以也具有非常好的稳定性（失调量小）；并且最小化了系统可控与可观Grammian矩阵最大与最小特征值的比值，所以又具有非常好的收敛性。本课题基于离散时间系统的参数，通过研究参数化系统的平衡实现方法，利用系统的传递函数和状态空间方程分别建立了一种新的自适应滤波结构，使得系统可控与可观Grammian矩阵最大与最小特征值的比值和迭代参数的敏感度最小化，并有效降低了计算复杂度；接着利用系统降维的方法，通过删除系统可控与可观Grammian矩阵的较小奇异值，进一步促进了系统可控与可观Grammian矩阵最大与最小特征值的比值，提高了所设计的自适应滤波器抗干扰的性能，促进了自适应滤波器的收敛性能。. 本项目的研究成果拓展了自适应滤波器迭代方向的概念和自适应滤波结构的设计方法，对系统辨识、信道均衡等理论的发展具有重要的意义。研究结论也可以应用在目标跟踪领域，以实现对变化频率系统的快速跟踪。