STUDY OF GLOBAL CONVERGENCE OF ADAPTIVE IIR FILTER BASED ON PSEUDO LINEAR REGRESSION METHOD

对伪线性回归法（即带遗忘因子的输出误差RLS法）进行了收敛性分析，并与归一化LMS法、带遗忘因子的方程误差RLS法、单位范数方程误差LMS法等自适应IIR滤波算法进行了性能比较。(1)高信噪比环境下初始阶段收敛速度最快的是带遗忘因子的方程误差RLS方法。我们用一个简单的统计模型从理论上解释了这一结果。(2)未知系统特性的突变导致方程误差RLS方法收敛缓慢。对于带有遗忘因子的输出误差RLS方法，即使在未知系统的初始阶段或特性突变后的一段时间内，其收敛速度也几乎相等。(3)方程误差RLS方法在突变后收敛缓慢，通过对信号矢量的协方差矩阵进行重新初始化，可以改善未知系统的更多特性。该技术也适用于输出RLS方法。(4)对于带有遗忘因子的输出误差RLS方法，在参数向量更新公式中采用小步长可以防止ERLE的退化。但是，对于方程误差法，不能通过步长控制来防止ERLE的退化。(5)当未知系统不满足严格正实条件时，输出误差RLS方法的参数向量收敛于局部最优解。对这个难题的理论解决以失败告终。然而，如果将自适应IIR滤波器的阶数提高到未知系统阶数的3倍，则可以在高信噪比环境下以足够的精度识别脉冲响应。因此，如果将系统约简技术应用于该结果，就可以对不满足严格正实条件的未知系统进行近似辨识。(6)利用NLMS自适应滤波器和作者的公式可以检测未知系统的信噪比和特性的突变。自适应IIR滤波器的发展，使这些检测是未来的任务。少

The convergence analysis of the pseudo linear regression method (which is equal to output error RLS method with forgetting factor) has been done and its performance has been compared with other adaptive IIR filtering algorithms such as normalized LMS method, equation error RLS method with forgetting factor, unit norm equation error LMS method.(1) The fastest convergence velocity in the initial phase under high SNR environment is marked by equation error RLS method with forgetting factor. We have explained this result by using a simple statistical model theoretically.(2) The abrupt change of the characteristics of unknown system causes the slow convergence of equation error RLS method. In case of the output error RLS method with forgetting factor, the convergence velocity is always almost equal even in the initial phase or in the period just after the abrupt change of the characteristics of unknown system.(3) The slow convergence of the equation error RLS method after the abrupt change … More of the characteristics of the unknown system can be improved by reinitializing the covariance matrix of the signal vector. This technique is also valid to output RLS method.(4) In case of output error RLS method with forgetting factor, the degradation of ERLE can be prevented by using small step size in the update formula for parameter vector. But, in case of equation error method, the degradation of ERLE can not be prevented by step size control.(5) The parameter vector of the output error RLS method converges to a local optimal solution if unknown system does not satisfy the strictly positive real condition. The theoretical resolution to this hard problem has resulted in fail. However, if we increase the order of the adaptive IIR filter up to three times of that of unknown system, the identification of impulse response is possible within sufficient accuracy under high SNR environment. Hence if we apply the system reduction technique to this result, we can carry out the approximate identification of the unknown systems which do not satisfy strictly positive real condition.(6) Detection of the abrupt change of SNR and characteristic of unknown system is possible by using NLMS adaptive filter and the author's formula. The development of the adaptive IIR filter which enables those detection is left as the future task. Less