A Unified Approach with New Algorithms for Adaptive Recursive (IIR) Digital Filters

众所周知，对应于无限冲激响应(IIR)滤波器的零极点结构比全零结构或有限冲激响应(FIR)滤波结构更好地模拟物理系统。然而，由于问题的非线性，目前很少有自适应IIR滤波算法存在。稳定和全球趋同等许多问题没有得到很好的解决。此外，缺乏足够的分析工具使得对这些算法的分析成为一项困难的任务。另一方面，递归系统辨识作为一个久负盛名的学科，已经有了许多性能良好的算法。虽然最近人们已经认识到这两个领域之间的相似之处，但文献中只看到一些系统辨识算法直接应用于自适应滤波问题。本文重点研究了一种新的自适应IIR滤波算法和一种具有一定全局收敛特性的统一解析方法。该方法在系统辨识领域采用了与常微分方程法平行的常增益自适应IIR滤波算法的常微分方程组方法。因此，可以进一步了解恒定增益自适应IIR滤波算法与它们的系统辨识对应物之间的关系。利用这种新的方法，对现有的许多自适应IIR滤波算法进行了统一的分析和比较。此外，更多的系统辨识算法正在以简化的计算和恒定的增益被定制用于自适应IIR滤波应用。观察到的全局收敛现象也在使用相同的方法进行研究，并应导致为某些环境类别开发新的全局收敛算法。自适应滤波器被广泛应用于噪声消除、信道均衡、谱估计和天线阵列处理等应用。本方案主要针对这些自适应滤波器进行研究。

It is well known that a pole-zero structure which corresponds to an infinite impulse response (IIR) filter models a physical system better than an all-zero structure, or a finite impulse response (FIR) filtering structure. However, few adaptive IIR filtering algorithms are in existence today due to the non-linearity of the problem. Many issues such as stability and global convergence are not well-solved. Besides, lack of adequate analytic tools makes the analysis of these algorithms a difficult task. On the other hand, recursive system identification, as a well-established subject, has available many well-behaved algorithms. Although the similarities between these two fields have recently been realized, only direct application of some system identification algorithms to adaptive filtering problems is seen in literature. This research focuses on a new adaptive IIR filtering algorithm and a unified analytical method which shows some global convergence. The new method uses an ordinary differential equation (ODE) approach for constant-gain adaptive IIR filtering algorithms which is parallel to the ODE approach in the field of system identification. Hence further insight into the connection between constant-gain adaptive IIR filtering algortihms and their system identification counterparts can be gained. Using this new approach, many existing adaptive IIR filtering algorithms are being analyzed and compared in a unified way. Furthermore, more system identification algorithms are being tailored for adaptive IIR filtering applications with simplified calculation and constant gains. The observed global convergence phenomenon is also being investigated using the same approach, and should lead to the development of new globally convergent algorithms for certain classes of environment. Adaptive filers are used in a wide spectrum of applications such as noise cancellation, channel equalization, spectral estimation, and antenna array processing. This proposal focuses research on these adaptive filters.