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

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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.