Analysis of subband adaptive filters and minor component extraction algorithms by using the averaging method

由于子带自适应滤波器系统包括一些抽取器和一个分析滤波器组，因此很难通过直接将平均方法应用于时域系统来检查系统的某些特性。因此，我们提出了一种新的分析方法，其中平均法应用于离散傅立叶变换转换的算法的频域表达式。我们将这种分析方法应用于Hadamard变换的无延迟子带自适应滤波器（DLSADF）和Pradhan和Reddy提出的新结构的子带自适应滤波器（SADF）。结果，获得了以下性质。1.推导了基于Hadamard变换的DLSADF的稳定性条件。找出了导致算法收敛缓慢的一种模式.从理论上推导了用Hadamard变换估计DLSADF中误差信号方差的公式.将同样的方法应用到Pradhan的SADF中，结果表明，该算法始终是稳定的，误差信号的方差小于传统的全带自适应滤波器，并提出了一种新的提取多个次要分量的算法。该算法是将道格拉斯等人提出的单一微量成分提取算法与紧缩技术和Gram-Schmidt正交化相结合而得。通过将ODE方法应用于所提出的方法，得到了以下结果1。该算法具有较好的稳定性.评估提取的次要成分的协方差矩阵。然后求解李雅普诺夫方程。在此过程中，出现了一些不确定项，因此我们引入了适当的约束条件来求解。

Since a system of subband adaptive filters includes some decimators and an analysis filter bank, it is difficult to examine some properties of the system by directly applying the averaging method to the system in the time domain. Therefore we developed a new analysis method in which the averaging method was applied to the frequency domain expression of the algorithm converted by the discrete Fourier transform. We applied this analysis method to both the delay-less subband adaptive filter (DLSADF) with Hadamard transform and the new structure of subband adaptive filter (SADF) proposed by Pradhan and Reddy. As a result, the following properties have been obtained.1. The stability condition of the DLSADF with Hadamard transform is derived. A mode that causes the slow convergence of the algorithm is found out.2. A formula that evaluates the variance of the error signal in the DLSADF with Hadamard transform is theoretically derived.3. By applying the same method to the Pradhan's SADF, it is shown that the algorithm is always stable and that the variance of the error signal is less than that of the conventional fullband adaptive filter.Also, we proposed a new algorithm to extract multiple minor components. The proposed algorithm was derived by combining a single minor component extraction algorithm proposed by Douglas et al. with the deflation technique and the Gram-Schmidt orthogonalization. By applying the ODE method to the proposed method, the following results have been obtained1. The proposed algorithm is always stable.2. The covariance matrix of the extracted minor components is evaluated. Then it is required to solve the Lyapunov equation. In this process, some indeterminate terms appear so that we introduce some appropriate constraints to obtain the solution.All the above results were summarized and have been submitted to international journals.